Nehari manifold for non-local elliptic operator with concave convex nonlinearities and sign-changing weight functions

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1 Proc. Indian Acad. Sci. (Math. Sci.) Vol. 125, No. 4, November 2015, pp c Indian Academy of Sciences Nehari manifold for non-local elliptic operator with concave convex nonlinearities and sign-changing weight functions SARIKA GOYAL and K SREENADH Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi , India Corresponding Author. sarika1.iitd@gmail.com; sreenadh@gmail.com MS received 17 December 2013; revised 26 January 2015 Abstract. In this article, we study the existence and multiplicity of non-negative solutions of the following p-fractional equation: u(y) u(x) p 2 (u(y) u(x)) 2 R n x y n+pα dy = λh(x) u q 1 u+b(x) u r 1 u in, u = 0 inr n \, u W α,p (R n ) where is a bounded domain in R n with continuous boundary, p 2, n>pα, α (0, 1),0<q<p 1 <r<p 1 with p = np(n pα) 1, λ>0andh, b are signchanging continuous functions. We show the existence and multiplicity of solutions by minimization on the suitable subset of Nehari manifold using the fibering maps. We find that there exists λ 0 such that for λ (0,λ 0 ), it has at least two non-negative solutions. Keywords. Non-local operator; p-fractional Laplacian; sign-changing weight functions; Nehari manifold; fibering maps Mathematics Subject Classification. 35J35, 35J60, 35R Introduction We consider the following p-fractional Laplace equation 2 R n u(y) u(x) p 2 (u(y) u(x)) x y n+pα dy =λh(x) u q 1 u+b(x) u r 1 u in, u 0 in, u W α,p (R n ), u = 0 onr n \, where is a bounded domain in R n with continuous boundary, p 2, n>pα,0< q<p 1 <r<p 1 with p = np(n pα) 1, λ>0, h and b are sign-changing continuous functions. 545

2 546 Sarika Goyal and K Sreenadh Recently a lot of attention has been given to the study of fractional and non-local operators of elliptic type due to concrete real world applications in finance, thin obstacle problem, optimization, quasi-geostrophic flow etc. The Dirichlet boundary value problem in the case of fractional Laplacian with polynomial type nonlinearity using variational methods has been studied in [7, 13 17, 24] recently. Also, existence and multiplicity results for non-local operators with convex concave type nonlinearity are shown in [18]. In case of square root of Laplacian, existence and multiplicity results for sublinear and superlinear type of nonlinearity with sign-changing weight functions are studied in [24]. In [24], the author used the idea of Caffarelli and Silvestre [8], which gives a formulation of the fractional Laplacian through Dirichlet Neumann maps. Recently, eigenvalue problem related to p-laplacian has been studied in [11, 12]. In particular, for α = 1, a lot of work has been done for multiplicity of positive solutions of semilinear elliptic problems with positive nonlinearities [1 3, 19]. Moreover multiplicity results with polynomial type nonlinearity with sign-changing weight functions using Nehari manifold and fibering map analysis is also studied in many papers (see [4 6, 10, 19 23]. In this work, we use fibering map analysis and Nehari manifold approach to solve the problem (1.1). Our work is motivated by the works of Servadei and Valdinoci [13] and by Brown and Wu [6]. The aim of this paper is to study the existence and multiplicity of non-negative solutions for the following equation driven by non-local operator L K with convex concave type nonlinearities L K u = λh(x) u q 1 u + b(x) u r 1 u in, u = 0inR n \. The non-local operator L K is defined as L K u(x) = 2 u(y) u(x) p 2 (u(y) u(x))k(x y)dy for all x R n, R n where K : R n \{0} (0, ) satisfying: (i) mk L 1 (R n ), where m(x) = min{1, x p }, (ii) there exist θ>0 and α (0, 1) such that K(x) θ x (n+pα), } (1.1) More precisely, we study the problem to find u W α,p (R n ) such that for every v W α,p (R n ), u(x) u(y) p 2 (u(x) u(y))(v(x) v(y))k(x y)dxdy = λ h(x) u q 1 uvdx + b(x) u r 1 uvdx holds. Now we first introduce a space and some notations, then we have the existence result. Define the space X 0 ={u u : R n R is measurable,u L p (), (u(x) u(y)) p K(x y) L p (),u = 0inR n \ },

