Nonlinear Nonhomogeneous Dirichlet Equations Involving a Superlinear Nonlinearity

Transkript

1 Results. Math , c 2015 Springer Basel /16/ published online May 19, 2015 DOI /s Results in Mathematics Nonlinear Nonhomogeneous Dirichlet Equations Involving a Superlinear Nonlinearity Nikolaos S. Papageorgiou and Patrick Winkert Abstract. We consider a nonlinear elliptic Dirichlet equation driven by a nonlinear nonhomogeneous differential operator involving a Carathéodory function which is p 1-superlinear but does not satisfy the Ambrosetti Rabinowitz condition. First we prove a three-solutions-theorem extending an earlier classical result of Wang Ann Inst H Poincaré Anal Non Linéaire 81:43 57, Subsequently, by imposing additional conditions on the nonlinearity fx,, we produce two more nontrivial constant sign solutions and a nodal solution for a total of five nontrivial solutions. In the special case of p, 2-equations we prove the existence of a second nodal solution for a total of six nontrivial solutions given with complete sign information. Finally, we study a nonlinear eigenvalue problem and we show that the problem has at least two nontrivial positive solutions for all parameters λ>0 sufficiently small where one solution vanishes in the Sobolev norm as λ 0 + and the other one blows up again in the Sobolev norm as λ 0 +. Mathematics Subject Classification. 35J20, 35J60, 35J92, 58E05. Keywords. Superlinear nonlinearity, Ambrosetti Rabinowitz condition, nonlinear regularity, nodal solutions, tangency principle, critical groups, nonlinear eigenvalue problem. 1. Introduction Let R N be a bounded domain with a C 2 -boundary and let 1 <p<. In this paper, we study the following nonlinear nonhomogeneous Dirichlet problem

2 32 N. S. Papageorgiou and P. Winkert Results. Math. div a u =fx, u in, u =0 on, 1.1 where a: R N R N is a continuous, strictly monotone map which is C 1 on R N \{0}. The precise conditions on a are given in hypotheses Ha below. These conditions are general enough to incorporate some differential operators of interest in our framework like the p-laplacian 1 <p<, the p, q- Laplacian 1 <q<p< and the generalized p-mean curvature differential operator 1 <p<. The nonlinearity f : R R is assumed to be a Carathéodory function i.e., x fx, s is measurable for all s R and s fx, s is continuous for a.a. x which exhibits p 1-superlinear growth near ± but without satisfying the usual in such cases Ambrosetti Rabinowitz condition. Our goal is to prove multiplicity theorems for such problems. For equations driven by the p-laplacian, such multiplicity results were proved by BartschandLiu[6], Bartsch et al. [7], Liu [28], Papageorgiou et al. [35] and Sun [38]. Recall that, if f : R R is a Carathéodory function and F x, s = s fx, tdt, we say that fx, satisfies the Ambrosetti Rabinowitz condition 0 if there exist μ>pand M>0such that 0 <μfx, s fx, ss for a.a. x and for all s M, < essinf F, ±M, 1.3 see Ambrosetti and Rabinowitz [4]. Integrating 1.2 and using 1.3, we obtain the following growth conditions for the primitive F x, η s μ F x, s for a.a. x, for all s M, and some η > Thanks to 1.4 we have the much weaker condition F x, s lim s ± s μ =+ uniformly for a.a. x. 1.5 This means that the primitive F x, isp 1-superlinear for a.a. x. In this paper we employ 1.5 combined with another asymptotic condition see Hf 1 iii, which together are weaker than the Ambrosetti Rabinowitz condition see 1.2, 1.3 and fit in our analysis superlinear nonlinearities with slower growth near ±. The Ambrosetti Rabinowitz condition, although very convenient in checking the Palais Smale condition for the energy functional, is rather restrictive as revealed in the discussion above. So there have been efforts to relax it. For an overview of the relevant literature we refer to the recent works of Liu [28], Li and Yang [29], and Miyagaki and Souto [30]. Our tools come from critical point theory and from Morse theory critical groups and involve also truncation and comparison techniques. In the next section, for the reader s convenience, we review the main definitions and facts

3 Vol Nonlinear Nonhomogeneous Dirichlet Equations 33 which will employ in this work. We also introduce the hypotheses on the map a and establish some useful consequences of these conditions. 2. Preliminaries and Hypotheses Let X be a Banach space and X its topological dual while, denotes the duality brackets to the pair X,X. We have the following definition. Definition 2.1. The functional ϕ C 1 X fulfills the Cerami condition the C-condition for short if the following holds: every sequence u n n 1 X such that ϕu n n 1 is bounded in R and 1+ u n X ϕ u n 0inX as n, admits a strongly convergent subsequence. This compactness type condition on ϕ is more general than the wellknown Palais Smale condition which we encounter more often in the literature. Nevertheless, the C-condition suffices to have a deformation theorem from which one derives the minimax theory of certain critical values of ϕ. One result of this theory is the so-called mountain pass theorem. Theorem 2.2. Let ϕ C 1 X be a functional satisfying the C-condition and let u 1,u 2 X, u 2 u 1 >ρ>0, max{ϕu 1,ϕu 2 } < inf{ϕu: u u 1 X = ρ} =: η ρ and c =inf γ Γ max 0 t 1 ϕγt with Γ = {γ C[0, 1],X: γ0 = u 1,γ1 = u 2 }.Thenc η ρ with c being a critical value of ϕ. By L p or L p ; R N andw 0 we denote the usual Lebesgue and Sobolev spaces with their norms p and W 0. Thanks to the Poincaré inequality we have u W 0 = u p for all u W 0. The norm of R N is denoted by and, R N stands for the inner product in R N.Fors R, wesets ± = max{±s, 0} and for u W 0 we define u ± =u ±. It is well known that u ± W 0, u = u + + u, u = u + u. The Lebesgue measure on R N is denoted by N and for a measurable function h : R R for example, a Carathéodory function, we define the Nemytskij operator corresponding to the function h by N h u =h,u for all u W 0. Evidently, x N h ux is measurable. In the analysis of problem 1.1 in addition to the Sobolev space W 0 we will also use the ordered Banach space C0 1 = { u C 1 :u =0 }

4 34 N. S. Papageorgiou and P. Winkert Results. Math. and its positive cone C = { u C 1 0 :ux 0 for all x }. This cone has a nonempty interior given by int C0 1 + { = u C0 1 + : ux > 0 for all x ; u x < 0 for all x n where n stands for the outward unit normal on. Let ϑ C 1 0, + be a function satisfying 0 < ĉ tϑ t ϑt c 0 and c 1 t p 1 ϑt c t p for all t>0 and with some constants ĉ, c 0,c 1,c 2 > 0. Then the hypotheses on a are the following. Ha: aξ =a 0 ξ ξ for all ξ R N with a 0 t > 0 for all t>0and i a 0 C 1 0,,t ta 0 t is strictly increasing, lim t 0 + ta 0 t = 0, and lim t 0 + ta 0 t a > 1; 0t ϑ ξ ii aξ c 3 ξ for all ξ R N \{0} and some c 3 > 0; }, iii aξy, y R N ϑ ξ ξ y 2 for all ξ R N \{0} and all y R N. Remark 2.3. Owing to hypothesis Hai it follows that a C 1 R N \{0}, R N CR N, R N and hence, hypotheses Haii, iii make sense. Let G 0 t = t 0 sa 0sds and let Gξ =G 0 ξ for all ξ R N. Then Gξ =G 0 ξ ξ ξ = a 0 ξ ξ = aξ for all ξ R N \{0}, which means that G is the primitive of a. Obviously, G isconvexand since G0 = 0 we have the estimate Gξ aξ,ξ R N for all ξ R N. 2.2 These hypotheses have some interesting consequences on the map a. Lemma 2.4. Let the hypotheses Ha be satisfied. Then there hold a ξ aξ is maximal monotone and strictly monotone; b aξ c ξ p 1 for all ξ R N and some c 4 > 0; c aξ,ξ R N c1 p 1 ξ p for all ξ R N. Taking into account Lemma 2.4 combined with 2.2 we infer the following growth estimates for the primitive G.

