Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 856932, 18 pages doi:10.1155/2010/856932 Research Article Multiple Positive Solutions for a Class of Concave-Convex Semilinear Elliptic Equations in Unbounded Domains with Sign-Changing Weights Tsing-San Hsu Center for General Education, Chang Gung University, Kwei-Shan, Tao-Yuan 333, Taiwan Correspondence should be addressed to Tsing-San Hsu, tshsu@mail.cgu.edu.tw Received September 2010; Accepted 18 October 2010 Academic Editor: Julio Rossi Copyright q 2010 Tsing-San Hsu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We study the existence and multiplicity of positive solutions for the following Dirichlet equations: −Δu u λa x |u|q−2 u b x |u|p−2 u in Ω, u on ∂Ω, where λ > 0, < q < < p < 2∗ 2∗ ∞ if N 1, , Ω is a smooth unbounded domain in ÊN , a x , b x 2N/ N − if N ≥ 3; 2∗ satisfy suitable conditions, and a x maybe change sign in Ω Introduction and Main Results In this paper, we deal with the existence and multiplicity of positive solutions for the following semilinear elliptic equation: −Δu u λa x |u|q−2 u u>0 u b x |u|p−2 u in Ω, in Ω, Eλa,b on ∂Ω, where λ > 0, < q < < p < 2∗ 2∗ 2N/ N − if N ≥ 3, 2∗ ∞ if N 1, , Ω ⊂ ÊN is an unbounded domain, and a, b are measurable functions and satisfy the following conditions: ∗ A1 a ∈ C Ω ∩ Lq Ω q∗ B1 b ∈ C Ω ∩ L∞ Ω and b p/ p − q with a max{a, 0} / in Ω ≡ max{b, 0} / in Ω ≡ Boundary Value Problems Semilinear elliptic equations with concave-convex nonlinearities in bounded domains are widely studied For example, Ambrosetti et al considered the following equation: −Δu in Ω, λuq−1 up−1 u>0 in Ω, u Eλ on ∂Ω, where λ > 0, < q < < p < 2∗ They proved that there exists λ0 > such that Eλ admits at least two positive solutions for all λ ∈ 0, λ0 and has one positive solution for λ λ0 and no positive solution for λ > λ0 Actually, Adimurthi et al , Damascelli et al , Ouyang and Shi , and Tang proved that there exists λ0 > such that Eλ in the unit ball BN 0; has exactly two positive solutions for λ ∈ 0, λ0 and has exactly one positive solution for λ λ0 and no positive solution exists for λ > λ0 For more general results of Eλ involving sign-changing weights in bounded domains, see Ambrosetti et al , Garcia Azorero et al , Brown and Wu , Brown and Zhang , Cao and Zhong 10 , de Figueiredo et al 11 , and their references However, little has been done for this type of problem in unbounded domains For Ω ÊN , we are only aware of the works 12–15 which studied the existence of solutions for some related concave-convex elliptic problems not involving sign-changing weights Wu in 16 has studied the multiplicity of positive solutions for the following equation involving sign-changing weights: −Δu u fλ x uq−1 u>0 u ∈ H1 gμ x up−1 in ÊN , ÊN in ÊN , Efλ ,gμ , where < q < < p < 2∗ , the parameters λ, μ ≥ He also