3 Non-local operator with sign-changing weight functions 547 where = R 2n \(C C). In the next section, we study the properties of X 0 in detail. { } { } B ± := u X 0 : b(x) u r+1 dx 0,B 0 := u X 0 : b(x) u r+1 dx =0, { H ± := u X 0 : and H ± 0 := H ± H 0, B ± 0 := B± B 0. } h(x) u q+1 dx 0,H 0 := { u X 0 : } h(x) u q+1 dx =0, Theorem 1.1. There exists λ 0 > 0 such that for λ (0,λ 0 ), (1.1) admits at least two non-negative solutions. Here λ 0 is the maximum of λ such that for 0 <λ<λ 0, the fibering map t J λ (tu) has exactly two critical points for each u B + H +. The paper is organized as follows: In 1, we study the properties of the space X 0.In 2, we introduce Nehari manifold and study the behavior of Nehari manifold by carefully analysing the associated fibering maps. Section 3 contains the existence of non-trivial solutions. 2. Functional analytic settings In this section, we first define the function space and prove some properties which are useful to find the solution of problem { (1.1). For this, we define W α,p (), } the usual fractional Sobolev space W α,p () := u L p (); (u(x) u(y)) x y p n L p ( ) endowed with the +α norm ( u(x) u(y) p ) 1 p u W α,p () = u L p () + dxdy x y n+pα. (2.1) To study fractional Sobolev space in detail, we refer to [9]. Due to the non-localness of the operator L K, we define the linear space as follows: X ={u u : R n R is measurable,u L p () and (u(x) u(y)) p K(x y) L p ()}, where = R 2n \ (C C) and C := R n \. In case of p = 2, the space X was firstly introduced by Servadei and Valdinoci [13]. The space X is a normed linear space endowed with the norm X defined as ( 1 u X = u L p () + u(x) u(y) p p K(x y)dxdy). (2.2) It is easy to check that. X is a norm on X. For this, we first show that if u X = 0, then u = 0a.e.inR n. Indeed, if u X = 0, then u L p () = 0 which implies that u = 0 a.e. in (2.3)

4 548 Sarika Goyal and K Sreenadh and u(x) u(y) p K(x y)dxdy = 0. Thus u(x) = u(y) a.e in means u is a constant in. Hence by (2.3), we have u = 0a.e.inR n. Also triangle inequality follows from the inequality a + b p a + b p 1 a + a + b p 1 b a,b R, p 1 and Hölders inequality. Moreover, other properties of norms are obvious. Now we define X 0 ={u X : u = 0 a.e. inr n \ } with the norm ( 1 u X0 = u(x) u(y) p p K(x y)dxdy) (2.4) which is a reflexive Banach space. We note that, if K(x) = x n+pα, then norms in (2.1) and (2.2) are not the same, because is strictly contained in. Observation. The spaces X and X 0 are non-empty as Cc 2() X 0. Such type of spaces were introduced for p = 2 by Servadei and Valdinoci [13]. For a proof of this, we consider ( ) u(x) u(y) p K(x y)dxdy = +2 u(x) u(y) p K(x y)dxdy R 2n c 2 u(x) u(y) p K(x y)dxdy. R n As u C 2 c,wehave u(x) u(y) 2 u L (R n ), u(x) u(y) u L (R n ) x y. Thus Hence u(x) u(y) 2 u C 1 (R n ) min{1, x y }. R 2n u(x) u(y) p K(x y)dxdy 2 p+1 u p C 1 (R n ) as required. R n m(x y)k(x y)dxdy 2 p+1 u p C 1 (R n ) R n m(z)k(z)dz<, Now we prove some properties of the spaces X and X 0. Proofs of these are easy to extend as in [13] but for completeness, we give the proof. Lemma 2.1. Let K : R n \{0} (0, ) be a function satisfying (ii). Then (1) if u X, then u W α,p () and moreover u W α,p () c(θ) u X ; (2) if u X 0, then u W α,p (R n ) and moreover u W α,p () u W α,p (R n ) c(θ) u X. In both the cases, c(θ) = max{1,θ 1/p }, where θ is given as in (ii).