5 Vol Nonlinear Nonhomogeneous Dirichlet Equations 35 Corollary 2.5. If hypotheses Ha hold, then c 1 pp 1 ξ p Gξ c ξ p for all ξ R N and some c 5 > 0. Example 2.6. The following maps satisfy hypotheses Ha: a aξ = ξ p 2 ξ with 1 <p<. This map corresponds to the p-laplacian defined by Δ p u =div u p 2 u for all u W 0. b aξ = ξ p 2 ξ + ξ q 2 ξ with 1 <q<p<. This map corresponds to the p, q-differential operator defined by Δ p u + Δ q u for all u W 0. Note that this operator arises in problems of mathematical physics such as quantum physics see Benci et al. [8] and in plasma physics and biophysics see Cherfils and Il yasov [12]. c aξ =1+ ξ 2 p 2 2 ξ with 1 <p<. This operator represents the generalized p-mean curvature differential operator defined by div [1 + u 2 p 2 2 u ] d aξ = ξ p 2 ξ ξ with 1 <p<. for all u W 0. Now, let f 0 : R R be a Carathéodory function with subcritical growth in s R, thatis f 0 x, s ax 1+ s r 1 for a.a. x, and all s R, with a L, and 1 <r<p, where p is the critical exponent of p given by p = { Np N p if p<n, + if p N. Let F 0 x, s = s 0 f 0x, tdt and let ϕ 0 : W 0 R be the C 1 -functional defined by ϕ 0 u = G udx F 0 x, udx. The following result, originally due to Brezis and Nirenberg [10], can be found in Gasiński and Papageorgiou [22]. We also refer to earlier results in this direction in García Azorero et al. [19] and more recently, in Motreanu and Papageorgiou [32] and Winkert [41].

6 36 N. S. Papageorgiou and P. Winkert Results. Math. Proposition 2.7. Let the assumptions in Ha be satisfied. If u 0 W 0 is alocalc 1 0-minimizer of ϕ 0, i.e., there exists ρ 0 > 0 such that ϕ 0 u 0 ϕ 0 u 0 + h for all h C 1 0 with h C 1 0 ρ 0, then u 0 C 1,β 0 for some β 0, 1 and u 0 is also a local W 0 - minimizer of ϕ 0, i.e., there exists ρ 1 > 0 such that ϕ 0 u 0 ϕ 0 u 0 + h for all h W 0 with h W 0 ρ 1. Now, let 1 p + 1 p =1andletA : W 0 W 0 = W be the nonlinear map defined by Au,v = a u, v R N dx for all u, v W Thanks to the results of Gasiński and Papageorgiou [21] the operator A has the following properties. Proposition 2.8. Under hypotheses Ha the operator A : W 0 W defined by 2.3 is bounded, continuous, monotone hence maximal monotone and of type S +, i.e., if u n uin W 0 and lim sup n Au n,u n u 0, then u n u in W 0. Given 1 <r<, ther-laplacian Δ r is a special case of A which is defined by Δ r u,v = u r 2 u, v R N dx for all u, v W 1,r 0. If r = 2, then Δ r = Δ becomes the well-known Laplace operator. Let us recall some basic facts about the spectrum of the r-laplacian with Dirichlet boundary condition. Consider the nonlinear eigenvalue problem Δ r u = ˆλ u r 2 u in, 2.4 u =0 on, we say that a number ˆλ R is an eigenvalue of Δ r,w 1,r 0 if problem 2.4 possesses a nontrivial solution û W 0 which is said to be an eigenfunction corresponding to the eigenvalue ˆλ. The set of all eigenvalues of 2.4 is denoted by ˆσr and it is known that ˆσr has a smallest element ˆλ 1 r which has the following properties: ˆλ1 r is positive; ˆλ1 r is isolated, that is, there exists ε>0 such that ˆλ 1 r, ˆλ 1 r+ε ˆσr = ; ˆλ1 r is simple, that is, if u, v are two eigenfunctions corresponding to ˆλ 1 r, then u = kv for some k R\{0};

7 Vol Nonlinear Nonhomogeneous Dirichlet Equations 37 [ ] u r ˆλ1 r =inf r u r : u W 1,r 0,u The infimum in 2.5 is realized on the one dimensional eigenspace corresponding to ˆλ 1 r > 0. In what follows we denote by û 1 r thel r -normalized eigenfunction i.e. û 1 r r = 1 associated to ˆλ 1 r. From the representation in 2.5 we easily see that û 1 r does not change sign in and so we may assume that û 1 r 0. The nonlinear regularity theory implies that û 1 r C0 1 and the usage of Vazquez s strong maximum principle [39] provides that û 1 r int C As a consequence of the properties above we have the following simple lemma see Papageorgiou and Kyritsi Yiallourou [34, p. 356]. Lemma 2.9. Let η L + be such that ηx ˆλ 1 p a.e. in and η ˆλ 1 p. Then there exists a positive number κ such that u p p ηx u p dx κ u p p for all u W 0. The Lusternik Schnirelmann minimax scheme produces a strictly increasing sequence ˆλ k r k 1 of eigenvalues such that ˆλ k r + as k.we do not know if this sequence exhausts the whole spectrum of Δ r,w 1,r 0 but if N = 1 ordinary differential equations or if r = 2 linear eigenvalue problem, then the Lusternik Schnirelmann sequence of eigenvalues is the whole spectrum. In the case r = 2 we denote by Eˆλ k 2,k 1, the eigenspace corresponding to the eigenvalue ˆλ k 2 and we have a direct sum decomposition of the form H0 1 = E ˆλk 2. k 1 Next, let us recall some basic definitions and facts about Morse theory. Let X be a Banach space and let Y 1,Y 2 be a topological pair such that Y 2 Y 1 X. For every integer k 0thetermH k Y 1,Y 2 stands for the k =-relative th singular homology group with integer coefficients. Recall that H k Y 1,Y 2 = Z ky 1,Y 2 / B k Y 1,Y 2 for all k N 0, where Z k Y 1,Y 2 is the group of relative singular k-cycles of Y 1 mod Y 2 that is, Z k Y 1,Y 2 = ker k with k being the boundary homomorphism and B k Y 1,Y 2 is the group of relative singular k-boundaries of Y 1 mod Y 2 that is, B k Y 1,Y 2 =im k+1. We know that k 1 k = 0 for all k N, hence B k Y 1,Y 2 Z k Y 1,Y 2 and so the quotient Z k Y 1,Y 2 / B k Y 1,Y 2

8 38 N. S. Papageorgiou and P. Winkert Results. Math. makes sense. Note that H k Y 1,Y 2 = 0 for all k<0. Given ϕ C 1 X and c R, we introduce the following sets: ϕ c = {u X : ϕu c} the sublevel set of ϕ at c, K ϕ = {u X : ϕ u =0} the critical set of ϕ, Kϕ c = {u K ϕ : ϕu =c} the critical set of ϕ at the level c. For every isolated critical point u Kϕ c the critical groups of ϕ at u Kϕ c are defined by C k ϕ, u =H k ϕ c U, ϕ c U\{u} for all k 0, where U is a neighborhood of u such that K ϕ ϕ c U = {u}. The excision property of singular homology theory implies that the definition of critical groups above is independent of the particular choice of the neighborhood U. Suppose that ϕ C 1 X satisfies the C-condition and that inf ϕk ϕ >. Letc<inf ϕk ϕ. The critical groups of ϕ at infinity are defined by C k ϕ, =H k X, ϕ c for all k see Bartsch and Li [5]. This definition is independent of the choice of the level c<inf ϕk ϕ which is a consequence of the second deformation theorem see, for example, Gasiński and Papageorgiou [20, p. 628]. We now assume that K ϕ is finite and introduce the following series in t R: Mt, u = rank C k ϕ, ut k for all u K ϕ, k 0 P t, = k 0 rank C k ϕ, t k. Then, the Morse relation see [11, Theorem ] reads as follows: Mt, u =P t, +1+tQt for all t R, 2.7 u K ϕ with Qt being a formal series in t R with nonnegative integer coefficients. Suppose next that X = H is a Hilbert space and let U be a neighborhood of a given point x H. We further assume that ϕ C 2 U, K ϕ is finite and u K ϕ. The Morse index of u, denoted by μ = μu, is defined to be the supremum of the dimensions of the vector subspaces of H on which ϕ u L H is negative definite. The nullity of u, denoted by ν = νu, is defined to be the dimension of ker ϕ U. We say that u K ϕ is nondegenerate if ϕ u is invertible, that is, ν = νu = 0. At a nondegenerate critical point u we have C k ϕ, u =δ k,μ Z for all k 0, where δ k,μ stands for the well-known Kronecker symbol.