assumed that fλ x λf x f− x is sign-changing and gμ x ax μb x , where a and b satisfy suitable conditions, and proved Efλ ,gμ has at least four positive solutions When Ω Ω × Ê Ω ⊂ ÊN−1 , N ≥ is an infinite strip domain, Wu in 17 considered ≤ Eλa,b not involving sign-changing weights assuming that / a ∈ L2/ 2−q Ω , ≤ b ∈ C Ω satisfies lim|xN √ ∞ b x , xN in Ω and there exist δ > and < C0 < such that |→ b x , xN ≥ − C0 e−2 θ1 δ|xN | for all x x , xN ∈ Ω, where θ1 is the first eigenvalue of the Dirichlet problem −Δ in Ω The author proved that there exists a positive constant Λ0 such that for λ ∈ 0, Λ0 , Eλa,b possesses at least two positive solutions Miotto and Miyagaki in 18 have studied Eλa,b in Ω Ω × Ê, under the assumption ≡ that a ∈ Lγ / γ −q Ω q < γ ≤ 2∗ with a / and a− is bounded and has a compact support and there exists C0 > such that in Ω and ≤ b ∈ L∞ Ω satisfies lim|xN | → ∞ b x , xN √ x , xN ∈ Ω, where θ1 is the first eigenvalue of the b x , xN ≥ − C0 e−2 θ1 |xN | for all x Dirichlet problem −Δ in Ω It was obtained there existence of Λ0 > such that for λ ∈ 0, Λ0 , Eλa,b possesses at least two positive solutions In a recent work 19 , Hsu and Lin have studied Eλa,b in ÊN under the assumptions A1 - A2 , B1 , and Ωb They proved that there exists a constant Λ0 > such that for Boundary Value Problems λ ∈ 0, q/2 Λ0 , Eλa,b possesses at least two positive solutions The main aim of this paper is to study Eλa,b on the general unbounded domains see the condition Ωb and extend the results of 19 to more general unbounded domains We will apply arguments similar to those used in 20 and prove the existence and multiplicity of positive solutions by using Ekeland’s variational principle 21 Set Λ0 2−q p−q b 2−q / p−2 p−2 p−q a L∞ Lq Sp Ω ∗ p 2−q /2 p−2 q/2 > 0, 1.1 ∗ ∗ where b L∞ supx∈Ω b x , a Lq∗ |a |q dx 1/q , and Sp Ω is the best Sobolev Ω constant for the imbedding of H0 Ω into Lp Ω Now, we state the first main result about the existence of positive solution of Eλa,b Theorem 1.1 Assume that (A1) and (B1) hold If λ ∈ 0, Λ0 , then Eλa,b admits at least one positive solution Associated with Eλa,b , we consider the energy functional Jλa,b in H0 Ω : Jλa,b u u 2 H1 − λ q Ω a x |u|q dx − p Ω b x |u|p dx, 1.2 1/2 where u H u2 dx By Rabinowitz 22, Proposition B.10 , Jλa,b ∈ Ω ∇u| 1 C H0 Ω , Ê It is well known that the solutions of Eλa,b are the critical points of the energy functional Jλa,b in H0 Ω Under the assumptions A1 , B1 , and λ > 0, Eλa,b can be regarded as a perturbation problem of the following semilinear elliptic equation: −Δu u u>0 u in Ω, b x up−1 in Ω, Eb on ∂Ω, where b x ∈ C Ω ∩ L∞ Ω and b x > for all x ∈ Ω We denote by Sb Ω the best constant p which is given by Sb p Ω u inf u∈H0 Ω \{0} Ω H1 p b x |u| dx 2/p 1.3 Boundary Value Problems A typical approach for solving problem of this