Hindawi Publishing Corporation Boundary Value Problems Volume 2009, Article ID 185319, 19 pages doi:10.1155/2009/185319 Research Article A Class of p-q-Laplacian Type Equation with Potentials Eigenvalue Problem in RN Mingzhu Wu1 and Zuodong Yang1, School of Mathematics Science, Institute of Mathematics, Nanjing Normal University, Jiangsu, Nanjing 210097, China College of Zhongbei, Nanjing Normal University, Jiangsu, Nanjing 210046, China Correspondence should be addressed to Zuodong Yang, zdyang jin@263.net Received 20 October 2009; Accepted December 2009 Recommended by Wenming Zou λa x |u|p−2 u The nonlinear elliptic eigenvalue problem −div |∇u|p−2 ∇u − div |∇u|q−2 ∇u q−2 1,p 1,q N N/p RN , b x ∈ λb x |u| u f x, u , u ∈ W ∩ W R , where ≤ q ≤ p < N and a x ∈ L LN/q RN , a x , b x > are studied The key ingredient is a special constrained minimization method Copyright q 2009 M Wu and Z Yang 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 Introduction In this paper, we are interested in finding nontrivial weak solutions for the nonlinear eigenvalue problem − div |∇u|p−2 ∇u − div |∇u|q−2 ∇u a x |u|p−2 u b x |u|q−2 u f x, u , 1.1 u∈W 1,p ∩W 1,q R N , u / 0, where ≤ q ≤ p < N and a x ∈ LN/p RN , b x ∈ LN/q RN , a x , b x inf a x , inf b x / 0, f x, u satisfy the following conditions: A f ∈ C RN × R, R , limt → f x, t /|t|p−1 uniformly in x ∈ RN , B lim|x| → ∞ f x, t 0, and lim|t| → ∞ f x, t /|t|p−1 f t uniformly for t in bounded subsets of R p2 /N > 0, Boundary Value Problems Remark 1.1 We can see if a x ∈ LN/p RN , b x ∈ LN/q RN , then p RN RN where p∗ a x |u| dx < b x |u|q dx < Np/ N − p and q∗ RN RN ax b x N/p 1−p/p∗ RN N/q 1−q/q∗ RN u p∗ uq ∗ p/p∗ q/q∗ < ∞, 1.2 < ∞, Nq/ N − q Problem 1.1 comes, for example, from a general reaction-diffusion system: ut div D u ∇u c x, u , 1.3 where D u |∇u|p−2 |∇u|q−2 This system has a wide range of applications in physics and related sciences such as biophysics, plasma physics, and chemical reaction design In such applications, the function u describes a concentration, the first term on the right-hand side of 1.3 corresponds to the diffusion with a diffusion coefficient D u ; whereas the second one is the reaction and relates to source and loss processes Typically, in chemical and biological applications, the reaction term c x, u is a polynomial of u with variable coefficients When p q 2, problem 1.1 is a normal Schrodinger equation which has been extensively studied, for example, 1–8 The authors used many different methods to study the equation In , the authors established some embedding results of weighted Sobolev spaces of radially symmetric functions which are used to obtain ground state solutions In , the authors studied the equation depending upon the local behavior of V near its global minimum In , the authors used spectral properties of the Schrodinger operator to study nonlinear