9 Vol Nonlinear Nonhomogeneous Dirichlet Equations Three Nontrivial Solutions In this section, using a combination of variational and Morse theoretic methods, we prove a multiplicity theorem producing three nontrivial solutions for problem 1.1 when the nonlinearity fx, isp 1-superlinear but does not necessarily satisfies the Ambrosetti Rabinowitz condition. Our result in this section improves significantly the well-known multiplicity theorem of Wang [40]. We point out that the results in this section are basically obtained by Gasiński and Papageorgiou in [23]. We decided to add these results since we need some steps in later sections and in order to give a complete analysis of superlinear equations involving nonhomogeneous operators. Furthermore, we note that our assumptions on the differential operator are slightly different than those in [23, see Hai]. First we slightly strengthen the assumptions on the map a. Ha 1 : aξ =a 0 ξ ξ for all ξ R N with a 0 t > 0 for all t>0, hypotheses Ha 1 i iii are the same as the corresponding hypotheses Hai iii and iv pg 0 t t 2 a 0 t c 6 and t 2 a 0 t G 0 t ˆηt p for all t>0and for some c 6, ˆη >0. Remark 3.1. Note that the examples given in Example 2.6 satisfy this new condition stated in Ha 1 iv. The hypotheses on the mapping f are the following: Hf 1 : f : R R is a Carathéodory function with fx, 0 = 0 for a.a. x such that i fx, s ax1 + s r 1 for a.a. x, for all s R, with a L + and p<r<p ; ii if F x, s = s fx, tdt, then 0 F x, s lim s ± s p =+ uniformly for a.a. x ; iii there exist τ r p max{ N p, 1},p andβ 0 > 0 such that fx, ss pf x, s lim inf s ± s τ β 0 uniformly for a.a. x ; iv there exists η L + with ηx c1 ˆλ p 1 1 p a.e. in and η c 1 ˆλ p 1 1 p such that pf x, s lim sup s 0 s p ηx uniformly for a.a. x ; v for every ρ>0 there exists κ ρ > 0 such that fx, ss + κ ρ s p 0 for a.a. x and all s ρ.

10 40 N. S. Papageorgiou and P. Winkert Results. Math. Remark 3.2. Hypothesis Hf 1 ii amounts to the superlinearity of the primitive F x,. This condition together with Hf 1 iii implies that fx, is p 1-superlinear. We point out that the assumptions in Hf 1 ii, iii are weaker than the Ambrosetti Rabinowitz condition see 1.2, 1.3 which is the usual hypothesis when dealing with superlinear problems see for example Wang [40]. Indeed, assume that fx, satisfies the Ambrosetti Rabinowitz condition and note that we may suppose r p max{ N p, 1} <μ. Hence, we have fx, s pf x, s fx, ss μf x, s μ pf x, s s μ = s μ + s μ see 1.2 and 1.4. μ pη for all x and for all s M Example 3.3. For the sake of simplicity we drop the x-dependence and consider the following two functions satisfying hypotheses Hf 1 : { ηs p if s 1, f 1 s = ηs r with η 0, ˆλ 1 p and p<r<p ; if s > 1 f 2 s = s p 2 s ln1 + s. Note that f 1 satisfies the Ambrosetti Rabinowitz condition but f 2 does not. Let ϕ : W 0 R be the energy functional of problem 1.1 given by ϕu = G udx F x, udx, which is of class C 1. Furthermore, we define the positive and negative truncations of fx,, namely f ± x, s =fx, ±s ±, and consider the C 1 -functionals ϕ ± : W 0 R defined by ϕ ± u = G udx F ± x, udx, with F ± x, s = s 0 f ±x, tdt. Proposition 3.4. If Ha 1 and Hf 1 are satisfied, then the functionals ϕ and ϕ ± fulfill the C-condition. Proof. We start with the proof for ϕ +. To this end, let u n n 1 W 0 be a sequence such that ϕ + u n M 1 for all n with some M 1 > 0and 1+ u n W 0 ϕ +u n 0inW. 3.2

11 Vol Nonlinear Nonhomogeneous Dirichlet Equations 41 By means of 3.2 we obtain ϕ +u n,v ε n v W 0 1+ u n W 0 for all v W 0 andε n 0 which means that a u n, v R N dx f + x, u n vdx ε n v W 1+ u n W for all n 1. Acting on 3.3 with v = u n W 0 and applying Lemma 2.4c yields c 1 p 1 u n p ε n, for all n 1 which means that u n 0 in W 0 asn Then, from 3.1 and 3.4 we obtain pg u + n dx pf x, u + n dx M 2, 3.5 with some M 2 > 0. Taking v = u + n W 0 in3.3 gives a u + n, u + n dx + fx, u + R N n u + n dx ε n, 3.6 for all n 1. Now, adding 3.5 and 3.6, we get [ M 3 pg u + n a u + n, u + ] n R dx N [ + fx, u + n u + n pf x, u + n ] dx, 3.7 for all n 1 and some M 3 > 0. By virtue of hypothesis Ha 1 iv we derive from 3.7 fx, u + n u + n pf x, u + n dx M Taking into account hypotheses Hf 1 i and iii, there is a number β 1 0,β 0 and a constant M 5 > 0 such that β 1 s τ M 5 fx, ss pf x, s for a.a. x and for all s Combining 3.8 and 3.9 gives u + n n 1 is bounded in Lτ. 3.10

12 42 N. S. Papageorgiou and P. Winkert Results. Math. Let us first consider the case N>p. Without loss of generality we may suppose that 1 < τ r < p cf. hypothesis Hf 1 iii. Then, we find a number t [0, 1 such that 1 r = 1 t + t τ p 3.11 and the usage of the interpolation theory implies that u + r n u + n 1 t τ u + n t 3.12 p see Gasiński and Papageorgiou [20, p. 905]. Combining 3.10, 3.12, and the Sobolev embedding theorem yields u + n r M r 6 u + n tr for all n W 0 with some positive constant M 6. Applying again v = u + n in 3.3 one has a u + n, u + n dx fx, u + R N n u + n dx ε n for all n Taking into account the growth condition of hypothesis Hf 1 i we infer fx, ss âx+m 7 s r for a.a. x, for all s R, 3.15 with â L andm 7 > 0. With the aid of Lemma 2.4c and 3.15 we obtain from 3.14 c 1 u + p 1 n p M p 8 1+ u + n r for all n 1 r with M 8 > 0. This estimate in conjunction with 3.13 yields u + n p M W u + n tr W 0 for all n and for some M 9 > 0. Taking into account the choice of τ see hypothesis Hf 1 iii and relation 3.11 we see that tr < p which implies that u + n n 1 W 0 is bounded see Now, let N p and note that in this case we have p = and the Sobolev embedding theorem gives W 0 L q for all q [1, +. Let ˆq be a number such that 1 <τ r<ˆq. As before, we find t [0, 1 such that Hence Moreover, we observe that 1 r = 1 t + ṱ τ q. tr = ˆqr τ ˆq τ. ˆqr τ tr = r τ as ˆq + = p ˆq τ By the choice of τ and since N p we have r τ <p. Combining this fact with 3.17 we see that tr < p if ˆq is chosen large enough. Now we may apply the