kind is to use the following Minimax method: αb Ω Γ b inf max J0 γ t , 1.4 γ ∈Γ Ω t∈ 0,1 where ΓΩ γ ∈ C 0, , H0 Ω :γ 0, γ e , 1.5 b J0 e and e / By the Mountain Pass Lemma due to Ambrosetti and Rabinowitz 23 , b we called the nonzero critical point u ∈ H0 Ω of J0 a ground state solution of Eb in Ω if b b αΓ Ω We remark that the ground state solutions of Eb in Ω can also be obtained J0 u by the Nehari minimization problem αb Ω where Mb Ω {u ∈ H0 Ω \ {0} : u nonzero solution of Eb in Ω, αb Ω Γ H1 αb Ω b inf J0 v , 1.6 v∈Mb Ω Ω b x |u|p dx} Note that Mb Ω contains every p−2 b S Ω 2p p p/ p−2 >0 1.7 b see Willem 24 , and if b x ≡ b∞ > is a constant, then J0 and αb Ω replace J0 and α∞ Ω , 0 respectively The existence of ground state solutions of Eb is affected by the shape of the domain Ω and b x that satisfies some suitable conditions and has been the focus of a great deal of research in recent years By the Rellich compactness theorem and the Minimax method, it is easy to obtain a ground state solution for Eb in bounded domains When Ω is an unbounded domain and b x ≡ b∞ , the existence of ground state solutions has been established by several authors under various conditions We mention, in particular, results by Berestycki and Lions 25 , Lien et al 26 , Chen and Wang 27 , and Del Pino and Felmer 28, 29 In 25 , Ω ÊN Actually, Kwong 30 proved that the positive solution of Eb in ÊN is unique In 26 , for Ω is a periodic domain In 26, 27 , the domain Ω is required to satisfy Ω1 Ω Ω1 ∪ Ω2 , where Ω1 , Ω2 are domains in ÊN and Ω1 ∩ Ω2 is bounded; Ω2 α∞ Ω < min{α∞ Ω1 , α∞ Ω2 } 0 y, z , In 28, 29 for ≤ l ≤ N − 1, ÊN Êl × ÊN−l For a point x ∈ ÊN , we have x where y ∈ Êl and z ∈ ÊN −l Let y ∈ Êl , we denote by Ωy ⊂ ÊN−l the projection of Ω onto ÊN−l , that is, Ωy z ∈ ÊN−l : y, z ∈ Ω 1.8 Boundary Value Problems The domain Ω satisfies the following conditions: Ω3 Ω is a smooth subset of ÊN and the projections Ωy are bounded uniformly in y ∈ Êl ; Ω4 there exists a nonempty closed set D ⊂ ÊN−l such that D ⊂ Ωy for all y ∈ Êl ; Ω5 for each δ > 0, there exists R0 > such that Ωy ⊂ z ∈ ÊN−l : dist z, D < δ 1.9 for all |y| ≥ R0 When b x / b∞ and b x ∈ C Ω ∩ L∞ Ω , the existence of ground state solutions ≡ of Eb has been established by the condition b x ≥ b∞ and the existence of ground state solutions of limit equation −Δu b∞ up−1 u u>0 u in Ω, in Ω, Eb∞ on ∂Ω In order to get the second positive solution of Eλa,b , we need some additional assumptions for a x , b x , and Ω We assume the following conditions on a x , b x , and Ω: Ωb b x > for all x ∈ Ω and Eb in Ω has a ground state solution w0 such that b αb Ω J w0 A2 Ω a x |w0 |q dx > where w0 is a positive ground state solution of Eb in Ω Theorem 1.2 Assume that (A1)-(A2), (B1), and (Ωb ) hold If λ ∈ 0, q/2 Λ0 , Eλa,b admits at least two positive solutions