On the Fučik spectrum of non-local elliptic operators

Transkript

1 Nonlinear Differ. Equ. Appl. 21 (2014), c 2014 Springer Basel /14/ published online January 7, 2014 DOI /s Nonlinear Differential Equations and Applications NoDEA On the Fučik spectrum of non-local elliptic operators Sarika Goyal and K. Sreenadh Abstract. In this article, we study the Fučik spectrum of the fractional Laplace operator which is defined as the set of all (α, β) R 2 such that } ( Δ) s u = αu + βu in u =0 inr n \. has a non-trivial solution u, where is a bounded domain in R n with Lipschitz boundary, n>2s, s (0, 1). The existence of a first nontrivial curve C of this spectrum, some properties of this curve C, e.g. Lipschitz continuous, strictly decreasing and asymptotic behavior are studied in this article. A variational characterization of second eigenvalue of the fractional eigenvalue problem is also obtained. At the end, we study a nonresonance problem with respect to the Fučik spectrum. Mathematics Subject Classification (2010). 35R11, 35R09, 35A15. Keywords. Non-local operator Fractional Laplacian Fučik spectrum Nonresonance. 1. Introduction The Fučik spectrum of fractional Laplace operator is defined as the set of all (α, β) R 2 such that { ( Δ) s u = αu + βu in u =0 onr n \. has a non-trivial solution u, where s (0, 1) and ( Δ) s is the fractional Laplacian operator defined as ( Δ) s u(x) = 1 u(x + y)+u(x y) 2u(x) 2 R y n+2s dy for all x R n. n In general, we study the Fučik spectrum of an equation driven by the non-local operator L K which is defined as L K u(x) = 1 (u(x + y)+u(x y) 2u(x))K(y)dy for all x R n, 2 R n where K : R n \{0} (0, ) satisfies the following:

2 568 S. Goyal and K. Sreenadh NoDEA (i) mk L 1 (R n ), where m(x) = min{ x 2, 1}, (ii) There exist λ>0ands (0, 1) such that K(x) λ x (n+2s), (iii) K(x) =K( x) for any x R n \{0}. In case K(x) = x (n+2s), L K is the fractional Laplace operator ( Δ) s. The Fučik spectrum of the non-local operator L K is defined as the set K of (α, β) R2 such that } L K u = αu + βu in u =0 inr n (1.1) \. (1.1) has a nontrivial solution u. Here u ± = max(±u, 0) and R n is a bounded domain with Lipshitz boundary. For α = β, the Fučik spectrum of (1.1) becomes the usual spectrum of } L K u = λu in u =0 inr n (1.2) \. Let 0 <λ 1 <λ 2 λ k denote the distinct eigenvalues of (1.2). It is proved in [22] that the first eigenvalue λ 1 of (1.2) is simple, isolated and can be characterized as follows { } λ 1 = inf (u(x) u(y)) 2 K(x y)dxdy : u 2 =1. u X 0 The author also proved that the eigenfunctions corresponding to λ 1 are of constant sign. We observe that K clearly contains (λ k,λ k )foreachk N and two lines λ 1 R and R λ 1. K is symmetric with respect to the diagonal. In this paper we will prove that the two lines R λ 1 and λ 1 R are isolated in K and give a variational characterization of the second eigenvalue λ 2 of L K. When s = 1, the fractional Laplacian operator becomes the usual Laplace operator. The Fučik spectrum was introduced by Fučik (1976) who studied the negative Laplacian in one dimension with periodic boundary conditions, and by Dancer [6]. The Fučik spectrum in the case of Laplacian, p-laplacian equation with Dirichlet, Neumann and Robin boundary condition has been studied by many authors [2 5,8,9,12,15,16,18 20]. A nonresonance problem with respect to Fučik spectrum is also discussed in many papers [4,14,15]. To the best of our knowledge, no work has been done to find the Fučik spectrum for non-local operators. Recently a lot of attention is given to the study of fractional and non-local operator of elliptic type due to concrete real world applications in finance, thin obstacle problem, optimization, quasi-geostrophic flow etc [21,23 25]. Here we use a similar approach to find the Fučik spectrum as the one used in [4]. In [21], Servadei and Valdinoci discussed the Dirichlet boundary value problem in case of fractional Laplacian using the Variational techniques. We also use similar variational techniques to find K. Due to non-localness of

3 Vol. 21 (2014) On the Fučik spectrum of non-local ellipticoperators 569 the fractional Laplacian, the space (X 0,. X0 ) is introduced by Servadei. We introduce this space as follows: { X = u u : R n R is measurable,u L 2 () and (u(x) u(y)) } K(x y) L 2 (), where = R 2n \ (C C) and C :=R n \. The space X is endowed with the norm defined as ( ) 1 u X = u L 2 () + u(x) u(y) 2 2 K(x y)dxdy. Then we define X 0 = {u X : u = 0 a.e. in R n \ } with the norm ( ) 1 u X0 = u(x) u(y) 2 2 K(x y)dxdy is a Hilbert space. Note that the norm. X0 involves the interaction between andr n \. Remark 1.1. (i) Cc 2 () X 0, X H s () and X 0 H s (R n ),whereh s () denotes the usual fractional Sobolev space endowed with the norm ( (u(x) u(y)) 2 ) 1 2 u Hs () = u L 2 + x y n+2s dxdy. (ii) The embedding X 0 L 2 (R n )=L 2 () is continuous, where 2 = To see the details of these embeddings, one can refer [10,21]. 2n n 2s. Definition 1.2. A function u X 0 is a weak solution of (1.1), if for every v X 0, u satisfies (u(x) u(y))(v(x) v(y))k(x y)dxdy = α u + vdx β u vdx. Weak solutions of (1.1) are exactly the critical points of the functional J : X 0 R defined as J(u) = 1 u(y)) 2 (u(x) 2 K(x y)dxdy α 2 (u + ) 2 dx β (u ) 2 dx. 2 Then J is Fréchet differentiable in X 0 and J (u),φ = (u(x) u(y))(φ(x) φ(y))k(x y)dxdy α u + φdx β u φdx.