13 Vol Nonlinear Nonhomogeneous Dirichlet Equations 43 same arguments as in the case N>pwhere p is replaced by ˆq >rsufficiently large. This yields the boundedness of the sequence u + n n 1 in W 0 inthe case N p as well. We have shown in both cases that u + n n 1 is bounded in W 0 and due to 3.4 wehavethatu n n 1 is bounded in W 0 as well. Now we may suppose that for a subsequence if necessary u n uin W 0 and u n u in L p Using again 3.3 with the special choice v = u n u and passing to the limit as n goes to +, we derive, thanks to 3.18, lim Au n,u n u =0. n Since A satisfies the S + -property see Proposition 2.8 we finally conclude u n u in W 0. This proves that ϕ fulfills the C-condition. Analogously, applying similar arguments, one can prove the same result for the functionals ϕ and ϕ. That finishes the proof. Now we are going to show that the functionals ϕ and ϕ ± satisfy the mountain pass geometry. Proposition 3.5. Assume Ha 1 and Hf 1, then u =0is a local minimizer of the functionals ϕ and ϕ ±. Proof. We only show this proposition for ϕ +, the proofs for ϕ and ϕ can be done similarly. By means of hypothesis Hf 1 iv we find for every ε>0a number δ = δε > 0 such that F x, s 1 p ηx+ε s p for a.a. x and for all s δ Let u C0 1 be such that u C 1 0 δ. With regards to Corollary 2.5, 3.19, Lemma 2.9, and 2.5 we obtain ϕ + u = G udx F + x, udx c 1 pp 1 u p p 1 ηx u + p ε dx u + p p p p 1 c1 p p 1 u p p ηx u p dx ε pˆλ 1 p u p p 1 κ ε u p ˆλ p p p Choosing ε>0 small enough such that ε 0,κˆλ 1 p we see from 3.20 that ϕ + u 0=ϕ + 0 for all u C0 1 with 0 u C 1 0 δ.

14 44 N. S. Papageorgiou and P. Winkert Results. Math. This implies that u = 0 is a local C0-minimizer 1 of ϕ +. Invoking Proposition 2.7 yields that u = 0 is a local W 0 -minimizer of ϕ + as well. It is easy to see that the critical points of ϕ + resp. of ϕ are positive resp. negative. Therefore, we may assume that u = 0 is an isolated critical point of the functionals ϕ ±, otherwise there would exist a sequence of distinct positive, resp. negative, solutions of 1.1. Consequently, we find small numbers ρ ± 0, 1 such that { } inf ϕ ± u : u W 0 = ρ ± =: m ± > 0=ϕ ± see Aizicovici et al. [1, Proof of Proposition 29]. Now we are going to prove the existence of two constant sign solutions of problem 1.1. Proposition 3.6. Under the assumptions Ha 1 and Hf 1 problem 1.1 possesses at least two constant sign solutions u 0 intc and v 0 int C Proof. We start with the proof of the existence of the positive solution. Recall that û 1 p intc0 1 + denotes the L p -normalized i.e. û 1 p p =1 eigenfunction corresponding to the first eigenvalue ˆλ 1 p of Δ p,w 0. First, we show that ϕ + tû 1 p as t By means of hypotheses Hf 1 i and ii, for every ε>0 there exists a constant M 10 = M 10 ε > 0 such that F x, s ε s p M 10 for a.a. x and for all s R From Corollary 2.5 and 3.23 we obtain for t>0 ϕ + tû 1 p c 5 N + t p û 1 p p p εt p + M 10 N = t p ˆλ1 p ε +c 5 + M 10 N Choosing ε>ˆλ 1 p in3.24 and letting t + implies Taking into account 3.22 and 3.21 we find a number t>0large enough such that ϕ + tû 1 p ϕ + 0 = 0 <m + and ρ + < tû 1 p W Thanks to 3.21, 3.25 and Proposition 3.4 we may apply Theorem 2.2 mountain pass theorem which provides the existence of an element u 0 W 0 such that ϕ + 0 = 0 <m + ϕ + u 0 and ϕ +u 0 = The first relation in 3.26 ensures that u 0 0 and the second one results in Au 0,v = N f+ u 0,v for all v W

15 Vol Nonlinear Nonhomogeneous Dirichlet Equations 45 Choosing v = u 0 as test function in 3.27 gives a u0, u 0 dx = R N Combining 3.28 and Lemma 2.4c we have c 1 u p p 1 0 p 0. Hence, u 0 0,u 0 0. Then, 3.27 becomes div a u 0 =fx, u 0 in, u =0 on. From the nonlinear regularity theory we obtain u 0 L see Ladyzhenskaya and Ural tseva [26, p. 286] and then u 0 C0 1 see Lieberman [27]. By means of hypothesis Hf 1 v we find, for ρ = u 0 C, a constant κ ρ > 0 such that div a u 0 x + κ ρ u 0 x p 1 =fx, u 0 x+κ ρ u 0 x p 1 0 for a.a. x. Hence, div a u 0 x κ ρ u 0 x p 1 for a.a. x Let γt =ta 0 t fort>0. We have tγ t =t 2 a 0t+ta 0 t Integration by parts and applying Ha 1 iv yields t 0 sγ sds = tγt t 0 γsds = t 2 a 0 t G 0 t ˆηt p Then, due to 3.29 and 3.31, we may apply the strong maximum principle of Pucci and Serrin [37, p. 111] which implies that u 0 x > 0 for all x. In addition, the boundary point theorem of Pucci and Serrin [37, p. 120] yields u 0 int C Using similar arguments one could easily verify the assertion for the existence of the constant sign solution v 0 int C0 1 + working with the functional ϕ instead of ϕ +. Now, we are interested to find a third nontrivial solutions of 1.1 via Morse theory. To this end, we will compute certain critical groups of the functionals ϕ and ϕ ±. We start with the computation of the critical groups of ϕ at infinity. Proposition 3.7. Assume Ha 1 and Hf 1, then C k ϕ, =0for all k 0.

16 46 N. S. Papageorgiou and P. Winkert Results. Math. Proof. By means of Hf 1 i and ii, for every ε>0, there exists a constant M 11 > 0 such that F x, s ε s p M 11 for a.a. x and for all s R By virtue of Corollary 2.5 and 3.32 there holds for u W 0 \{0} and for every t>0 ϕtu = Gt udx F x, tudx c 5 N + t p u p p εt p u p p + M 11 N = t p c 5 u p p ε u p p + M12, with M 12 =c 5 + M 11 N. Choosing ε> c5 u p p implies that u p p ϕtu as t Thanks to the hypotheses Hf 1 i and iii, there is a number β 2 0,β 0 and a constant M 13 > 0 such that pf x, s fx, ss M 13 β 2 s τ for a.a. x and for all s R Taking into account hypothesis Ha 1 iv and 3.34 we obtain d dt ϕtu = ϕ tu,u = 1 t ϕ tu,tu = 1 [ at u,t u t R N dx 1 [ t ] fx, tutudx pgt udx +c 6 + M 13 N ] pf x, tudx = 1 t [pϕtu+m 14] 3.35 with M 14 =c 6 + M 13 N. Combining 3.33 and 3.35 we conclude that d dt ϕtu < 0 for t>0 sufficiently large. Therefore, for every u B 1 = {y W 0 : y W a unique ψu > 0 such that 0 =1}, there exists ϕψuu =ρ < M 14 p see Moreover, the implicit function theorem implies that ψ C B 1. Now we extend ψ on W 0 \{0} by setting ψu = 1 u W 0 ψ u u W 0 for all u W 0 \{0}.