Throughout this paper, A1 and B1 will be assumed H0 Ω denotes the standard Sobolev space, whose norm · H is induced by the standard inner product The dual space of 1 H0 Ω will be denoted by H −1 Ω ·, · denote the usual scalar product in H0 Ω We denote s the norm in L Ω by · Ls for ≤ s ≤ ∞ on denotes on → as n → ∞ C, Ci will denote various positive constants, the exact values of which are not important This paper is organized as follows In Section 2, we give some properties of Nehari manifold In Sections and 4, we complete proofs of Theorems 1.1 and 1.2 Nehari Manifold In this section, we will give some properties of Nehari manifold As the energy functional Jλa,b is not bounded below on H0 Ω , it is useful to consider the functional on the Nehari manifold Mλa,b Ω u ∈ H0 Ω \ {0} : Jλa,b u ,u 2.1 Boundary Value Problems Thus, u ∈ Mλa,b Ω if and only if Jλa,b u ,u u H1 −λ a x |u|q dx − Ω Ω b x |u|p dx 2.2 Note that Mλa,b Ω contains every nonzero solution of Eλa,b Moreover, we have the following results Lemma 2.1 The energy functional Jλa,b is coercive and bounded below on Mλa,b Ω Proof If u ∈ Mλa,b Ω , then by A1 , 2.2 , Holder and Sobolev inequalities ă p2 u 2p Ja,b u H1 −λ p−q pq p−2 u 2p H1 −λ p−q Sp Ω pq Ω a x |u|q dx −q/2 a 2.3 Lq ∗ u q H1 2.4 Thus, Jλa,b is coercive and bounded below on Mλa,b Ω Define ψλa,b u Jλa,b u ,u 2.5 Then for u ∈ Mλa,b Ω , ψλa,b u ,u u H1 − λq 2−q u λ p−q H1 Ω Ω a x |u|q dx − p − p−q Ω b x |u|p dx 2.6 p Ω b x |u| dx a x |u|q dx − p − u H1 2.7 Similar to the method used in Tarantello 20 , we split Mλa,b Ω into three parts: Mλa,b Ω u ∈ Mλa,b Ω : ψλa,b u ,u > , M0 Ω λa,b u ∈ Mλa,b Ω : ψλa,b u ,u M− Ω λa,b u ∈ Mλa,b Ω : ψλa,b u ,u < Then, we have the following results , 2.8 Boundary Value Problems ∈ λa,b Lemma 2.2 Assume that uλ is a local minimizer for Jλa,b on Mλa,b Ω and uλ / M0 Ω Then −1 Jλa,b uλ in H Ω Proof Our proof is almost the same as that in Brown and Zhang 9, Theorem 2.3 Binding et al 31 or see Lemma 2.3 We have the following i If u ∈ Mλa,b Ω ∪ M0 Ω , then λa,b ii If u ∈ M− Ω , then λa,b Ω Ω a x |u|q dx > 0; b x |u|p dx > Proof The proof is immediate from 2.6 and 2.7 Moreover, we have the following result Lemma 2.4 If λ ∈ 0, Λ0 , then M0 Ω λa,b ∅ where Λ0 is the same as in 1.1 Proof Suppose the contrary Then there exists λ ∈ 0, Λ0 such that M0 Ω / ∅ Then for λa,b u ∈ M0 Ω by 2.6 and Sobolev inequality, we have λa,b 2−q u p−q H1 Ω b x |u|p dx ≤ b L∞ Sp Ω −p/2 u p H1 2.9 and so u H1 2−q p−q b ≥ 1/ p−2 Sp Ω p/2 p−2 2.10 L∞ Similarly, using 2.7 and Holder and Sobolev inequalities, we have ă u H1 p−q p−2 Ω a x |u|q dx ≤ λ p−q a p−2 Lq ∗ Sp Ω −q/2 u q , H1 2.11 which implies u H1 ≤ λ p−q a p−2 1/ 2−q ∗ Lq Sp Ω −q/2 2−q 2.12 Hence, we must have λ≥ 2−q p−q b 2−q / p−2 L∞ p−2 p−q a ∗ Lq which is a contradiction This completes the proof Sp Ω p 2−q /2 p−2 