4 570 S. Goyal and K. Sreenadh NoDEA The paper is organized as follows: In Sect. 2 we construct a first nontrivial curve in K, described as (p + c(p),c(p)). In Sect. 3 we prove that the lines λ 1 R and R λ 1 are isolated in K, the curve that we obtained in Sect. 2 is the first nontrivial curve and give the variational characterization of second eigenvalue of L K. In Sect. 4 we prove some properties of the first curve. A nonresonance problem with respect to the Fučik spectrum is also studied in Sect. 5. We shall throughout use the function space X 0 with the norm. X0 and we use the standard L p () space whose norms are denoted by u L p.alsoφ 1 is the eigenfunction of L K corresponding to λ Fučik spectrum K for L K In this section we study the existence of the first nontrivial curve in the Fučik spectrum K of L K. We find that the points in K are associated with the critical value of some restricted functional. For this we fix p R and for p 0, consider the functional J p : X 0 R by J p (u) = (u(x) u(y)) 2 K(x y)dxdy p (u + ) 2 dx. Then J p C 1 (X 0, R) andforanyφ X 0 J p(u),φ =2 (u(x) u(y))(φ(x) φ(y))k(x y)dxdy 2p Also J p := J p P is C 1 (X 0, R), where P is defined as { } P = u X 0 : I(u) := u 2 dx =1. u + (x)φ(x)dx. We first note that u Pis a critical point of Jp if and only if there exists t R such that (u(x) u(y))(v(x) v(y))k(x y)dxdy p u + vdx = t uvdx, (2.1) for all v X 0. Hence u P is a nontrivial weak solution of the problem L K u =(p + t)u + tu in ; u =0 onr n \, which exactly means (p + t, t) K. Putting v = u in (2.1), we get t = J p (u). Thus we have the following result, which describes the relationship between the critical points of J p and the spectrum K. Lemma 2.1. For p 0, (p + t, t) R 2 belongs to the spectrum K if and only if there exists a critical point u P of J p such that t = J p (u), a critical value. Now we look for the minimizers of J p.

5 Vol. 21 (2014) On the Fučik spectrum of non-local ellipticoperators 571 Proposition 2.2. The first eigenfunction φ 1 is a global minimum for J p with J p (φ 1 )=λ 1 p. The corresponding point in K is (λ 1,λ 1 p) which lies on the vertical line through (λ 1,λ 1 ). Proof. It is easy to see that J p (φ 1 )=λ 1 p and J p (u) = (u(x) u(y)) 2 K(x y)dxdy p (u + ) 2 dx λ 1 u 2 dx p (u + ) 2 dx λ 1 p. Thus φ 1 is a global minimum of J p with J p (φ 1 )=λ 1 p. Now we have a second critical point of Jp at φ 1 corresponding to a strict local minimum. Proposition 2.3. The negative eigenfunction φ 1 is a strict local minimum for J p with J p ( φ 1 )=λ 1. The corresponding point in K is (λ 1 + p, λ 1 ),which lies on the horizontal line through (λ 1,λ 1 ). Proof. Let us suppose by contradiction that there exists a sequence u k P, u k φ 1 with J p (u k ) λ 1, u k φ 1 in X 0. Firstly, we show that u k changes sign for sufficiently large k. Since u k φ 1,itmustbe 0for some x X 0. If u k 0 for a.e x, then J p (u k )= (u k (x) u k (y)) 2 K(x y)dxdy > λ 1, since u k ±φ 1 and we get contradiction as J p (u k ) λ 1.Sou k changes sign for sufficiently large k. Define w k := u+ k and u + k L 2 r k := (w k (x) w k (y)) 2 K(x y)dxdy. Now we claim that r k. Let us suppose by contradiction that r k is bounded. Then there exists a subsequence of w k still denoted by w k and w X 0 such that w k wweakly in X 0 and w k w strongly in L 2 (R n ). Therefore w2 dx =1,w 0 a.e. and so for some ɛ>0, δ = {x X 0 : w(x) ɛ} > 0. As u k φ 1 in X 0 and hence in L 2 (). Therefore {x :u k (x) ɛ} 0 as k and so {x :w k (x) ɛ} 0ask which is a contradiction as δ>0. Hence the claim. Next, (u k (x) u k (y)) 2 =((u + k (x) u+ k (y)) (u k (x) u k (y)))2 =(u + k (x) u+ k (y))2 +(u k (x) u k (y))2 2(u + k (x) u+ k (y))(u k (x) u k (y)) =(u + k (x) u+ k (y))2 +(u k (x) u k (y))2 +2u + k (x)u k (y)+2u k (x)u+ k (y), wherewehaveusedu + k (x)u k (x) = 0. Using K(x) =K( x) wehave u + k (x)u k (y)k(x y)dxdy = u + k (y)u k (x)k(x y)dxdy. (2.2)