17 Vol Nonlinear Nonhomogeneous Dirichlet Equations 47 It is clear that ψ CW 0 \{0} andϕ ψuu =ρ for all u W 0 \ {0}. Note that ϕu =ρ implies ψu = 1. Then, putting { 1 if ϕu ρ, ˆψu = 3.36 ψu if ϕu >ρ, we have ˆψ CW 0 \{0}. Next, we introduce the deformation h :[0, 1] W 0 \{0} W 0 \ {0} defined by ht, u =1 tu + t ˆψuu. It is easy to see that h0,u=u and h1,u ϕ ρ for all u W 0 \{0}. Moreover, thanks to 3.36 there holds ht, =id ϕ ρ ϕ for all t [0, 1]. ρ This means that the sublevel set ϕ ρ is a deformation retract of W 0 \{0}. u Because of the radial retraction u u for all u W 0 \{0} we W 0 see that B 1 is a retract of W 0 \{0} while the deformation u h 0 t, u =1 tu + t for all t, u [0, 1] W 0 \{0}, u W 0 points out that W 0 \{0} is deformable into B 1 over W 0. Then, we may apply Theorem 6.5 of Dugundji [16, p. 325] which implies that B 1 is a deformation retract of W 0 \{0}. We conclude that ϕ ρ and B 1 are homotopy equivalent. Hence, H k W 0,ϕ ρ = H k W 0, B 1 for all k Since the space W 0 is infinite dimensional, it follows that B 1 is contractible in itself. Then, from Granas and Dugundji [24, p. 389] we have H k W 0, B 1 = 0 for all k 0, which in view of 3.37 gives H k W 0,ϕ ρ = 0 for all k Choosing ρ < M14 p see 2.6. even smaller if necessary, we conclude from 3.38 that C k ϕ, = 0 for all k 0 A similar reasoning leads to the following result. Proposition 3.8. Assume Ha 1 and Hf 1, then C k ϕ ±, =0 for all k 0.

18 48 N. S. Papageorgiou and P. Winkert Results. Math. Proof. We do the proof only for the functional ϕ +, the assertion for ϕ can be done similarly. Let B 1 + := {u B 1 : u + 0} and t>0. As in the proof of Proposition 3.7 we can show that for all u B 1 + there holds ϕ + tu as t Taking into account Ha 1 iv and 3.34 yields, for all u B 1 +, d dt ϕ +tu = ϕ +tu,u = 1 t ϕ +tu,tu = 1 [ ] at u,t u t R N dx f + x, tutudx 1 [ ] pgt udx +c 6 + M 15 N pf x, tu + dx t 1 t [pϕ +tu+m 16 ] 3.40 where M 16 =c 6 + M 15 N and M 15 > 0. Regarding 3.39 and 3.40, we conclude that d dt ϕ +tu < 0 for all t>0 sufficiently large. As before, for every u B 1 +, we find an unique ψ +u > 0 such that ϕ + ψ + uu =ρ + < M16 p and the implicit function theorem implies that ψ + C B 1 +. Let E + = {u W 0 :u + 0} and set for all u E + 1 u ψ + u = ψ +. u W 0 u W 0 Obviously, ψ+ CE + andϕ + ψ + uu =ρ +. Moreover, if ϕ + u =ρ +, then ψ + u = 1. Hence, { 1 if ϕ + u ρ ˆψ +, + u := ψ + u if ϕ + u >ρ , belongs to CE +. Consider the deformation h + :[0, 1] E + E + defined by h + t, u =1 tu + t ˆψ + uu. We see at once that h + 0,u=u, h + 1,u ϕ + ρ+ for all u E+,and h + t, ϕ =id for all t [0, 1] + ρ+ ϕ + ρ+ cf Consequently, ϕ + ρ+ is a strong deformation retract of E+.

19 Vol Nonlinear Nonhomogeneous Dirichlet Equations 49 Let us consider the deformation ĥ+ :[0, 1] E + E + defined by ĥ + t, u =1 tu + tu 0, where u 0 E + is fixed. Then, ĥ+0,u=u and ĥ+1,u=u 0 which means that id E+ is homotopic to the constant map u u 0.Thus,E + is contractible to itself see Bredon [9, Proposition 14.5] and from Granas and Dugundji [24, p. 389], it follows H k W 0,E + = 0 for all k 0. Then we infer H k W 0, ϕ + ρ+ = 0 for all k As before, we choose ρ + < M16 p sufficiently small. Thus, 3.42 implies C k ϕ +, = 0 for all k 0. This yields the assertion of the proposition. Recall that u 0 intc andv 0 intc are the constant sign solutions of 1.1 obtained in Proposition 2.7. We may assume that K ϕ = {0,u 0,v 0 }, otherwise we would find another nontrivial solution of 1.1 which would belong to C 1 0 as a consequence of the nonlinear regularity theory see Ladyzhenskaya and Ural tseva [26] and Lieberman [27] and therefore we would have done. Note that K ϕ = {0,u 0,v 0 } ensures that K ϕ+ = {0,u 0 } and K ϕ = {0,v 0 }. Proposition 3.9. Assume Ha 1 and Hf 1, then C k ϕ +,u 0 =C k ϕ,v 0 =δ k,1 Z for all k 0. Proof. We only compute C k ϕ +,u 0, the computation of C k ϕ,v 0 isdone in a similar way. Let ς 1,ς 2 R be two numbers such that ς 1 < 0=ϕ + 0 <ς 2 <m + ϕ + u see 3.21 and 3.26 and consider the following triple of sets ϕ + ς1 ϕ + ς2 W 0. Concerning this triple of sets we study the corresponding long exact sequence of homology groups which is given by... H k W 0, ϕ + ς1 i Hk W 0, ϕ + ς2 Hk 1 ϕ + ς2, ϕ + ς1..., 3.44

20 50 N. S. Papageorgiou and P. Winkert Results. Math. where i denotes the group homomorphism induced by the inclusion mapping i :ϕ + ς1 ϕ + ς2 and stands for the boundary homomorphism. Recall that K ϕ+ = {0,u 0 } and thanks to 3.43 as well as Proposition 3.8 it follows H k W 0, ϕ + ς1 = C k ϕ +, = 0 for all k Furthermore, from Chang [11, p. 338], 3.43, and Proposition 3.5 we have H k W 0, ϕ + ς2 = C k ϕ +,u 0 for all k and H k 1 ϕ + ς2, ϕ + ς1 =C k 1 ϕ +, 0 = δ k,1 Z for all k Taking into account 3.45 and 3.47 one observes that only the tail k =1in 3.44 is nontrivial. Applying the rank theorem yields rank H 1 W 0, ϕ + ς2 =rankker + rankim. Then from it follows rank C 1 ϕ +,u 0 = rank ker + rank im = rank im i + rank im However, the proof of Proposition 3.6 has shown that u 0 intc0 1 + isa critical point of ϕ + of mountain pass type. Thus, C 1 ϕ +,u Combining 3.48 and 3.49 yields C k ϕ +,u 0 =δ k,1 Z for all k 0. With the aid of Proposition 3.9 we are now in the position to compute the critical groups of ϕ at u 0 and v 0. Proposition Assume Ha 1 and Hf 1, then C k ϕ, u 0 =C k ϕ, v 0 =δ k,1 Z for all k 0. Proof. As before, we only compute C k ϕ, u 0, the other one works similarly. We consider the homotopy h :[0, 1] W 0 W 0 defined by ht, u =tϕu+1 tϕ + u. Recall that K ϕ = {0,u 0,v 0 }. We are going to prove the existence of a number ρ>0 such that u 0 is the only critical point of ht, in { } B ρ = u W 0 : u u 0 W 0 <ρ