q/2 Λ0 , 2.13 Boundary Value Problems For each u ∈ H0 Ω with Ω b x |u|p dx > 0, we write 2−q u tmax u p−q Ω 1/ p−2 H1 b x |u|p dx 2.14 > Then the following lemma holds Lemma 2.5 Let λ ∈ 0, Λ0 For each u ∈ H0 Ω with i If Ω a x |u|q dx ≤ 0, then there is a unique t− and Jλa,b t− u Ω b x |u|p dx > 0, we have the following t− u > tmax u such that t− u ∈ M− Ω λa,b supJλa,b tu 2.15 t≥0 ii If Ω a x |u|q dx > 0, then there are unique 0 d0 for some d0 > λa,b In particular, for each λ ∈ 0, q/2 Λ0 , we have αλa,b αλa,b inf u∈M− Ω λa,b Jλa,b u 3.2 Boundary Value Problems Proof i Let u ∈ Mλa,b Ω By 2.6 , 2−q u p−q H1 > Ω b x |u|p dx 3.3 and so 1 − q Jλ u − 1 − q p H1 u p−2 2−q 2pq u H1 Ω 1 − q b x |u|p dx < 1 − q p 2−q p−q u H1 < 3.4 Therefore, αλa,b < ii Let u ∈ M− Ω By 2.6 , λa,b 2−q u p−q H1 < Ω b x |u|p dx 3.5 Moreover, by B1 and Sobolev inequality theorem, Ω b x |u|p dx ≤ b L∞ Sp Ω −p/2 u p H1 3.6 ∀ u ∈ M− Ω λa,b 3.7 This implies u H1 2−q p−q b > 1/ p−2 L∞ Sp Ω p/2 p−2 By 2.4 and 3.7 , we have Jλa,b u p−2 u 2p ≥ u q H1 > 2−q p−q b ⎡ p−2 Sp Ω ×⎣ 2p 2−q H1 −λ p−q Sp Ω pq −q/2 a Lq ∗ q/ p−2 Sp Ω pq/2 p−2 L∞ p 2−q /2 p−2 2−q p−q b 2−q / p−2 L∞ p−q −λ Sp Ω pq ⎤ −q/2 a Lq ∗ ⎦ 3.8 10 Boundary Value Problems Thus, if λ ∈ 0, q/2 Λ0 , then Jλa,b u > d0 ∀ u ∈ M− Ω , λa,b 3.9 for some positive constant d0 This completes the proof We define the Palais-Smale simply by P S conditions in H0 Ω for Jλa,b as follows sequences, P S -values, and P S - Definition 3.2 (i) For c ∈ Ê, a sequence {un } is a P S c -sequence in H0 Ω for Jλa,b if Jλa,b un −1 on strongly in H Ω as n → ∞ c on and Jλa,b un (ii) c ∈ Ê is a P S -value in H0 Ω for Jλa,b if there exists a P S c -sequence in H1 Ω for Jλa,b 1 (iii) Jλa,b satisfies the P S c -condition in H0 Ω if any P S c -sequence {un } in H0 Ω for Jλa,b contains a convergent subsequence Now, we use the Ekeland variational principle 21 to get the following results Proposition 3.3 (i) If λ ∈ 0, Λ0 , then there exists a P S αλa,b -sequence {un } ⊂ Mλa,b Ω in H0 Ω for Jλa,b (ii) If λ ∈ 0, q/2 Λ0 , then there exists a P S α− -sequence {un } ⊂ M− Ω in H0 Ω for λa,b λa,b Jλa,b Proof The proof is almost the same as that in Wu 32, Proposition Now, we establish the existence of a local minimum for Jλa,b on Mλa,b Ω Theorem 3.4 Assume (A1) and (B1) hold If λ ∈ 0, Λ0 , then Jλa,b has a minimizer uλ in Mλa,b Ω and it satisfies the following i Jλa,b uλ αλa,b αλa,b ii uλ is a positive solution of Eλa,b in Ω iii uλ H1 → as λ → Proof By Proposition 3.3 i , there is a minimizing sequence {un } for Jλa,b on Mλa,b Ω such that Jλa,b un αλa,b on , Jλa,b un on in H −1 Ω 3.10 Since Jλ is coercive on Mλa,b Ω see Lemma 2.1 , we get that {un } is bounded in H0 Ω Going if necessary to a subsequence, we can assume that there exists uλ ∈ H0 Ω such that un uλ weakly in H0 Ω , un −→ uλ almost every where in