Schrodinger equations with steep potential well In , the author imposed on functions k and K conditions ensuring that this problem can be written in a variational form We know that W 1,p RN is not a Hilbert space for < p < N, except for p The space W 1,p RN with p / does not satisfy the Lieb lemma e.g., see And RN results in the loss of compactness So there are many difficulties to study equation 1.1 of p q / by the usual methods There seems to be little work on the case p q / for problem 1.1 , to the best of our knowledge In this paper, we overcome these difficulties and study 1.1 of p ≥ q ≥ Recently, when p q, a x b x , and f x, u then the problem is the following eigenvalue problem has been studied by many authors: − div |∇u|p−2 ∇u 1,p u ∈ D0 Ω , V x |u|p−2 u, 1.4 u / 0, where Ω ⊆ RN We can see 10–13 In 13 , Szulkin and Willem generalized several earlier results concerning the existence of an infinite sequence of eigenvalues Boundary Value Problems When p q and a x , b x is constant then the problem is the following quasilinear elliptic equation: − div |∇u|p−2 ∇u u∈ λ|u|p−2 u 1,p W0 Ω, f x, u , in Ω, 1.5 u / 0, where < p < N, N ≥ 3, λ is a parameter, Ω is an unbounded domain in RN There are many results about it we can see 14–18 Because of the unboundedness of the domain, the Sobolev compact embedding does not hold There are many methods to overcome the difficulty In 15 , the authors used the concentration-compactness principle posed by P L Lions and the mountain pass lemma to solve problem 1.5 In 17, 18 , the authors studied the problem in symmetric Sobolev spaces which possess Sobolev compact embedding By the result and a min-max procedure formulated by Bahri and Li 16 , they considered the existence of positive solutions of − div |∇u|p−2 ∇u up−1 q x uα in RN , 1.6 where q x satisfies some conditions We can see if λ is function, then it cannot easily be proved by the above methods When a x , b x is positive constant, He and Li used the mountain pass theorem and concentration-compactness principle to study the following elliptic problem in 19 : − div |∇u|p−2 ∇u − div |∇u|q−2 ∇u m|u|p−2 u n|u|q−2 u f x, u in RN , u ∈ W 1,p ∩ W 1,q RN , 1.7 where m, n > 0, N ≥ 3, and < q < p < N, f x, u /up−1 tends to a positive constant l as u → ∞ The authors prove in this paper that the problem possesses a nontrivial solution even if the nonlinearity f x, t does not satisfy the Ambrosetti-Rabinowitz condition In 20 , Li and Liang used the mountain pass theorem to study the following elliptic problem: − div |∇u|p−2 ∇u − div |∇u|q−2 ∇u |u|p−2 u u ∈ W 1,p ∩ W 1,q RN , |u|q−2 u f x, u in RN , 1.8 where < q < p < N They generalized a similar result for p-Laplacian type equation in 15 It is our purpose in this paper to study the existence of ground state to the problem 1.1 in RN We call any minimizer a ground state for 1.1 We inspired by 9, 