5 Non-local operator with sign-changing weight functions 549 Proof. (1) Let u X, then by (ii) we have u(x) u(y) p x y n+pα dxdy 1 θ 1 θ Thus ( u W α,p () = u L p () + u(x) u(y) p K(x y)dxdy u(x) u(y) p K(x y)dxdy <. u(x) u(y) p ) 1 p dxdy x y n+pα c(θ) u X. (2) Let u X 0 then u = 0onR n \. So u L 2 (R n ) = u L 2 (). Hence R2n u(x) u(y) p u(x) u(y) p dxdy = dxdy x y n+pα x y n+pα 1 u(x) u(y) p K(x y)dxdy <+, θ as required. Lemma 2.2. Let K : R n \{0} (0, ) be a function satisfying (ii). Then there exists a positive constant c depending on n and α such that for every u X 0, we have u p L p () = u p L p (R n ) c u(x) u(y) p dxdy, R 2n x y n+pα where p = np(n pα) 1 is the fractional critical Sobolev exponent. Proof. Let u X 0, then by Lemma 2.1, u W α,p (R n ). Also we know that W α,p (R n ) L p (R n ) (see [9]). Then we have u p L p () = u p L p (R n ) c u(x) u(y) p dxdy R 2n x y n+pα and hence the result. Lemma 2.3. Let K : R n \{0} (0, ) be a function satisfying (ii). Then there exists some C>1, depending only on n, α, p, θ and such that for any u X 0, u(x) u(y) p K(x y)dxdy u p X C u(x) u(y) p K(x y)dxdy i.e. u p X 0 = u(x) u(y) p K(x y)dxdy (2.5) is a norm on X 0 and is equivalent to the norm on X.

6 550 Sarika Goyal and K Sreenadh Proof. Clearly u p X u(x) u(y) p K(x y)dxdy. Now by Lemma 2.2 and (ii), we get ( ) ) 1/p p u p X ( u = L p () + u(x) u(y) p K(x y)dxdy 2 p 1 u p L p () + 2p 1 u(x) u(y) p K(x y)dxdy 2 p 1 1 p p u p L p () + 2p 1 u(x) u(y) p K(x y)dxdy 2 p 1 c 1 p u(x) u(y) p p dxdy + 2p 1 R 2n x y n+pα u(y) p K(x y)dxdy ( 2 p 1 c 1 p ) p + 1 θ = C u(x) u(y) p K(x y)dxdy, u(x) u(x) u(y) p K(x y)dxdy where C>1asrequired. Now we show that (2.5) is a norm on X 0. For this we need only to show that if u X0 = 0, then u = 0a.e.inR n as other properties of norm are obvious. Indeed, if u X0 = 0, then u(x) u(y) p K(x y)dxdy = 0 which implies that u(x) = u(y) a.e in. Therefore, u is a constant in and hence u = c R a.e in R n. Also by definition of X 0,wehaveu = 0onR n \. Thus u = 0a.e.inR n. Lemma 2.4. The space (X 0,. X0 ) is a reflexive Banach space. Proof. Let {u k } be a Cauchy sequence in X 0. Then by Lemma 2.3, (ii) and Lemma 2.2, {u k } is a Cauchy sequence in L p () and so {u k } has a convergent subsequence. Thus we assume that u k u strongly in L p (). Since u k = 0inR n \, we define u = 0in R n \. Then u k u strongly in L p (R n ) as k. So, there exists a subsequence denoted by u k such that u k u a.e. in R n. Therefore, one can easily show by Fatou s lemma and using the fact that u k is a Cauchy sequence that u X 0. Moreover, using the same fact one can verify that u k u X0 0ask. Hence X 0 is a Banach space. Reflexivity of X 0 follows from the fact that X 0 is a closed subspace of reflexive Banach space W α,p (R n ). Thus we have X 0 ={u X : u = 0 a.e. inr n \ } with the norm ( 1 u X0 = u(x) u(y) p p K(x y)dxdy) (2.6) is a reflexive Banach space. Note that the norm. X0 involves the interaction between and R n \.