Ω, un −→ uλ strongly in Ls Ω ∀ ≤ s < 2∗ loc 3.11 By A1 , Egorov theorem, and Holder inequality, we have ă a x |un |q dx λ Ω a x |uλ |q dx on as n −→ ∞ 3.12 Boundary Value Problems 11 First, we claim that uλ is a nonzero solution of Eλa,b By 3.10 and 3.11 , it is easy to see that uλ is a solution of Eλa,b From un ∈ Mλa,b Ω and 2.3 , we deduce that λ Ω q p−2 a x |un |q dx un p−q H1 − pq Jλa,b un p−q 3.13 Let n → ∞ in 3.13 ; by 3.10 , 3.12 , and αλa,b < 0, we get λ Ω a x |uλ |q dx ≥ − pq αλa,b > p−q 3.14 Thus, uλ ∈ Mλa,b Ω is a nonzero solution of Eλa,b Now we prove that un → uλ strongly in αλa,b By 3.13 , if u ∈ Mλa,b Ω , then H0 Ω and Jλa,b uλ p−2 u 2p Jλa,b u H1 − p−q λ pq Ω a x |u|q dx 3.15 In order to prove that Jλa,b uλ αλa,b , it suffices to recall that un , uλ ∈ Mλa,b Ω , by 3.15 and by applying Fatou’s lemma to get p−2 uλ 2p αλa,b ≤ Jλa,b uλ ≤ lim inf n→∞ p−2 un 2p ≤ lim inf Jλa,b un n→∞ H1 H1 − − p−q λ pq H1 un Ω Ω a x |uλ |q dx a x |un |q dx 3.16 αλa,b This implies that Jλa,b uλ αλa,b and limn → ∞ un Br´ zis and Lieb, lemma 33 implies that e un p−q λ pq H1 − uλ H1 H1 uλ H1 on Let un − uλ ; then by 3.17 Therefore, un → uλ strongly in H0 Ω Moreover, we have uλ ∈ Mλa,b Ω On the contrary, if − uλ ∈ Mλa,b Ω , then by Lemma 2.5, there are unique t0 and t− such that t0 uλ ∈ Mλa,b Ω and t− uλ ∈ M− Ω In particular, we have t0 < t− Since 0 λa,b d Jλa,b t0 uλ dt 0, d2 Jλa,b t0 uλ > 0, dt2 3.18 12 Boundary Value Problems there exists t0 < t ≤ t− such that Jλa,b t0 uλ < Jλa,b tuλ By Lemma 2.5, Jλa,b t0 uλ < Jλa,b tuλ ≤ Jλa,b t− uλ Jλa,b uλ , 3.19 Jλa,b |uλ | and |uλ | ∈ Mλa,b Ω , by Lemma 2.2 we which is a contradiction Since Jλa,b uλ may assume that uλ is a nonzero nonnegative solution of Eλa,b By Harnack inequality 34 , we deduce that uλ > in Finally, by 2.3 and Holder and Sobolev inequalities, ă uλ and so uλ H1 2−q H1 Thus we conclude that l > Furthermore, by the definition of Sb Ω we obtain p H1 un ≥ Sb Ω p Ω b x |un |p dx 2/p 4.12 Then as n → ∞ we have l H1 lim un n→∞ ≥ Sb Ω l2/p , p 4.13 which implies that l ≥ Sb Ω p p/ p−2 4.14 Hence, from 1.7 and 4.8 – 4.14 we get, c lim Jλa,b un n→∞ lim un 2n→∞ p−2 b 1 − S Ω l≥ p 2p p H1 − λ lim qn→∞ p/ p−2 αb Ω a x |un |q dx − lim pn→∞ Ω b x |un |p dx 4.15 Ω This is a contradiction to c < αb Ω Therefore u0 is a nonzero solution of Eλa,b Lemma 4.3 Assume that (A1)-(A2), (B1), and (Ωb ) hold Let w0 be a positive ground state solution of Eb ; then i supt≥0 Jλa,b tw0 < αb Ω for all λ > 0; ii α− < αb Ω for all λ ∈ 0, Λ0 λa,b Proof i First, we consider the functional Q : H0 Ω → Q u u 2 H1 − p Ω b x |u|p dx Ê defined by ∀ u ∈ H0 Ω 4.16 Boundary Value Problems 15 Then, from 1.3 and 1.7 , we conclude that supQ tw0 t≥0 ⎛ p − 2⎝ 2p ⎞p/ p−2 H1 ⎠ 2/p |w0 |p dx p−2 b S Ω 2p p w0 Ωb x p/ p−2 αb Ω , 4.17 where the following fact has been used: sup t≥0 p−2 A 2p