6 572 S. Goyal and K. Sreenadh NoDEA Then from above estimates, we get J p (u k )= (u k (x) u k (y)) 2 K(x y)dxdy p (u + k )2 dx = (u + k (x) u+ k (y))2 K(x y)dxdy + (u k (x) u k (y))2 K(x y)dxdy +4 u + k (x)u k (y)k(x y)dxdy p (u + k )2 dx (r k p) (u + k )2 dx + λ 1 (u k )2 dx +4 u + k (x)u k (y)k(x y)dxdy (r k p) (u + k )2 dx + λ 1 (u k )2 dx. As u k P,weget J p (u k ) λ 1 = λ 1 (u + k )2 dx + λ 1 (u k )2 dx. Combining both the inequalities we have, (r k p) (u + k )2 dx + λ 1 (u k )2 dx λ 1 (u + k )2 dx + λ 1 (u k )2 dx. This implies (r k p λ 1 ) (u+ k )2 dx 0, and hence r k p λ 1,which contradicts the fact that r k +, as required. We will now find the third critical point based on the mountain pass theorem by Ambrosetti Robinowitz. A norm of derivative of the restriction J p of J p at u P is defined as J p (u) = min{ J p(u) ti (u) X0 : t R}. Definition 2.4. We say that J p satisfies the Palais Smale [in short, (P.S)] condition on P if for any sequence u k P such that J p (u k ) is bounded and J p(u k ) 0, then there exists a subsequence that converges strongly in X 0. Now we state here the version of mountain pass theorem which will be used later: Proposition 2.5. [1] Let E be a Banach space, g, f C 1 (E,R), M = {u E g(u) =1} and u 0, u 1 M. Letɛ>0 such that u 1 u 0 >ɛand inf{f(u) :u M and u u 0 E = ɛ} > max{f(u 0 ),f(u 1 )}. Assume that f satisfies the (P.S) condition on M and that Γ={γ C([ 1, 1],M):γ( 1) = u 0 and γ(1) = u 1 } is non empty. Then c =inf γ Γ max f(u) is a critical value of f M. Lemma 2.6. J p satisfies the (P.S) condition on P.

7 Vol. 21 (2014) On the Fučik spectrum of non-local ellipticoperators 573 Proof. Let {u k } be a (P.S) sequence. i.e., there exists K>0andt k such that J p (u k ) K, (2.3) (u k (x) u k (y))(v(x) v(y))k(x y)dxdy p u + k v t k u k v = o k (1) v X0. (2.4) From (2.3), we get u k is bounded in X 0. So we may assume that up to a subsequence u k u 0 weakly in X 0,andu k u 0 strongly in L 2 (). Putting v = u k in (2.4), we get t k is bounded and up to a subsequence t k converges to t. We now claim that u k u 0 strongly in X 0.Asu k u 0 weakly in X 0,we have (u k (x) u k (y))(v(x) v(y))k(x y)dxdy (u 0 (x) u 0 (y))(v(x) v(y))k(x y)dxdy (2.5) for all v X 0.Also J p(u k )(u k u 0 )=o k (1). Therefore we get (u k (x) u k (y)) 2 K(x y)dxdy (u k (x) u k (y))(u 0 (x) u 0 (y))k(x y)dxdy o k (1) + p u + k L 2 u k u 0 L 2 + t k u k L 2 u k u 0 L 2 0 as k. Taking v = u 0 in (2.5), we get (u k (x) u k (y))(u 0 (x) u 0 (y))k(x y)dxdy (u 0 (x) u 0 (y)) 2 K(x y)dxdy. From above two equations, we have (u k (x) u k (y)) 2 K(x y)dxdy (u 0 (x) u 0 (y)) 2 K(x y)dxdy. Thus u k 2 X 0 u 0 2 X 0. Now using this and v = u 0 in (2.5), we get u k u 0 2 X 0 = u k 2 X 0 + u k 2 X 0 2 (u k (x) u k (y))(u 0 (x) u 0 (y))k(x y)dxdy 0 as k. Hence u k u 0 strongly in X 0. Lemma 2.7. Let ɛ 0 > 0 be such that J p (u) > J p ( φ 1 ) (2.6) for all u B( φ 1,ɛ 0 ) P with u φ 1, where the ball is taken in X 0.Then for any 0 <ɛ<ɛ 0, inf{ J p (u) :u Pand u ( φ 1 ) X0 = ɛ} > J p ( φ 1 ). (2.7)

8 574 S. Goyal and K. Sreenadh NoDEA Proof. Assume by contradiction that the infimum in (2.7) is equal to J p ( φ 1 )= λ 1 for some ɛ with 0 <ɛ<ɛ 0. Then there exists a sequence u k Pwith u k ( φ 1 ) X0 = ɛ such that J p (u k ) λ k 2. Consider the set C = {u P: ɛ δ u ( φ 1 ) X0 ɛ + δ}, where δ is chosen such that ɛ δ>0andɛ + δ<ɛ 0. In view of our contradiction hypothesis and (2.6), it follows that inf{ J p (u) :u C} = λ 1. Now for each k, we apply Ekeland s variational principle to the functional Jp on C to get the existence of v k C such that J p (v k ) J p (u k ), v k u k X0 1 k, J p (v k ) J p (u)+ 1 k v v k X0 v C (2.8) We claim that v k is a (P.S) sequence for J p on P i.e. Jp (v k ) is bounded and J p(v k ) 0. Once this is proved we get by Lemma 2.6, up to a subsequence v k v strongly in X 0. Clearly v Pand satisfies v ( φ 1 ) X0 ɛ + δ<ɛ 0 and J p (v) =λ 1 which contradicts the given hypotheses. Clearly J p (v k )isa bounded. So we only need to prove that J p(v k ) 0. Let k> 1 δ and take w X 0 tangent to P at v k i.e v k wdx =0, and for t R, define u t := v k + tw. v k + tw L 2 For sufficiently small t, u t Cand take v = u t in (2.8), we get J p(v k ),w 1 k w X 0. (2.9) For complete details refer to Lemma 2.9 of [4]. Since w is arbitrary in X 0, we choose α k such that v k(w α k v k )dx =0. Replacing w by w α k v k in (2.9), we have J p (v k ),w α k J p(v k ),v k 1 k w α kv k X0, since α k v k X0 C w X0,weget J p(v k ),w t k v k wdx C k w X 0 where t k = J p(v k ),v k. Hence as we required. J p(v k ) 0 as k,