7 Non-local operator with sign-changing weight functions 551 Lemma 2.5. Let K : R n \{0} (0, ) be a function satisfying (ii) and let {u k } be a bounded sequence in X 0. Then, there exists u L β (R n ) such that up to a subsequence, u k u in L β (R n ) as k for any β [1,p ). Proof. Let {u k } be bounded in X 0. Then by Lemmas 2.1 and 2.3, {u k } is bounded in W α,p () and in L p (). Also by assumption on and Corollary 7.2 of [4], there exists u L β () such that up to a subsequence u k u strongly in L β () as k for any β [1,p ). Since u k = 0onR n \, we can define u := 0inR n \. Then we get u k u in L β (R n ). 3. Nehari manifold and Fibering map analysis for (1.1) The Euler functional J λ : X 0 R associated to the problem (1.1) is defined as J λ (u) = 1 u(x) u(y) p K(x y)dxdy λ h(x) u q+1 dx p q b(x) u r+1 dx. r + 1 Then J λ is Fréchet differentiable and J λ (u),v = u(x) u(y) p 2 (u(x) u(y))(v(x) v(y))k(x y)dxdy λ h(x) u q 1 uvdx b(x) u r 1 uvdx, which shows that the weak solutions of (1.1) are critical points of the functional J λ. It is easy to see that the energy functional J λ is not bounded below on the space X 0, but we show that it is bounded below on an appropriate subset of X 0 and a minimizer on subsets of this set gives rise to solutions of (1.1). In order to obtain the existence results, we introduce the Nehari manifold N λ := { u X 0 : J λ (u),u =0}, where, denotes the duality between X 0 and its dual space. Therefore u N λ if and only if u(x) u(y) p K(x y)dxdy λ h(x) u q+1 dx b(x) u r+1 dx = 0. (3.1) We note thatn λ contains every solution of (1.1). Now we know that the Nehari manifold is closely related to the behavior of the functions φ u : R + R defined as φ u (t) = J λ (tu). Such maps are called fiber maps and were introduced by Drabek and Pohozaev in [10]. For u X 0,wehave φ u (t) = tp p u p X 0 λtq+1 q + 1 φ u (t) = tp 1 u p X 0 λt q h(x) u q+1 dx tr+1 b(x) u r+1 dx, r + 1 h(x) u q+1 dx t r b(x) u r+1 dx,

8 552 Sarika Goyal and K Sreenadh φ u (t) = (p 1)tp 2 u p X 0 qλt q 1 h(x) u q+1 dx rt r 1 b(x) u r+1 dx. Then it is easy to see that tu N λ if and only if φ u (t) = 0 and in particular, u N λ if and only if φ u (1) = 0. Thus it is natural to splitn λ into three parts corresponding to local minima, local maxima and points of inflection. For this we set N λ ± := { u N λ : φ u (1) 0} ={tu X 0 : φ u (t) = 0, φ u (t) 0}, Nλ 0 := { u N λ : φ u (1) = 0} ={tu X 0 : φ u (t) = 0, φ u (t) = 0}. Now we study the fiber map φ u according to the sign of h(x) u q+1 dx and b(x) u r+1 dx. Case 1. u H 0 B 0. In this case φ u(0) = 0, φ u (t) > 0 for all t>0which implies that φ u is strictly increasing and hence no critical point. Case 2. u H 0 B+. In this case, firstly we define m u : R + R by m u (t) = t p 1 q u p X 0 t r q b(x) u r+1 dx. Clearly, for t>0, tu N λ if and only if t is a solution of m u (t) = λ h(x) u q+1 dx. (3.2) Then we have m u (t) as t and m u (t) = (p 1 q)tp 2 q u p X 0 (r q)t r 1 q b(x) u r+1 dx. Therefore m u (t) > 0 near zero. Since u H, there exists t (u) such that m u (t ) = λ h(x) u q+1 dx. Thus for 0 <t<t, φ u (t) = tq (m u (t) λ h(x) u q+1 dx) > 0 and for t>t, φ u (t) < 0. Hence φ u is increasing on (0,t ) and decreasing on (t, ). Since φ u (t) > 0fort close to 0 and φ u (t) as t, φ u has exactly one critical point t 1 (u), which is a global maximum point. Hence t 1 (u)u N λ. Case 3. u H + B 0. In this case m u(0) = 0, m u (t) > 0 for all t>0which implies that m u is strictly increasing. Since u H +, there exists a unique t 1 = t 1 (u) > 0 such that m u (t 1 ) = λ h(x) u q+1 dx. This implies that φ u (t) is decreasing on (0,t 1 ), and increasing on (t 1, ) and φ u (t 1) = 0. Thus φ u has exactly one critical point t 1 (u), corresponding to global minimum point. Hence t 1 (u)u N λ +. Case 4. u H + B +.Ifλ is sufficiently large, then (3.2) has no solution and so, φ u has no critical point. Moreover in this case φ u is a decreasing function. If λ is sufficiently small then there are exactly two solutions t 1 (u) < t 2 (u) of (3.2) with m u (t 1(u)) > 0 and m u (t 2(u)) < 0. It follows that φ u has exactly two critical points t 1 (u) and t 2 (u) corresponding to a local minimum and a local maximum point respectively such that t 1 (u)u N + λ and t 2(u)u N λ. Moreover, φ u is decreasing in (0,t 1 ), increasing in (t 1,t 2 ) and decreasing in (t 2, ).