B2/p t2 A− B p p/ p−2 where A, B > 4.18 Using the definitions of Jλa,b , w0 and b x > for all x ∈ Ω, for any λ > 0, we have Jλa,b tw0 −→ −∞ as t −→ ∞ 4.19 From this we know that there exists t0 > such that supJλa,b tw0 t≥0 sup Jλa,b tw0 4.20 0≤t≤t0 By the continuity of Jλa,b tw0 as a function of t ≥ and Jλa,b t1 ∈ 0, t0 such that 0, we can find some sup Jλa,b tw0 < αb Ω 4.21 sup Jλa,b tw0 < αb Ω 4.22 0≤t≤t1 Thus, we only need to show that t1 ≤t≤t0 To this end, by A2 and 4.17 , we have q sup Jλa,b tw0 ≤ supQ tw0 t1 ≤t≤t0 t≥0 t − q Ω a x |w0 |q dx < αb Ω 4.23 Hence i holds ii By A1 , A2 , and the definition of w0 , we have Ω b x |w0 |p dx > 0, Ω a x |w0 |q dx > 4.24 16 Boundary Value Problems Combining this with lemma 2.5 ii , from the definition of α− and part i , for all λ ∈ 0, Λ0 , λa,b we obtain that there exists t0 > such that t0 w0 ∈ M− Ω and λa,b α− ≤ Jλa,b t0 w0 ≤ supJλa,b tw0 < αb Ω λa,b 4.25 t≥0 Therefore, ii holds Now, we establish the existence of a local minimum of Jλ on M− Ω λa,b Theorem 4.4 Assume that (A1)-(A2), (B1), and (Ωb ) hold If λ ∈ 0, q/2 Λ0 , then Jλa,b has a minimizer Uλ in M− Ω and it satisfies the following λa,b i Jλa,b Uλ α− λa,b ii Uλ is a positive solution of Eλa,b in Ω Proof If λ ∈ 0, q/2 Λ0 , then by Theorem 3.1 ii , Proposition 3.3 ii , and Lemma 4.3 ii , there exists a P S α− -sequence {un } ⊂ M− Ω in H0 Ω for Jλa,b with α− ∈ 0, αb Ω λa,b λa,b λa,b From Lemma 4.2, there exist a subsequence still denoted by {un } and a nonzero solution 1 Uλ weakly in H0 Ω Now we prove that un → Uλ Uλ ∈ H0 Ω of Eλa,b such that un − strongly in H0 Ω and Jλa,b Uλ αλa,b By 3.15 , if u ∈ Mλa,b Ω , then Jλa,b u p−2 u 2p H1 p−q λ pq − Ω a x |u|q dx 4.26 First, we prove that Uλ ∈ M− Ω On the contrary, if Uλ ∈ Mλa,b Ω , then by M− Ω being λa,b λa,b closed in H0 Ω , we have Uλ < lim infn → ∞ un From Lemma 2.3 i and b x > for H H all x ∈ Ω, we get Ω a x |Uλ |q dx > 0, Ω b x |Uλ |p dx > 4.27 By Lemma 2.5 ii , there exists a unique t− such that t− Uλ ∈ M− Ω Since un ∈ M− Ω , λ λ λa,b λa,b Jλa,b un ≥ Jλa,b tun for all t ≥ and by 4.26 , we have α− ≤ Jλa,b t− Uλ < lim Jλa,b t− un ≤ lim Jλa,b un λa,b λ λ n→∞ n→∞ α− λa,b 4.28 and this is a contradiction In order to prove that Jλa,b Uλ α− , it suffices to recall that un , λa,b − Uλ ∈ Mλa,b for all n, by 4.26 and applying Fatou’s lemma to get α− ≤ Jλa,b Uλ λa,b ≤ lim inf n→∞ p−2 Uλ 2p p−2 un 2p ≤ lim inf Jλa,b un n→∞ H1 H1 − − p−q λ pq p−q λ pq α− λa,b Ω Ω a x |Uλ |q dx a x |un |q dx 4.29 Boundary Value Problems 17 α− and limn → ∞ un This implies that Jλa,b Uλ λa,b Br´ zis and Lieb, lemma 33 implies that e H1 un H1 − Uλ H1 H1 Uλ H1 on Let un − Uλ ; then by 4.30 Therefore, un → Uλ strongly in H0 Ω Jλa,b |Uλ | and |Uλ | ∈ M− Ω , by Lemma 2.2 we may assume that Since Jλa,b Uλ λa,b Uλ is a 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