21 Vol Nonlinear Nonhomogeneous Dirichlet Equations 51 for all t [0, 1]. We proceed by contradiction. If we assume that this assertion is not true, then we find a sequence t n,u n n 1 [0, 1] W 0 such that t n t, in [0, 1], u n u 0 in W 0, and h ut n,u n = 0 for all n Relation 3.50 gives Au n,v =t n fx, u n vdx+1 t n f + x, u n vdx for all v W 0, which means that u n solves the problem div a u n x = t n fx, u n x + 1 t n f + x, u n x in, 3.51 u =0 on. Because of 3.50, from Ladyzhenskaya and Ural tseva [26, p. 286], there exists M 17 > 0 such that u n L M 17 for all n 1 and due to Lieberman [27, p. 320] we find β 0, 1 and M 18 > 0 such that u n C 1,α 0 M 18 for all n 1. Due to the compact embedding C 1,α 0 C0, 1 we may assume that u n u 0 in C0 1 for a subsequence if necessary. Recalling u 0 intc0 1 + there exists a number n 0 1 such that u n n n0 intc Thus 3.51 reduces to div a u n =fx, u n in, u =0 on. Hence, u n n n0 is a sequence of distinct solutions of 1.1 which contradicts the fact that K ϕ = {0,u 0,v 0 }. Therefore, we find a number ρ>0 such that h ut, u 0 for all t [0, 1] and all u B ρ u 0 \{u 0 }. Similar to the proof of Proposition 3.4 one could verify that ht, fulfills the C-condition for every t [0, 1]. Thus, we can invoke the homotopy invariance of critical groups to get C k h0,,u 0 =C k h1,,u 0 for all k 0, which is equivalent to C k ϕ, u 0 =C k ϕ +,u 0 for all k 0. Combining this with Proposition 3.9 implies that C k ϕ, u 0 =C k ϕ +,u 0 =δ k,1 Z for all k 0. Similarly, we show that C k ϕ, v 0 =δ k,1 Z for all k 0. Now we are ready to produce a third nontrivial solution of problem 1.1. We have the following multiplicity theorem.

22 52 N. S. Papageorgiou and P. Winkert Results. Math. Theorem Under hypotheses Ha 1 and Hf 1 problem 1.1 has at least three nontrivial solutions u 0 int C0 1 +, v0 int C0 1 + and y 0 C0. 1 Proof. The existence of the two constant-sign solutions of 1.1 follows directly from Proposition 3.6, that is u 0 int C0 1 +, v0 int C Suppose that K ϕ = {0,u 0,v 0 } and recall that C k ϕ, u 0 =C k ϕ, v 0 =δ k,1 Z for all k see Proposition Thanks to Proposition 3.5 we know that C k ϕ, 0 = δ k,0 Z for all k Finally, Proposition 3.7 implies C k ϕ, = 0 for all k Combining and the Morse relation with t = 1 see2.7 yields =0, which is a contradiction. Thus, we can find y 0 K ϕ \{0,u 0,v 0 } which means that y 0 is a third nontrivial solution of 1.1 and as before, the nonlinear regularity theory guarantees that y 0 C0. 1 That finishes the proof. Remark The first multiplicity result three-solutions-theorem for superlinear elliptic equations has been proved by Wang [40]. In that work p = 2,aξ =ξ for all ξ R N hence the differential operator is the Laplacian, semilinear equation and fx, =f i.e., the nonlinearity is x-independent, f C 1 R,f 0 = 0 and it satisfies the Ambrosetti Rabinowitz condition see 1.2, 1.3. We point out that Theorem 3.11 extends significantly the multiplicity result of Wang [40]. Other multiplicity results for p-laplacian equations with a superlinear nonlinearity satisfying more restrictive conditions than Hf 1 were proved by Liu [28] and Sun [38]. For Neumann problems driven by the p-laplacian we refer to Aizicovici et al. [2]. 4. Five Nontrivial Solutions In this section we produce additional nontrivial solutions for problem 1.1 by changing the geometry of the problem near the origin. Roughly speaking we require that fx, exhibits an oscillatory behavior near zero. We also suppose some stronger assumptions on the map a which allows us to prove the existence of five nontrivial solutions of 1.1 given with complete sign information. The results in this section extend the recent work of Aizicovici et al. [3]. The new hypotheses on the map a are the following.

23 Vol Nonlinear Nonhomogeneous Dirichlet Equations 53 Ha 2 : aξ =a 0 ξ ξ for all ξ R N with a 0 t > 0 for all t>0, hypotheses Ha 2 i iii are the same as the corresponding hypotheses Ha 1 i iii and iv pg 0 t t 2 a 0 t c 6 for all t>0 and some c 6 > 0; v there exists q such that t G 0 t 1 q isconvexin0, +, lim sup t 0 + qg 0 t t q < +, and t 2 a 0 t qg 0 t ˆηt p for all t>0 and some ˆη >0. Remark 4.1. The examples given in Example 2.6 still satisfy the new hypotheses Ha 2. Note that hypothesis Ha 2 v implies Gξ c 7 ξ q + ξ p for all ξ R N, 4.1 with some c 7 > 0. Furthermore, we suppose new hypotheses on the nonlinearity f : R R as follows. Hf 2 : f : R R is a Carathéodory function such that fx, 0 = 0 for a.a. x, hypotheses Hf 2 i iii are the same as the corresponding hypotheses Hf 1 i iii and iv there exist ζ 1,qq as in hypothesis Ha 2 v and δ>0such that ζfx, s fx, ss >0 for a.a. x and for all 0 < s δ and essinf F, ±δ > 0; v there exist real numbers ξ < 0 <ξ + such that fx, ξ + η 1 < 0 <η 2 fx, ξ for a.a. x ; vi for every ρ>0, there exists ξ ρ > 0 such that s fx, s+ξ ρ s p 2 s is nondecreasing on [ ρ, ρ] for a.a. x. Remark 4.2. Hypothesis Hf 2 iv implies that F x, s M 19 s ζ for a.a. x, for all s δ, and some M 19 > 0. We also point out that fx, exhibits an oscillatory behavior near zero which follows directly from hypothesis Hf 2 v. Example 4.3. As before, we drop the x-dependence. The following function satisfies hypotheses Hf 2. { s τ 2 s 2 s p 2 s if s 1, fs = s p 2 s ln s s q 2 with 1 <q,τ<p. s if s > 1 Note that this f does not satisfy the Ambrosetti Rabinowitz condition.

24 54 N. S. Papageorgiou and P. Winkert Results. Math. First we produce two nontrivial constant sign solutions. Proposition 4.4. Let the hypotheses Ha 2 and Hf 2 be satisfied. Then problem 1.1 has at least two nontrivial constant sign solutions u 0 intc0 1 and v 0 intc0 1 such that ξ <v 0 x 0 u 0 x <ξ + for all x. Moreover, both solutions are local minimizers of the energy functional ϕ. Proof. Let ˆf + : R R be the truncation function defined by 0 if s<0 ˆf + x, s = fx, s if 0 s ξ +, 4.2 fx, ξ + if ξ + <s which is known to be a Carathéodory function. We introduce the C 1 -functional ˆϕ + : W 0 R through ˆϕ + u = G udx ˆF + x, udx with ˆF + x, s = s ˆf 0 + x, tdt. It is clear that ˆϕ + : W 0 R is coercive see Corollary 2.5, 4.2 and sequentially weakly lower semicontinuous. Hence, its global minimizer u 0 W 0 exists, that is { } ˆϕ + u 0 =inf ˆϕ + u :u W 0 =ˆm +. By virtue of hypothesis Hf 2 v we know that we can find β>0andδ 0 0, min{δ, ξ + } such that Gξ β ξ q for all ξ δ Recall that hypothesis Hf 2 iv implies F x, s M 20 s ζ for a.a. x and for all s δ 0, 4.4 with some M 20 > 0. Since û 1 q int C0 1 + we can choose t 0, 1 sufficiently small such that tû 1 qx [0,δ 0 ] for all x. Taking into account 4.3, 4.4 and û 1 q q = 1, we obtain ˆϕ + tû 1 q = G tû 1 dx ˆF + x, tû 1 dx βt q û 1 q q q M 20t ζ û 1 q ζ ζ = βt qˆλ1 q M 20 t ζ û 1 q ζ ζ. 4.5 Since ζ<q, choosing t 0, 1 small enough, 4.5 gives ˆϕ + tû 1 q < 0, meaning ˆϕ + u 0 = ˆm + < 0= ˆϕ + 0.