9 Vol. 21 (2014) On the Fučik spectrum of non-local ellipticoperators 575 Proposition 2.8. Let ɛ>0 such that φ 1 ( φ 1 ) X0 >ɛand inf{ J p (u) :u Pand u ( φ 1 ) X0 = ɛ} > max{ J p ( φ 1 ), J p (φ 1 )}. Then Γ={γ C([ 1, 1], P) :γ( 1) = φ 1 and γ(1) = φ 1 } is non empty and is a critical value of Jp. Moreover c(p) >λ 1. c(p) = inf max J p(u) (2.10) γ Γ Proof. Let φ X 0 be such that φ Rφ 1 and consider the path γ(t) = tφ 1+(1 t )φ tφ, then γ(t) Γ. Moreover by Lemmas 2.6 and 2.7, Jp 1+(1 t )φ satisfies (P.S) condition and the geometric assumptions. Then by Proposition L 2 2.5, c(p) is a critical value of Jp. Using the definition of c(p) wehavec(p) > max{ J p ( φ 1 ), J p (φ 1 )} = λ 1. Thus we have proved the following: Theorem 2.9. For each p 0, the point (p + c(p),c(p)), wherec(p) >λ 1 is defined by the minimax formula (2.10), then the point (p + c(p),c(p)) belongs to K. It is a trivial fact that K is symmetric with respect to the diagonal. The whole curve, which we obtain using Theorem 2.9 and symmetrizing, is denoted by C := {(p + c(p),c(p)), (c(p),p+ c(p)) : p 0}. 3. First nontrivial curve We start this section by establishing that the lines R {λ 1 } and {λ 1 } R are isolated in K. Then we state some topological properties of the functional J p and finally we prove that the curve C constructed in the previous section is the first nontrivial curve in the spectrum K. Proposition 3.1. The lines R {λ 1 } and {λ 1 } R are isolated in K. In other words, there exists no sequence (α k,β k ) K with α k >λ 1 and β k >λ 1 such that (α k,β k ) (α, β) with α = λ 1 or β = λ 1. Proof. Suppose by contradiction that there exists a sequence (α k,β k ) K with α k,β k >λ 1 and (α k,β k ) (α, β) with α or β = λ 1.Letu k X 0 be a solution of L K u k = α k u + k β ku k in, u k =0 onr n \ (3.1) with u k L 2 = 1. Then we have (u k (x) u k (y)) 2 K(x y)dxdy = α k (u + k )2 dx β k (u k )2 dx α k,

10 576 S. Goyal and K. Sreenadh NoDEA which shows that u k is bounded sequence in X 0. Therefore up to a subsequence u k uweakly in X 0 and u k u strongly in L 2 (). Then the limit u satisfies (u(x) u(y)) 2 K(x y)dxdy = λ 1 (u + ) 2 dx β (u ) 2 dx, since u k uweakly in X 0 and J p(u k ),u k u 0ask. i.e u is a weak solution of L K u = αu + βu in, u =0 onr n \ (3.2) where we have considered the case α = λ 1. Multiplying by u + in (3.2), integrate, using (u(x) u(y))(u + (x) u + (y))=(u + (x) u + (y)) 2 + u + (x)u (y)+u + (y)u (x) and (2.2), we get (u + (x) u + (y)) 2 K(x y)dxdy +2 u + (x)u (y)k(x y)dxdy = λ 1 (u + ) 2 dx. Using this we have, λ 1 (u + ) 2 dx Thus (u + (x) u + (y)) 2 K(x y)dxdy λ 1 (u + (x) u + (y)) 2 K(x y)dxdy = λ 1 (u + ) 2 dx, (u + ) 2 dx so that either u + 0oru = φ 1.Ifu + 0 then u 0 and (3.2) implies that u is an eigenfunction with u 0 so that u = φ 1.Soinanycaseu k converges to either φ 1 or φ 1 in L 2 (). Thus for every ɛ>0 either {x :u k (x) ɛ} 0 or {x :u k (x) ɛ} 0. (3.3) On the other hand, taking u + k as test function in (3.1), we get (u + k (x) u+ k (y))2 K(x y)dxdy +2 u + k (x)u k (y)k(x y)dxdy = α k (u + k )2 dx. Using this, Hölders inequality and Sobolev embeddings we get (u + k (x) u+ k (y))2 K(x y)dxdy (u + k (x) u+ k (y))2 K(x y)dxdy +2 u + k (x)u k (y)k(x y)dxdy = α k (u + k )2 dx α k C {x :u k (x) > 0} 1 2 q u + k 2 X 0

11 Vol. 21 (2014) On the Fučik spectrum of non-local ellipticoperators 577 with a constant C>0, 2 <q 2 = {x :u k (x) > 0} 1 2 q Similarly, one can show that 2n n 2s. Then we have α 1 k C 1. {x :u k (x) < 0} 1 2 q β 1 k C 1. Since (α k,β k ) does not belong to the trivial lines λ 1 R and R λ 1 of K,by using (3.1) wehavethatu k changes sign. Hence, from the above inequalities, we get a contradiction with (3.3). Hence the trivial lines λ 1 R and R λ 1 are isolated in K. Lemma 3.2. [4] Let P = {u X 0 : u2 =1} then 1. P is locally arcwise connected. 2. Any open connected subset O of P is arcwise connected. 3. If O is any connected component of an open set O P, then O O =. Lemma 3.3. Let O = {u P: Jp (u) <r}, then any connected component of O contains a critical point of Jp. Proof. Proof follows in the same lines as Lemma 3.6 of [4] by replacing. 1,p by. X0. Theorem 3.4. Let p 0 then the point (p + c(p),c(p)) is the first nontrivial point in the intersection between K and the line (p, 0) + t(1, 1). Proof. Assume by contradiction that there exists μ such that λ 1 <μ<c(p) and (p+μ, μ) K. Using the fact that {λ 1} R and R {λ 1 } are isolated in K and K is closed we can choose such a point with μ minimum. In other words J p has a critical value μ with λ 1 <μ<c(p), but there is no critical value in the open interval (λ 1,μ). If we construct a path connecting from φ 1 to φ 1 such that J p μ, then we get a contradiction with the definition of c(p), which completes the proof. Let u P be a critical point of J p at level μ. Then u satisfies, (u(x) u(y))(v(x) v(y))k(x y)dxdy =(p + μ) u + vdx μ u vdx. for all v X 0. Replacing v by u + and u,wehave (u + (x) u + (y)) 2 K(x y)dxdy +2 u + (x)u (y)k(x y)dxdy =(p + μ) (u + ) 2 dx, and (u (x) u (y)) 2 K(x y)dxdy +2 = μ (u ) 2 dx. u + (x)u (y)k(x y)dxdy