9 Non-local operator with sign-changing weight functions 553 In the following lemma, we show that for small λ, φ u takes a positive value for all non zero u. Lemma 3.1. There exists λ 0 > 0 such that λ<λ 0, φ u takes a positive value for all non-zero u X 0. Proof. For u B +, we define F u : R + R, F u (t) := tp u(x) u(y) p K(x y)dxdy tr+1 p r + 1 Then F u (t) = tp 1 and F u attains its maximum value at t = F u (t ) = p 1 r + 1 u(x) u(y) p K(x y)dxdy t r b(x) u r+1 dx. b(x) u r+1 dx ( ) 1 u(x) u(y) p K(x y)dxdy r p+1 b(x) u r+1 dx. Moreover, ) ( ( u(x) u(y) p K(x y)dxdy) r+1 ( b(x) u r+1 dx) p ) 1 r p+1 and u (t ) = (p r 1) ( u(x) u(y) p K(x y)dxdy) r p+1 r 1 ( b(x) u r+1 dx ) p 2 F r p+1 < 0. Let S r+1 be the best constant of Sobolev embedding W α,p (R n ) L r+1 (R n ), then u L r+1 () = u L r+1 (R n ) S r+1 u W α,p (R n ). By Lemmas 2.1 and 2.3, we have u W α,p (R n ) C(θ) u X0 = M u X0. Combining the above two inequalities, we get 1 (MS r+1 ) p(r+1) ( u(x) u(y) p K(x y)dxdy) r+1 ( u r+1 dx) p. Hence F u (t ) r p + 1 p(r + 1) ( 1 b + p (MS r+1 ) p(r+1) ) 1 r p+1. (3.3) We now show that there exists λ 0 > 0 such that φ u (t )>0. Using Sobolev embedding of fractional order spaces, we get t q+1 h(x) u q+1 dx q q + 1 h (MS q+1 ) q+1 ( u(x) u(y) p K(x y)dxdy b(x) u r+1 dx ) q+1 r p+1 u q+1