25 Vol Nonlinear Nonhomogeneous Dirichlet Equations 55 We conclude u On the other hand, since u 0 is a critical point of ˆϕ + there holds Au 0,v = N ˆf+ u 0,v for all v W Choosing v = u 0 as test function in 4.7 and applying Lemma 2.4c as well as the definition of the truncation see 4.2 yields c 1 u p p 1 0 p 0. Hence, u Now, making use of hypothesis Hf 2 v and taking u 0 ξ + + W 0 as test function in 4.7 one gets a u 0, u 0 ξ + + dx = ˆf + x, u 0 u 0 ξ + + dx R N = fx, ξ + u 0 ξ + + dx From 4.9 it follows {u 0>ξ +} a u 0 a ξ +, u 0 ξ + R N dx 0, and by virtue of Lemma 2.4a, {u 0 >ξ + } N =0. Hence, u 0 x ξ + a.e. in Combining 4.6, 4.8 and 4.10 wehave 0 u 0 x ξ + a.e. in and u 0 0. Then, 4.7 becomes meaning that Au 0,v = N f u 0,v for all v W 0, div a u 0 =fx, u 0 in, u =0 on. The nonlinear regularity theory ensures that u 0 C 1 0 see Ladyzhenskaya and Ural tseva [26] and Lieberman [27, p. 320].

26 56 N. S. Papageorgiou and P. Winkert Results. Math. Thanks to hypothesis Hf 2 vi we find for ρ = ξ + a constant ξ ρ > 0 such that div a u 0 x + ξ ρ u 0 x p 1 =fx, u 0 x+ξ ρ u 0 x p 1 0 for a.a. x. Hence, div a u 0 x ξ ρ u 0 x p 1 for a.a. x. Due to Hypothesis Ha 2 iv the strong maximum principle implies that u 0 intc0 1 + see Pucci and Serrin [37, pp. 111 and 120]. Now, let δ>0 and set u δ = u 0 + δ C 1. Recall that u 0 x ξ + for all x, by means of hypotheses Hf 2 v, vi, we have div a u δ x + ξ ρ u δ x p 1 div a u 0 x + ξ ρ u 0 x p 1 + oδ = fx, u 0 x + ξ ρ u 0 x p 1 + oδ fx, ξ + +ξ ρ ξ p oδ η 1 + ξ ρ ξ p oδ Recall that η 1 < 0seeHf 2 v and oδ 0 + as δ 0 +. Then, for δ>0 sufficiently small there holds η 1 + oδ 0. Hence, from 4.11 we obtain div a u δ x + ξ ρ u δ x p 1 v div a ξ + +ξ ρ ξ p 1 +. Applying again Pucci and Serrin [37, p. 61] it follows u δ x ξ + for all x, consequently, ux <ξ + for all x. Therefore, we have u 0 int [0,ξ + ]. C0 1 Since ϕ [0,ξ+] =ˆϕ + [0,ξ+] we conclude that u 0 is a local C0-minimizer 1 of ϕ. So, Proposition 2.7 implies that u 0 is a local W 0 -minimizer of ϕ. For the nontrivial negative solution we introduce the following truncation of the nonlinearity fx, fx, ξ if s<ξ ˆf x, s = fx, s if ξ s 0, 0 if 0 <s which is a Carathéodory function. Setting ˆF x, s = s ˆf 0 x, tdt we consider the C 1 -functional ˆϕ : W 0 R defined by ˆϕ u = G udx ˆF x, udx.

27 Vol Nonlinear Nonhomogeneous Dirichlet Equations 57 Working as above via the direct method we produce a solution v 0 int C0 1 + being a local minimizer of ϕ. Remark 4.5. A careful inspection of the proof above reveals that we only needed hypotheses Hf 2 iv, v, vi, i.e., the asymptotic conditions at ± see Hf 2 ii, iii are irrelevant. Moreover, the global growth condition Hf 2 i can be replaced by the following local one. For every ρ>0 there exists a ρ L + such that fx, s a ρ x for a.a. x and for all s ρ. Using these two nontrivial constant sign solutions we can produce two more precisely localized with respect to u 0 and v 0. Now we need the asymptotic conditions at ±. Proposition 4.6. Under the hypotheses Ha 2 and Hf 2 problem 1.1 possesses two more nontrivial constant sign solutions u 1 int C0 1 + and v 1 int C0 1 + satisfying u 0 x u 1 x and v 1 x v 0 x for all x with u 1 u 0 and v 1 v 0. Proof. We begin with the proof for the existence of u 1.Foru 0 int C0 1 + being the constant sign solution obtained in Proposition 4.4 we define the truncation mapping e + : R through R { e + x, s = fx, u 0 x if s<u 0 x, fx, s if u 0 x s, 4.12 which is again a Carathéodory function. Setting E + x, s = s 0 e +x, tdt we introduce the C 1 -functional σ + : W 0 R by σ + u = G udx E + x, udx. First we note that σ + fulfills the C-condition which can be shown as in the proof of Proposition 3.4 with minor modifications by applying Claim: We may assume that u 0 int C0 1 + is a local minimizer of the functional σ +. Recalling u 0 x <ξ + for all x we introduce the subsequent Carathéodory truncation function ê + x, s = { e + x, s if s ξ + e + x, ξ + if s>ξ and consider the C 1 -functional ˆσ + : W 0 R ˆσ + u = G udx Ê + x, udx

28 58 N. S. Papageorgiou and P. Winkert Results. Math. with Ê+x, s= s 0 ê+x, tdt. Obviously, ˆσ + is coercive and sequentially weakly lower semicontinuous which implies due to the Weierstrass theorem that there is a global minimizer û 0 W 0 meaning { } ˆσ + û 0 =inf ˆσ + u :u W 0. In particular, this gives ˆσ +û 0 = 0 and hence, Aû 0,v = Nê+ û 0,v for all v W Taking v =u 0 û 0 + W 0 in the last equation and using 4.12, 4.13 we obtain It follows that meaning Aû 0, u 0 û 0 + = ê + x, û 0 u 0 û 0 + dx = fx, u 0 u 0 û 0 + dx = Au 0, u 0 û 0 +. {u 0>û 0} Au 0 Aû 0, u 0 û 0 + =0, a u 0 a û 0, u 0 û 0 R N dx =0. Hence, {u 0 > û 0 } N =0,thatis,u 0 û 0. Now, taking v =û 0 ξ + + in 4.14, applying 4.12, 4.13, Hf 2 v, and recalling u 0 x <ξ + for all x, weget Aû 0, û 0 ξ + + = ê + x, û 0 û 0 ξ + + dx = fx, ξ + û 0 ξ + + dx which implies {û 0>ξ +} 0, û 0 p dx 0 see Lemma 2.4c. As above we conclude that {û 0 >ξ + } N = 0, i.e., û 0 ξ +. Then, û 0 [u 0,ξ + ]andeq.4.14 becomes Aû 0,v = N f û 0,v for all v W 0, which means that û 0 solves our original problem 1.1. Applying again the nonlinear regularity theory we obtain that û 0 int C0 1 + see the proof of Proposition 4.4. If û 0 u 0, then the assertion of the proposition is proved and we are done.