12 578 S. Goyal and K. Sreenadh NoDEA Thus we obtain, ( ) u + J p (u) =μ, Jp u + = μ 2 u+ (x)u (y)k(x y)dxdy L 2 u + 2, L 2 ( ) u J p u = μ p 2 u+ (x)u (y)k(x y)dxdy L 2 u 2, L 2 and ( ) J p u u = μ 2 u+ (x)u (y)k(x y)dxdy L 2 u 2. L 2 Since u changes sign, the following paths are well-defined on P: u 1 (t) = (1 t)u + tu+ (1 t)u + tu +, u 2 (t) = tu +(1 t)u + L 2 tu +(1 t)u +, L 2 tu +(1 t)u u 3 (t) = tu. +(1 t)u L 2 Then by using the above calculation one can easily get that for all t [0, 1], J p (u 1 (t)) = [(u+ (x) u + (y)) 2 +(1 t) 2 (u (x) u (y)) 2 ]K(x y)dxdy u + (1 t)u 2 L 2 + 4(1 t) u (x)u (y)k(x y)dxdy p (u+ ) 2 dx u + (1 t)u 2 L 2 = μ 2t2 u+ (x)u (y)k(x y)dxdy u + (1 t)u 2 L 2. J p (u 2 (t)) = [(1 t)2 (u + (x) u + (y)) 2 + t 2 (u (x) u (y)) 2 ]K(x y)dxdy (1 t)u + + tu 2 L 2 J p (u 3 (t)) = 4t(1 t) u+ (x)u (y)k(x y)dxdy + p(1 t) 2 (u+ ) 2 + pt 2 (u ) 2 (1 t)u + + tu 2 L 2 = μ 2 u+ (x)u (y)k(x y)dxdy (1 t)u + + tu 2 L 2 pt 2 (u ) 2 dx (1 t)u + + tu 2 L 2. [(1 t)2 (u + (x) u + (y)) 2 +(u (x) u (y)) 2 ]K(x y)dxdy (1 t)u + u 2 L 2 + 4(1 t) u+ (x)u (y)k(x y)dxdy p(1 t) 2 (u+ ) 2 dx (1 t)u + u 2 L 2 = μ 2t2 u+ (x)u (y)k(x y)dxdy (1 t)u + u 2 L 2. Let O = {v P: Jp (v) <μ p}. Then clearly φ 1 O, while φ 1 Oif μ p>λ 1. Moreover φ 1 and φ 1 are the only possible critical points of J p in O because of the choice of μ.

13 Vol. 21 (2014) On the Fučik spectrum of non-local ellipticoperators 579 We note that J ( ) u u p u μ p, L 2 u does not change sign and L 2 vanishes on a set of positive measure, it is not a critical point of J p. Therefore there exists a C 1 path η :[ ɛ, ɛ] Pwith η(0) = and d J dt p (η(t)) t=0 u u L 2 0. Using this path we can move from u u to a point v with J L 2 p (v) <μ p. Taking a connected component of O containing v and applying Lemma 3.3 we have that either φ 1 or φ 1 is in this component. Let us assume that it is φ 1. ( So we continue by a path u 4 (t) from ( μ. Then the path u 4 (t) connects u u L 2 u u L 2 ) to φ 1 which is at level less than ) to φ 1. We observe that J p (u) J p ( u) p. Then it follows that J p ( u 4 (t)) J p (u 4 (t)) + p μ p + p = μ t. Connecting u 1 (t), u 2 (t) andu 4 (t), we get a path from u to φ 1 and joining u 3 (t) and u 4 (t) wegetapathfromu to φ 1. These yields a path γ(t) onp joining from φ 1 to φ 1 such that J p (γ(t)) μ for all t, which concludes the proof. Corollary 3.5. The second eigenvalue λ 2 of (1.2) has the variational characterization given as λ 2 =inf max (u(x) u(y)) 2 K(x y)dxdy, γ Γ where Γ is as in Proposition 2.8. Proof. Take p = 0 in Theorem 3.4. Then we have c(0) = λ 2 and (2.10) concludes the proof. 4. Properties of the curve In this section we prove that the curve C is Lipschitz continuous, has a certain asymptotic behavior and is strictly decreasing. Proposition 4.1. The curve p (p+c(p),c(p)), p R + is Lipschitz continuous. Proof. Proof follows as in Proposition 4.1 of [4]. For completeness we give details. Let p 1 <p 2 then J p1 (u) > J p2 (u) for all u P.Sowehavec(p 1 ) c(p 2 ). Now for every ɛ>0 there exists γ Γ such that max J p2 (u) c(p 2 )+ɛ, and so 0 c(p 1 ) c(p 2 ) max J p1 (u) max J p2 (u)+ɛ. Let u 0 γ[ 1, 1] such that max J p1 (u) = J p1 (u 0 )