10 554 Sarika Goyal and K Sreenadh [ = 1 ( q + 1 h (MS q+1 ) q+1 u(x) u(y) p K(x y)dxdy) r+1 ( b(x) u r+1 dx) p = 1 q + 1 h (MS q+1 ) q+1 ] q+1 p(r p+1) ( ) q+1 p(r + 1) p (Fu (t )) q+1 p = cf u (t ) q+1 p, r p + 1 where c is a constant independent of u. Thus by (3.3) and u H +, we get φ u (t ) F u (t ) λcf u (t ) q+1 p =F u (t ) q+1 p (F u (t ) p 1 q p ( where δ = r p+1 p(r+1) we get the required result. Lemma 3.2. Ifλ<λ 0, then inf N λ ) 1 1 b + p (MS r+1 ) p(r+1) J λ (u) > 0. δ q+1 p (δ p 1 q p λc), r p+1.letλ 0 = δ p 1 q p λc) c. Then for every λ<λ 0, Proof. Let u N λ. Then φ u has a positive global maximum at t = 1. If u H + B +, then J λ (u) = φ u (1) = φ u (t ) F u (t ) q+1 p (F u (t ) p 1 q p λc) δ q+1 p (δ p 1 q p λc) where δ is same as in Lemma 3.1. So the infimum of J λ over H + B + is positive. If u B + H 0, then by Case 2, φ u has a unique global maximum at t = 1 and J λ (u) = r + 1 p r q p(r + 1) u p X 0 λ h u q+1 > 0. (3.4) (q + 1)(r + 1) Moreover, inf J λ (u) > 0. Indeed, if this infimum is zero, then from (3.4), u Nλ B+ H + minimizing sequence {u k } converges strongly in X 0 to 0 and 0 Nλ, a contradiction. Since φ u has a unique global minimum if u B 0 H + (by Case 3), we infer u B 0 H + Nλ. Similarly, by Case 1, there is no critical point for φ u if u B 0 H 0. So u B 0 H 0 N λ. Therefore, inf J λ (u) > 0. Nλ Lemma 3.3. If0 <λ<λ 0, then N 0 λ ={0}. Proof. Let 0 u Nλ 0. Then 1 is the critical point of φ u.ifu H 0 B 0, then φ u has no critical point. If u H 0 B+ or u H + B 0 or u H + B +, then φ u has critical point(s) corresponding to local maxima or local minima. So 1 is the critical point corresponding to local minima or local maxima of φ u. That is, either u N λ + or N λ,a contradiction. Hence the proof. In the following lemma we show that the minimizers on subsets of N λ are solutions of (1.1).

11 Non-local operator with sign-changing weight functions 555 Lemma 3.4. Let u be a local minimizer for J λ on subsets N + λ or N λ of N λ such that u/ N 0 λ, then u is a critical point for J λ. Proof. Since u is a minimizer for J λ under the constraint I λ (u) := J λ (u),u =0, by the theory of Lagrange multipliers, there exists μ R such that J λ (u) = μi λ (u). Thus J λ (u),u =μ I λ (u),u =μφ u (1)=0, but u/ N λ 0 and so φ u (1) = 0. Hence μ = 0 completes the proof. Lemma 3.5. J λ is coercive and bounded below on N λ. Proof. OnN λ, J λ (u) = p 1 ) ( u p 1 X r λ q ) h(x) u q+1 dx r + 1 c 1 u p X 0 c 2 u q+1 X 0. Hence J λ is bounded below and coercive on N λ. 4. Existence of solutions In this section, we show the existence of minimizers in N + λ and N λ for λ (0,λ 0). Lemma 4.1. Ifλ<λ 0, then J λ achieves its minimum on N + λ. Proof. Since J λ is bounded below on N λ, and so also on N λ +. Then there exists a minimizing sequence {u k } N λ + such that lim J λ(u k ) = inf k u N λ + J λ (u). As J λ is coercive on N λ, {u k } is a bounded sequence in X 0. Therefore u k u λ weakly in X 0 andu k u λ strongly in L α (R n ) for 1 α<np(n pα) 1. If we choose u X 0 such that h(x) u q+1 dx>0, then there exist t 1 (u) > 0 such that t 1 (u)u N λ + and J λ (t 1 (u)u) < 0 and hence and so J λ (u k ) = inf u N + λ J λ (u) J λ (t 1 (u)u) < 0. Now on N λ, p 1 ) ( u k p 1 X r λ q ) h(x) u k q+1 dx r + 1 λ q ) h(x) u k q+1 dx = r + 1 p 1 ) u k p X r J λ (u k ). Letting k, we get h(x) u λ q+1 dx>0. Next we claim that u k u λ. Suppose this is not true, then u λ (x) u λ (y) p K(x y)dxdy<lim inf u k (x) u k (y) p K(x y)dxdy. k