29 Vol Nonlinear Nonhomogeneous Dirichlet Equations 59 Let us suppose that û 0 = u 0. By means of the truncations in 4.12, 4.13 wehave σ [0,ξ+] + =ˆσ +. [0,ξ+] Since û 0 = u 0 int C 1 0 [0,ξ +] we see that û 0 = u 0 is a local C0-minimizer 1 of σ + and with regard to Proposition 2.7 it is also a local W 0 -minimizer of σ +. This proves the claim. We may also assume that u 0 is an isolated critical point of σ +, otherwise we would find a sequence u n n 1 W 0 such that u n u 0 in W 0 and σ +u n = 0 for all n It follows Au n =N e+ u n for all n 1 meaning that div a u n x = e + x, u n x a.e. in Then, from 4.15, 4.16 and Ladyzhenskaya and Ural tseva [26] we can find M 21 > 0 such that u n L M 21. Applying the regularity results of Lieberman [27] we find γ 0, 1 and M 22 > 0 such that u n C 1,γ 0 and u n C 1,γ 0 M 22 for all n 1. Exploiting the compact embedding of C 1,γ intoc0 1 and by virtue of 4.15 one gets u n u 0, u n u 0 for all n 1. That means we have proved the existence of a whole sequence u n n 1 int C0 1 + of distinct nontrivial positive solutions of 1.1. Hence, we are done. Therefore, we may consider u 0 as an isolated critical point of σ +. Because of the claim there exists a number ρ 0, 1 such that { } σ + u 0 < inf σ + u : u u 0 W 0 = ρ =: η ρ see Aizicovici et al. [1, Proof of Proposition 29]. Recall that σ + satisfies the C-condition. Thanks to hypothesis Hf 2 ii we verify that if u int C0 1 +, then σ + tu as t +. These facts combined with 4.17 permit the usage of the mountain pass theorem stated in Theorem 2.2. This provides the existence of u 1 W 0 such that u 1 K σ+ and η ρ + e + u With a view to 4.17 and 4.18 we see that u 0 u 1,u 0 u 1 and u 1 int C0 1 + solves problem 1.1. The case of a second nontrivial negative solution v 1 int C0 1 + with v 1 v 0 and v 1 v 0 can be shown using similar arguments.

30 60 N. S. Papageorgiou and P. Winkert Results. Math. Now we are interested to find a fifth solution of 1.1 being a sign-changing one. In order to produce the nodal solution we will use some tools from Morse theory. For this purpose we start by computing the critical groups at the origin of the C 1 -energy functional ϕ : W 0 R N defined by ϕu = G udx F x, udx. Our proof uses ideas from Moroz [31] inwhichgξ = 1 2 ξ 2 for all ξ R N with more restrictive conditions on f : R R and from Jiu and Su [25] where Gξ = 1 p ξ p for all ξ R N. Proposition 4.7. Under the assumptions Ha 2 and Hf 2 i, iv there holds C k ϕ, 0 = 0 for all k 0. Proof. Note that from Hf 2 i and iv we have F x, s M 23 s ζ M 24 s r for a.a. x and for all s R 4.19 with positive constants M 23,M 24. Recall that hypothesis Ha 2 v implies Gξ c 7 ξ q + ξ p for all ξ R N 4.20 see also 4.1. Let u W 0 andt>0. Combining 4.19 and 4.20 gives ϕtu = G tudx F x, tudx c 7 t q u q q + c 7 t p u p p M 23 t ζ u ζ ζ + M 24t r u r r. Since ζ<q<p<rthere exists a small number t 0 > 0 such that ϕtu < 0 for all 0 <t<t 0. Now let u W 0 be such that ϕu = 0. Taking into account Ha 2 v, Hf 2 i, iv, and the Sobolev embedding theorem it follows d dt ϕtu = ϕ tu,u t=1 t=1 = a u, u R N dx fx, uudx ζ G udx + ζfx, udx ˆη u p p + [ζfx, u fx, uu] dx ˆη u p M W 0 25 u r 4.21 W 0 with some M 25 > 0. Since p<rwe can find ρ 0, 1 small enough such that d dt ϕtu > 0 u W 0 with ϕu = 0 and 0 < u W 0 ρ. t=1 4.22

31 Vol Nonlinear Nonhomogeneous Dirichlet Equations 61 Now, let u W 0 with 0 < u W 0 ρ and ϕu = 0. In the following we are going to show that ϕtu 0 for all t [0, 1] Arguing by contradiction, suppose that we can find a number t 0 0, 1 such that ϕt 0 u > 0. Since ϕ is continuous and ϕu = 0 there exists t 1 t 0, 1] such that ϕt 1 u=0.lett = min{t [t 0, 1] : ϕtu =0}. It is clear that t >t 0 > 0and ϕtu > 0 for all t [t 0,t Setting v = t u we have 0 < v W 0 u W 0 ρ and ϕv =0. Then, 4.22 gives d dt ϕtv > t=1 Moreover, from 4.24 we obtain ϕv =ϕt u=0<ϕtu for all t [t 0,t. Hence, d dt ϕtv d = t t=1 dt ϕtu ϕtu = t lim t=t t t t t Comparing 4.25 and 4.26 we reach a contradiction. This proves By taking ρ 0, 1 even smaller if necessary we may assume that K ϕ B ρ = {0} where B ρ = {u W 0 : u W 0 ρ}. Leth :[0, 1] ϕ0 B ρ ϕ 0 B ρ be the deformation defined by ht, u =1 tu. Thanks to 4.23 we verify that this deformation is well-defined and it implies that ϕ 0 B ρ is contractible in itself. Fix u B ρ with ϕu > 0. We show that there exists an unique tu 0, 1 such that ϕtuu =0. Since ϕu > 0 and the continuity of t ϕtu, 4.22 ensures the existence of such a tu 0, 1. It remains to show its uniqueness. Arguing by contradiction, suppose that for 0 <t 1 = tu 1 <t 2 = tu 2 < 1wehave ϕt 1u =ϕt 2u = 0. Then, 4.23 implies γt =ϕtt 2u 0 for all t [0, 1]. Therefore t 1 t 2 0, 1 is a maximizer of γ and thus, d dt γt t t= 1 t 2 =0,

32 62 N. S. Papageorgiou and P. Winkert Results. Math. which implies that t 1 t 2 d dt ϕtt 2u t= t 1 = d dt ϕtt 1u =0. t t=1 2 But this is a contradiction to 4.22 and the uniqueness of tu 0, 1 is proved. This uniqueness implies that ϕtu < 0 if t 0,tu and ϕtu > 0 for all t tu, 1]. Let T 1 : B ρ \{0} 0, 1] be defined by { 1 if ϕu 0, T 1 u = tu if ϕu > 0. It is easy to check that T 1 is continuous. Next, we consider a map T 2 : B ρ \{0} ϕ 0 B ρ \{0} defined by { u if ϕu 0, T 2 u = T 1 uu if ϕu > 0. Obviously, T 2 is a continuous function. We observe that ϕ T 2 =id 0 B ρ\{0} ϕ 0 B ρ\{0}, which proves that ϕ 0 B ρ \{0} is a retract of B ρ \{0}. Note that B ρ \{0} is contractible in itself. Therefore, the same is true for ϕ 0 B ρ \{0}. Previously, we proved that ϕ 0 B ρ is contractible in itself. From Granas and Dugundji [24, p. 389] it follows that H k ϕ 0 B ρ, ϕ 0 B ρ \{0} = 0 for all k 0. Hence, C k ϕ, 0 = 0 for all k 0. see Sect. 2. This completes the proof. Thanks to Proposition 4.7 we can now establish the existence of extremal nontrivial constant sign solutions, that means, we will produce the smallest nontrivial positive solution and the greatest nontrivial negative solution of 1.1. To this end, let S + resp. S be the set of all nontrivial positive resp. negative solutions of problem 1.1. As in Filippakis et al. [18] we can show that S + is downward directed, that means, if u 1,u 2 S +, then there exists u S + such that u u 1 and u u 2. S is upward directed, that means, if v 1,v 2 S, then there exists v S such that v 1 v and v 2 v.