14 580 S. Goyal and K. Sreenadh NoDEA then 0 c(p 1 ) c(p 2 ) J p1 (u 0 ) J p2 (u 0 )+ɛ p 2 p 1 + ɛ, as ɛ>0isarbitrary so the curve C is Lipschitz continuous with constant 1. Lemma 4.2. Let A, B be two bounded open sets in R n,witha B and B is connected then λ 1 (A) >λ 1 (B). Proof. From the Proposition 4 of [25] and Theorem 2 of [24], we see that φ 1 is continuous and is a solution of (1.2) in viscosity sense. Then from Lemma 12 of [13], φ 1 > 0. Now from the variational characterization, we see that for A B, λ 1 (A) λ 1 (B). Since φ 1 (B) > 0inB, we get the strict inequality as claimed. Lemma 4.3. Let (α, β) C,andletα(x), β(x) L () satisfying λ 1 α(x) α, λ 1 β(x) β. (4.1) Assume that λ 1 <α(x) and λ 1 <β(x) on subsets of positive measure. (4.2) Then any non-trivial solution u of L K u = α(x)u + β(x)u in, u =0 in R n \. (4.3) changes sign in and α(x) =α a.e. on {x :u(x) > 0}, β(x) =β a.e. on {x :u(x) < 0}. Proof. Let u be a nontrivial solution of (4.3). Replacing u by u if necessary. we can assume that the point (α, β) inc is such that α β. We first claim that u changes sign in. Suppose by contradiction that this is not true, we first consider the case u 0 (the case u 0 can be prove similarly). Then u solves L k u = α(x)u in u =0 onr n \. This implies that the first eigenvalue of L K on X 0 with respect to weight α(x) is equal to 1. i.e { inf (v(x) v(y))2 K(x y)dxdy α(x)v2 dx } : v X 0,v 0 =1. (4.4) We deduce from (4.1), (4.2) and (4.4) that 1= (φ1(x) φ1(y))2 K(x y)dxdy > (φ1(x) φ1(y))2 K(x y)dxdy 1, λ 1 α(x)φ2 1dx a contradiction and hence the claim. Again we assume by contradiction that either {x X 0 : α(x) <αand u(x) > 0} > 0 (4.5) or {x X 0 : β(x) <βand u(x) < 0} > 0. (4.6)

15 Vol. 21 (2014) On the Fučik spectrum of non-local ellipticoperators 581 Suppose that (4.5) holds [a similar argument will hold for (4.6)]. Put α β = p 0. Then β = c(p), where c(p) is given by (2.10). We show that there exists a path γ Γ such that max J p (u) <β, (4.7) which yields a contradiction with the definition of c(p). In order to construct γ we show that there exists a function v X 0 such that it changes sign and satisfies (v+ (x) v + (y)) 2 K(x y)dxdy (v+ ) 2 <α and dx (v (x) v (y)) 2 K(x y)dxdy (v ) 2 <β. dx (4.8) ForthisletO 1 be a component of {x :u(x) > 0} satisfying x O 1 : α(x) <α > 0, which is possible by (4.5). Let O 2 be a component of {x :u(x) < 0} satisfying x O 1 : β(x) <β > 0, which is possible by (4.6). Then we claim that λ 1 (O 1 ) <α and λ 1 (O 2 ) β, (4.9) where λ 1 (O i ) denotes the first eigenvalue of L k on X 0 Oi = {u X Oi : u = 0onR n \O i }. Clearly u Oi X 0 Oi then we have O1 (u(x) u(y)) 2 K(x y)dxdy O1 (u(x) u(y)) 2 K(x y)dxdy <α = α O 1 u 2 dx O 1 α(x)u 2 dx which implies that λ 1 (O 1 ) <α. The other inequality in (4.9) is proved similarly. Now with some modification on the sets O 1 and O 2, we construct the sets Õ1 and Õ2 such that Õ1 Õ2 = and λ 1 (Õ1) <αand λ 1 (Õ2) <β.for this, we consider for ν 0, O 1 (ν) ={x O 1 : dist(x, O c 1) >ν}. Then clearly λ 1 (O 1 (ν)) λ 1 (O 1 ) and moreover λ 1 (O 1 (ν)) λ 1 (O 1 )asν 0. Then there exists ν 0 > 0 such that λ 1 (O 1 (ν)) <α for all 0 ν ν 0. (4.10) Let x 0 O 2 (is not empty as O 1 O 2 = ) and choose 0 < ν < min{ν 0,dist(x 0, c )} and Õ1 = O 1 (ν)andõ2 = O 2 B(x 0, ν 2 ). Then Õ1 Õ2 = and by (4.10), λ 1 (Õ1) <α. Since Õ2 is connected then by (4.9) and Lemma 4.2, wegetλ 1 (Õ2) <β. Now we define v = v 1 v 2, where v i are the extension by zero outside Õi of the eigenfunctions associated to λ i (Õi). Then v satisfies

16 582 S. Goyal and K. Sreenadh NoDEA (4.8). Thus there exist v X 0 which changes sign and satisfies condition (4.8), and moreover we have ( ) J v p = (v+ (x) v + (y)) 2 K(x y)dxdy + (v (x) v (y)) 2 K(x y)dxdy v L 2 v 2 L v 2 2 L 2 2 v+ (x)v (y)k(x y)dxdy p (v+ ) 2 dx v 2 L v 2 2 L 2 < (α p) (v+ ) 2 dx + β (v ) 2 dx 2 v+ (x)v (y)k(x y)dxdy <β. v 2 L v 2 2 L v 2 2 L 2 ( ) ( ) v + v J p v + <α p = β, Jp L 2 v <β p. L 2 Using Lemma 3.3, we have that there exists a critical point in the connected component of the set O = {u P: Jp (u) <β p}. As the point (α, β) C, the only possible critical point is φ 1, then we can construct a path from φ 1 to φ 1 exactly in the same manner as in Theorem 3.4 only by taking v in place of u. Thus we have constructed a path satisfying (4.7), and hence the result follows. Corollary 4.4. Let (α, β) Cand let α(x),β(x) L () satisfying λ 1 α(x) α a.e, λ 1 β(x) β a.e. Assume that λ 1 <α(x) and λ 1 <β(x) on subsets of positive measure. If either α(x) <αa.e in or β(x) <βa.e. in, then (4.3) has only the trivial solution. Lemma 4.5. The curve p (p + c(p),c(p)) is strictly decreasing, (in the sense that p 1 <p 2 implies p 1 + c(p 1 ) <p 2 + c(p 2 ) and c(p 1 ) >c(p 2 )). Proof. Let p 1 <p 2 and suppose by contradiction that either (i) p 1 + c(p 1 ) p 2 + c(p 2 )or(ii) c(p 1 ) c(p 2 ). In case (i) we deduce from p 1 + c(p 1 ) p 2 +c(p 2 ) >p 1 +c(p 2 ) that c(p 1 ) c(p 2 ). If we take (α, β) =(p 1 +c(p 1 ),c(p 1 )) and (α(x),β(x)) = (p 2 + c(p 2 ),c(p 2 )), then by Corollary 4.4, the only solution of (4.3) with (α(x),β(x)) is the trivial solution which contradicts the fact that (p 2 +c(p 2 ),c(p 2 )) K.If(ii) holds then p 1 +c(p 1 ) p 1 +c(p 2 ) <p 2 +c(p 2 ), if we take (α, β) =(p 2 +c(p 2 ),c(p 2 )) and (α(x),β(x)) = (p 1 +c(p 1 ),c(p 1 )), then the only solution of (4.3) with (α(x),β(x)) is the trivial one which contradicts the fact that (p 1 + c(p 1 ),c(p 1 )) K and hence the result follows. As c(p) is decreasing and positive so the limit of c(p) existsasp. In the next Theorem we find the asymptotic behavior of the first nontrivial curve. Theorem 4.6. If n 2s then the limit of c(p) as p is λ 1. Proof. For n 2s, we can choose a function φ X 0 such that there does not exist r R such that φ(x) rφ 1 (x) a.e. in. For this it suffices to take φ X 0 such that it is unbounded from above in a neighborhood of some point x X 0. Then by contradiction argument, one can similarly show c(p) λ 1 as p as in Proposition 4.4 of [4].