12 556 Sarika Goyal and K Sreenadh Thus and φ u k (t) = t p 1 u k p X 0 λt q φ u λ (t) = t p 1 u λ p X 0 λt q h(x) u k q+1 dx t r h(x) u λ q+1 dx t r b(x) u k r+1 dx b(x) u λ r+1 dx. It follows that φ u k (t λ (u λ )) > 0 for sufficiently large k. So, we must have t λ > 1. But we have t λ (u λ )u λ N λ + and so J λ (t λ (u λ )u λ )<J λ (u λ )< lim k J λ(u k ) = inf u N + λ J λ (u), which is a contradiction. Hence we must have u k u λ in X 0. Moreover, u λ N λ +, since Nλ 0 =. Hence u λ is a minimizer for J λ on N λ +. Lemma 4.2. Ifλ<λ 0, then J λ achieves its minimum on N λ. Proof. Let u Nλ, then from Lemma 3.2 we have J λ(u) δ 1. So there exists a minimizing sequence {u k } Nλ such that lim J λ(u k ) = inf k u Nλ J λ (u) > 0. Since J λ (u k ) is coercive, {u k } is a bounded sequence in X 0. Therefore u k u λ weakly in X 0 and u k u λ strongly in L β for 1 β< np n pα. Then J λ (u k ) = p 1 ) u k p X q q ) b(x) u k r+1 dx. r + 1 b(x) u k r+1 dx = b(x) u λ r+1 dx, we must Since lim J λ(u k )> 0 and lim k k have b(x) u λ r+1 dx>0. Hence φ uλ has a global maximum at some point t so that t(u λ )u λ Nλ. On the other hand, u k Nλ implies that 1 is a global maximum point for φ uk. That is, φ uk (t) φ uk (1) for every t>0. Thus we have J λ ( t(u λ )u λ ) = 1 p ( t(u λ)) p u λ p X 0 λ( t(u λ)) q+1 h(x) u λ q+1 dx q + 1 ( t(u λ)) r+1 b(x) u λ r+1 dx, r + 1 < lim inf k p ( t(u λ)) p u k p X 0 λ( t(u λ)) q+1 h u k q+1 dx q + 1 ( t(u λ)) r+1 ) b u k r+1 dx, r + 1 lim J λ( t(u λ )u k ) lim J λ(u k ) = inf J λ (u), k k u Nλ

13 Non-local operator with sign-changing weight functions 557 which is a contradiction as t(u λ )u λ Nλ. Hence u k u λ strongly in X 0 and moreover u λ Nλ, since N λ 0 ={0}. Next, we prove the existence of non-negative solutions. For this, we first define some notations. where F + = t 0 f + (x,t) = f + (x,s)ds, { f(x,t), if t 0, 0, if t<0. Let J λ + (u) = u p X 0 F + (x,u)dx. Then the functional J λ + (u) is well defined and it is Fréchet differentiable in u X 0 and for any v X 0, J λ + (u),v = u(x) u(y) p 2 (u(x) u(y))(v(x) v(y))k(x y)dxdy f + (x,u)vdx. (4.1) If f(x,t) := λh(x) t q 1 t + b(x) t r 1 t, then J λ + (u) satisfies all the above lemmas. So for λ (0,λ 0 ), there exists two non-trivial critical points u λ N λ + and v λ Nλ. Now we claim that both u λ and v λ are non-negative inr n.takev = u in (4.1), then 0 = J λ + (u),u = u(x) u(y) p 2 (u(x) u(y))(u (x) u (y))k(x y)dxdy = = u p X 0. u(x) u(y) p 2 ((u (x) u (y)) 2 +2u (x)u + (y))k(x y)dxdy u (x) u (y) p K(x y)dxdy Thus u X0 = 0 and hence u = u +. So by taking u = u λ and u = v λ respectively, we get the non-negative solutions of (1.1). Proof of Theorem 1.1. Lemmas 4.1, 4.2, 3.4 and the above discussion complete the proof. References [1] Adimurthi and Giacomoni J, Multiplicity of positive solutions for a singular and critical elliptic problem in R 2, Commun. Contemp. Math. 8(50) (2006) [2] Ambrosetti A, Brezis H and Cerami G, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122(2) (1994) [3] Ambrosetti A, García Azorero J and Peral I, Multiplicity results for some nonlinear elliptic equations, J. Funct. Anal. 137(1) (1996)

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