17 Vol. 21 (2014) On the Fučik spectrum of non-local ellipticoperators Nonresonance between (λ 1,λ 1 ) and C In this section we study the following problem { LK u = f(x, u) in u =0 onr n (5.1) \, where f(x, u)/u lies asymptotically between (λ 1,λ 1 ) and (α, β) C.Letf : R R be a function satisfying L () Caratheodory conditions. Given a point (α, β) C, we assume following: f(x, s) f(x, s) γ ± (x) lim inf lim sup Γ ± (x) (5.2) s ± s s ± s hold uniformly with respect to x, where γ ± (x) and Γ ± (x) arel functions which satisfy { λ1 γ + (x) Γ + (x) α a.e. in (5.3) λ 1 γ (x) Γ (x) β a.e. in. Write F (x, s) = s f(x, t)dt, we also assume the following inequalities: 0 2F (x, s) 2F (x, s) δ ± (x) lim inf s ± s 2 lim sup s ± s 2 Δ ± (x) (5.4) hold uniformly with respect to x, where δ ± (x) and Δ ± (x) arel functions which satisfy λ 1 δ + (x) Δ + (x) α a.e. in λ 1 δ (x) Δ (x) β a.e. in (5.5) δ + (x) >λ 1 and δ (x) >λ 1 on subsets of positive measure, either Δ + (x) <αa.e. in or Δ (x) <βa.e. in. Theorem 5.1. Let (5.2), (5.3), (5.4), (5.5) hold and (α, β) C. Then problem (5.1) admits at least one solution u in X 0. Define the energy functional Ψ : X 0 R as Ψ(u) = 1 (u(x) u(y)) 2 K(x y)dxdy F (x, u)dx 2 Then Ψ is a C 1 functional on X 0 and v X 0 Ψ (u),v = (u(x) u(y))(v(x) v(y))k(x y)dxdy f(x, u)vdx and critical points of Ψ are exactly the weak solutions of (5.1). Lemma 5.2. Ψ satisfies the (P.S) condition on X 0. Proof. Let u k be a (P.S) sequence in X 0, i.e Ψ(u k ) c, Ψ (5.6) (u k ),v ɛ k v X0, v X 0, where c is a constant and ɛ k 0ask. It suffices to show that u k is a bounded sequence in X 0. Assume by contradiction that u k is not a bounded sequence. Then define v k = which is a bounded sequence. Therefore u k u k X0

18 584 S. Goyal and K. Sreenadh NoDEA there exists a subsequence v k of v k and v 0 X 0 such that v k v 0 weakly in X 0, v k v 0 strongly in L 2 () and v k (x) v 0 (x) a.e. in R n.alsoby using (5.2) and (5.3), we have f(x, u k )/ u k X0 f 0 (x) weakly in L 2 (). Take v = v k v 0 in (5.6) and divide by u k X0 we get v k v 0. In particular v 0 X0 = 1. One can easily see from (5.6) that (v 0 (x) v 0 (y))(v(x) v(y))k(x y)dxdy f 0 (x)vdx =0 v X 0. Now by a standard argument based on assumption (5.2), f 0 (x) =α(x)v 0 + β(x)v0 for some L functions α(x), β(x) satisfying (4.1). In the expression of f 0 (x), the value of α(x) (resp.β(x)) on {x : v 0 (x) 0} (resp. {x : v 0 (x) 0}) are irrelevant, and consequently we can assume that α(x) >λ 1 on {x : v 0 (x) 0} and β(x) >λ 1 on {x : v 0 (x) 0}. (5.7) So v 0 is a nontrivial solution of Eq. (4.3). It then follows from Lemma 4.3 that either (i) α(x) =λ 1 a.e in or (ii) β(x) =λ 1 a.e in, or (iii) v 0 is an eigenfunction associated to the point (α, β) C. We show that in each case we get a contradiction. If (i) holds then by (5.7), v 0 > 0 a.e. in and (4.3) gives (v 0(x) v 0 (y)) 2 K(x y)dxdy = λ 1 v2 0, which implies that v 0 is a multiple of φ 1. Dividing (5.6) by u k 2 X 0 and taking limit we get, λ 1 v0 2 = (v 0 (x) v 0 (y)) 2 K(x y)dxdy = lim k 2F (x, u k ) u k 2 X 0 δ + (x)v 2 0dx. This contradicts assumption (5.5). The case (ii) is treated similarly. Now if (iii) holds, we deduce from (5.4) that α(v 0 + )2 + β(v0 )2 = (v 0 (x) v 0 (y)) 2 2F (x, u k ) K(x y)dxdy = lim k u k 2 X 0 Δ + (x)(v 0 + )2 +Δ (x)(v0 )2. This contradicts assumption (5.5), since v 0 changes sign. Hence u k is a bounded sequence in X 0. Now we study the geometry of Ψ. Lemma 5.3. There exists R>0 such that max{ψ(rφ 1 ), Ψ( Rφ 1 )} < max Ψ(u) (5.8) for any γ Γ 1 := {γ C([ 1, 1],X 0 ):γ(±1) = ±Rφ 1 }. Proof. From (5.4), we have for any ɛ>0 there exists a ɛ (x) L 2 () such that for a.e x, { (δ + (x) ɛ) s2 2 a ɛ(x) F (x, s) (Δ + (x)+ɛ) s2 2 + a ɛ(x) s>0 (δ (x) ɛ) s2 2 a ɛ(x) F (x, s) (Δ (x)+ɛ) s2 2 + a (5.9) ɛ(x) s<0.

19 Vol. 21 (2014) On the Fučik spectrum of non-local ellipticoperators 585 Now consider the following functional associated to the functions Δ ± (x) as Φ(u) = (u(x) u(y)) 2 K(x y)dxdy Δ + (x)(u + ) 2 dx Δ (x)(u ) 2 dx. Then we claim that d =inf max γ Γ Ψ(u) > 0 (5.10) where Γ is the set of all continuous paths from φ 1 to φ 1 in P. Write p = α β 0. we can choose (α, β) Csuch that α β (by replacing u by u if necessary), we have for any γ Γ i.e. max which implies max J p (u) c(p) =β. ( (u(x) u(y)) 2 K(x y)dxdy max Φ(u) 0, by (5.5). So d 0. On the other hand, since δ ± (x) Δ ± (x), Φ(±φ 1 ) (λ 1 δ ± (x))φ 2 1dx < 0 ) α(u + ) 2 dx β(u ) 2 dx 0 by (5.5). Thus we have a mountain pass geometry for the restriction Φ ofφ to P, max{ Φ(φ 1 ), Φ( φ 1 )} < 0 max Φ(u) for any path γ Γ and moreover one can verify exactly as in Lemma 2.6 that Φ satisfies the (P.S.) condition on X 0. Then d is a critical value of Φ i.e there exists u P and μ R such that { Φ(u) =d Φ (u),v = μ I (u),v v X 0. Assume by contradiction that d = 0. Taking v = u in above, we get μ =0so u is a nontrivial solution of L K u =Δ + (x)u + Δ (x)u in, u =0 inr n \. Using (5.5), we get a contradiction with Lemma 4.3. This completes the proof of the claim. Next we show that (5.8) holds. From the left hand side of inequality (5.9), we have for R>0andη>0, Ψ(±Rφ 1 ) R2 2 (λ 1 δ ± (x))φ ηr2 2 + a η L 1,

20 586 S. Goyal and K. Sreenadh NoDEA Then, Ψ(±Rφ 1 ) as R +, by(5.5) and letting η to be sufficiently small. Fix ɛ with 0 <ɛ<d.wecanchooser = R(ɛ) sothat Ψ(±Rφ 1 ) < a ɛ L 1, (5.11) where a ɛ is associated to ɛ using (5.9). Consider a path γ Γ 1. Then if 0 γ[ 1, 1], then by (5.11), Ψ(±Rφ 1 ) < a ɛ L 1 0 = Ψ(0) max Ψ(u), so the Lemma is proved in this case. If 0 γ[ 1, 1], then we take the normalized path γ(t) = belonging to Γ. Since by (5.9) wehave γ(t) γ(t) L 2 Then we obtain Ψ(u) Φ(u) ɛ u 2 L 2 2 a ɛ L 1. 2Ψ(u)+ɛ u 2 L +2 a max 2 ɛ L 1 γ [ 1,1] u 2 max Φ(v) d, L 2 γ [ 1,1] and consequently, by the choice of ɛ, wehave 2Ψ(u)+2 a ɛ max L 1 γ [ 1,1] u 2 d ɛ>0. L 2 This implies that max Ψ(u) > a ɛ L 1 > Ψ(±Rφ 1 ), by (5.11) and hence the Lemma. ProofofTheorem5.1: Lemmas 5.2 and 5.3 complete the proof. References [1] Ambrosetti, A., Robinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, (1973) [2] Alif, M.: Fučik spectrum for the Neumann problem with indefinite weights, partial differential equations. In: Lecture notes, vol Pure Appl. Math. Dekker, New York, pp (2002) [3] Arias, M., Campos, J.: Radial Fučik spectrum of the Laplace operator. J. Math. Anal. Appl. 190, (1995) [4] Cuesta, M., de Figueiredo, D., Gossez, J.-P.: The beginning of the Fučik spectrum for the p-laplacian. J. Differ. Equ. 159, (1999) [5] Cuesta, M., Gossez, J.-P.: A variational approach to nonresonance with respect to the Fučik spectrum. Nonlinear Anal. 19, (1992) [6] Dancer, E.N.: On the Dirichlet problem for weakly non-linear elliptic partial differential equations. Proc. R. Soc. Edinburgh Sect. A. 76(4): (1976/77)

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22 588 S. Goyal and K. Sreenadh NoDEA [25] Servadei, R.: A Brezis Nirenberg result for non-local critical equations in low dimension. Commun. Pure Appl. Math. 12, (2013) Sarika Goyal and K. Sreenadh Department of Mathematics Indian Institute of Technology Delhi Hauz Khaz New Delhi-16 India sreenadh@gmail.com Sarika Goyal sarika1.iitd@gmail.com Received: 17 June Accepted: 3 December 2013.