16, 21 try to use constrained minimization method to study problem 1.1 Let us point out that although the idea was used before for other problems, the adaptation to the procedure to our problem is not trivial at all But since both p- and q-Laplacian operators are involved, careful analysis is needed A typical difficulty for problem 1.1 in RN is the lack of compactness of the Sobolev imbedding due to the invariance of RN under the translations and rotations However, our method has essential difference with the methods used in 19, 20 In order to obtain the Boundary Value Problems results, we have to overcome two main difficulties; one is that RN results in the loss of compactness; the other is that W 1,p RN is not a Hilbert space for < p < N and it does not satisfy the Lieb lemma, except for p The paper is organized as follows In Section 2, we state some condition and many lemmas which we need in the proof of the main theorem In Section 3, we give the proof of the main result of the paper Preliminaries Let t t F x, t f x, s ds, F t 2.1 f s ds 0 and we define variational functionals I : W 1,p ∩W 1,q RN → R and I ∞ : W 1,p ∩W 1,q RN → R by p I u I ∞ p u RN q |∇u|p dx q p RN RN |∇u| dx |∇u|q dx − RN q RN |∇u| dx − RN F x, u dx, 2.2 F u dx Solutions to problem 1.1 will be found as minimizers of the variational problem Iλ inf I u ; u ∈ W 1,p RN , RN a x |u|p b x |u|q dx λ , λ > Iλ To find a solution of problem Iλ we introduce the limit variational problem ∞ Iλ inf I ∞ u ; u ∈ W 1,p RN , 1,p Lemma 2.1 Let un ⊆ W0 one may assume that un subset Then, lim n→∞ Ω RN a x |u|p b x |u|q dx λ , λ > ∞ Iλ Ω a bounded sequence and p ≥ Going if necessary to a subsequence, 1,p u in W0 Ω , un → u almost everywhere, where Ω ⊆ RN is an open |∇un |p dx ≥ lim n→∞ Ω |∇un − ∇u|p dx lim n→∞ Ω |∇u|p dx 2.3 Boundary Value Problems Proof When p from Brezis-Lieb lemma see 21, Lemma 1.32 lim n→∞ Ω |∇un |2 dx lim n→∞ Ω |∇un − ∇u|2 dx lim we have n→∞ Ω |∇u|2 dx, 2.4 when ≥ p > 2, using the lower semicontinuity of the Lp -norm with respect to the weak u in W 1,p Ω , we deduce convergence and un |∇un |p−2 ∇un , ∇un ≥ |∇u|p−2 ∇u, ∇u o1, lim |∇un − ∇u|p−2 ∇un , ∇un ≥ lim |∇un − ∇u|p−2 ∇un , ∇u n→∞ n→∞ lim |∇un − ∇u|p−2 ∇u, ∇un 2.5 n→∞ lim |∇un − ∇u|p−2 ∇u, ∇u n→∞ Then, lim n→∞ Ω |∇un |p − |∇u|p dx lim n→∞ Ω lim n→∞ Ω lim |∇un |p−2 |∇un |2 − |∇u|2 dx |∇un | n→∞ Ω From un p−2 |∇u| p−2 lim n→∞ Ω |∇un |p−2 − |∇u|p−2 |∇u|2 dx 2.6 |∇un | − |∇u| dx |∇un |p−2 |∇u|2 − |∇u|p−2 |∇un |2 dx u in W 1,p Ω , lim n→∞ Ω |∇un |p−2 |∇u|2 − |∇u|p−2 |∇un |2 dx 2.7 So lim n→∞ Ω |∇un |p − |∇u|p dx lim n→∞ Ω ≥ lim n→∞ Ω |∇un |p−2 |∇u|p−2 |∇un |2 − |∇u|2 dx 2.8 |∇un − ∇u| p−2 2 |∇un | − |∇u| Boundary Value Problems So we have |∇un |p−2 ∇un , ∇un |∇un − ∇u|p−2 ∇u, ∇un |∇un − ∇u|p−2 ∇un , ∇u 2.9 ≥ |∇un − ∇u|p−2 ∇un , ∇un |∇un − ∇u|p−2 ∇u, ∇u |∇u|p−2 ∇u, ∇u o1 Then, |∇un |p−2 ∇un , ∇un ≥ |∇un − ∇u|p−2 ∇un − ∇u, ∇un − ∇u lim n→∞ Ω |∇un |p dx ≥ lim n→∞ Ω |∇un − ∇u|p dx |∇u|p−2 ∇u, ∇u lim n→∞ Ω o1 |∇u|p dx, 2.10 when p > 3, there exists a k ∈ N that < p − k ≤ Then, we only need to prove the following inequality: lim n→∞ Ω |∇un |p − |∇u|p dx ≥ lim n→∞ Ω |∇un − ∇u|p−k |∇un |k − |∇u|k 2.11 The proof of it is similar to the above, so we omit it here So, the lemma is proved Lemma 2.2 Let {un } be a bounded sequence in W 1,p RN such that q lim sup n→∞ y∈RN B y,R un dx 0, p ≤ q < p∗ for some R > Then un → in Ls RN for p < s < p∗ , where p∗ 2.12 Np/ N − p Proof We consider the case N ≥ Let q < s < p∗ and u ∈ W 1,p RN Holder and Sobolev inequalities imply that |u|Ls B y,R ≤ |u|1−λB Lq y,R |u|λ p∗ L B y,R 2.13 λ/p ≤ where λ s − q / p∗ − q C|u|1−λB y,R Lq p∗ /s Choosing λ B y,R 1−λ s B y,R |u|s ≤ Cs |u|Lq |u| p |∇u| p , B y,R p/s, we obtain |u|p B y,R |∇u|p 2.14 Boundary Value Problems Now, covering RN by balls of radius r, in such a way that each point of RN is contained in at most N balls, we find 1−λ s/q s RN |u| ≤ N |u| s C sup q |u|p B y,R y∈RN |∇u|p 2.15 B y,R Under the assumption of the lemma, un → in Ls RN , p < s < p∗ The proof is complete Corollary 2.3 Let {um } be a sequence in W 1,p RN satisfying < ρ |u |p dx and such that RN m 1,p N N in W R Then there exist a sequence {ym } ⊂ R and a function / u ∈ W 1,p RN um u in W 1,p RN such that up to a subsequence um · ym Lemma 2.4 Let f ∈ C RN × R and suppose that lim |s| → ∞ f x, s |s|p ∗ −1 2.16 uniformly in x ∈ RN and f x, s for all x ∈ RN and t ∈ R If um lim m→∞ where F x, u u f RN ≤ C |s|p−1 |s|p ∗ −1 2.17 u0 in W 1,p RN and um → u0 a.e on RN , then F x, um dx − RN F x, u0 dx − RN F x, um − u0 dx 0, 2.18 x, t dt Proof Let R > Applying the mean value theorem we have RN F x, um dx |x|≤R |x|≤R F x, um dx F x, um dx |x|≥R |x|≥R F x, u0 um − u0 dx F x, um − u0 f x, θu0 um − u0 u0 dx, 2.19 Boundary Value Problems where θ depends on x and R and satisfies < θ < We now write RN F x, um dx − ≤ RN F x, u0 dx − RN F x, um − u0 dx F x, um − F x, u0 dx |x|≤R |x|≤R F x, um − u0 dx |x|≥R |x|≥R F x, u0 dx f x, θu0 2.20 um − u0 u0 dx For each fixed R > lim m → ∞ |x|≤R F x, um − F x, u0 dx 0, 2.21 lim m → ∞ |x|≤R F x, um − u0 dx Applying 2.20 and the Holder inequality we get that |x|≥R um − u0 u0 dx f x, θu0 ≤C |x|≥R um − u0 |p−1 |u0 | |θu0 |θu0 um − u0 |p 1/p ≤C |x|≥R |u0 |p |x|≥R |θu0 p∗ |x|≥R −1 |u0 | dx 2.22 p−1 /p um − u0 |p 1/p∗ C ∗ |u0 | |x|≥R |θu0 um − u0 | p∗ p∗ −1 /p∗ Since {um } is bounded in W 1,p RN we see that lim R→∞ |x|≥R f x, θu0 um − u0 u0 dx 2.23 The result follows from 2.21 and 2.23 ∞ Lemma 2.5 Functions Iλ and Iλ are continuous on 0, ∞ and minimizing sequences for problems ∞ 1,p Iλ and Iλ are bounded in W RN Boundary Value Problems Proof From condition A , we observe that for each ε > there exists Cε > such that F u , |F x, u | ≤ ε RN |u|p dx ε RN p2 /N |u|p dx Cε RN |u|α dx, 2.24 where p < α < p p2 /N and ε > By the Holder and Sobolev inequalities we have RN p2 /N |u|p dx RN ≤ |u|p p RN ≤ S−1 p |u| ∗ −p−p2 /N / p∗ −p p RN p∗ p2 /N / p∗ −p p∗ −p−p2 /N / p∗ −p RN p/N |u|p RN |u| dx p∗ p2 /N/ p∗ −p 2.25 |∇u|p dx, p where |u|p∗ ≤ S−1 |∇u|p Similarly we have RN |u|α dx RN ≤ |u|p RN ≤ S−p ∗ p∗ −α / p∗ −p |u|p dx p∗ α−p / p∗ −p p∗ −α / p∗ −p RN α−p / p∗ −p ∗ RN α−p /p p∗ −p dx |u|p dx |u|p dx 2.26 p∗ −α / p∗ −p RN |∇u|p dx p∗ α−p /p p∗ −p Consequently by the Young inequality we have RN |u|α dx ≤ η RN |∇u|p dx K η RN |u|α dx p p∗ −α / p2 p∗ −p2 −p∗ α 2.27 for η > 0, where K η > is a constant a x |u|p Because u ∈ W 1,p ∩ W 1,q RN so we can by Sobolev embedding and λ RN b x |u|q dx letting λ |u|p dx < ∞, we derive the following estimates for I u and I ∞ u : RN I u , I∞ u ≥ − εS−1 λp/N − Cε η p q q RN RN |∇u|p dx |∇u| dx − ελ − K η Cε λ 2.28 p p∗ −α / p2 p∗ −p2 −p∗ α 10 Boundary Value Problems Choosing ε > and η > so that − εS−1 λp/N − Cε η > 0, p 2.29 ∞ we see that Iλ and Iλ are finite and moreover minimizing sequences for problems Iλ and ∞ ∞ Iλ are bounded It is easy to check that Iλ and Iλ are continuous on 0, ∞ ∞ ∞ We observe that Iμ ≤ for all μ > Indeed, let u ∈ C0 RN and RN a x p u x/σ σ N/q dx RN b x u x/σ σ N/q q dx 2.30 μ, then for each σ > we have ∞ Iμ ≤ pσ p p/q−1 N RN qσ q |∇u|p dx RN |∇u|q dx − σ N RN F σ −N/q u dx −→ 2.31 as σ → ∞ ∞ ∞ Lemma 2.6 Suppose that Iλ < for some λ > 0, then Iμ /μ is nonincreasing on 0, ∞ and ∞ ∗ Moreover there exists λ ≤ λ such that limμ → Iμ /μ ∞ Iμ μ > ∞ Iλ λ for μ ∈ 0, λ∗ 2.32 Proof We observe that inf I∞ u a x |u|p b x |u|q dx RN 2.33 I ∞ u x/σ 1/N inf RN p a x/σ 1/N u x/σ 1/N dx b x/σ 1/N u x/σ 1/N q dx So if RN a x |u|p b x |u|q dx k and RN a x/σ 1/N |u x/σ 1/N |p dx b x/σ 1/N |u x/ ∞ I ∞ u x/σ 1/N Ik σ 1/N |q dx k then I ∞ u x We have that if σ > and α > with RN a x |u|p b x |u|q dx α, then RN a x σ 1/N u x σ 1/N p dx b x σ 1/N u x σ 1/N q dx σα, I∞ u x σ 1/N ∞ Iσα 2.34 Boundary Value Problems 11 Consequently, for all α1 > and α2 > we have ∞ Iα1 α1 α2 p inf RN N−p /N RN a x |u|p b x |u|q dx N−q /N α1 α2 q |∇u|p dx RN |∇u|q dx − α1 α2 RN F u dx; 2.35 α2 If < α1 < α2 , then for each ε > there exists u ∈ W 1,p ∩ W 1,q RN with b x |u|q dx α2 such that ∞ Iα1 ε> p α1 ≥ α2 N−p /N α1 α2 p RN |∇u|p dx q p RN |∇u| dx N−q /N α1 α2 q RN q RN |∇u| dx − RN F u dx |∇u|q dx − α1 α2 RN RN a x |u|p F u dx 2.36 α1 ∞ ≥ Iα2 α2 This inequality yields ∞ ∞ Iα1 Iα > α1 α2 2.37 ∞ Since Iμ ≤ for all μ > 0, we see that lim μ→0 ∞ Iμ μ c ≤ 2.38 We claim that c Indeed, it follows from 2.36 and from the estimate obtained in the Lemma 2.1 that for every < μ < λ there exists an uμ ∈ W 1,p ∩ W 1,q RN , with RN a x |uμ |p b x |uμ |q dx λ such that μ2 > μ p λ ≥ μ λ p ≥ ∞ Iμ μ C1 λ λ N−p /N p RN ∇uμ dx q p RN ∇uμ dx p RN ∇uμ dx μ q λ N−q /N q RN ∇uμ dx − C2 λ q RN RN ∇uμ dx − RN ∇uμ dx − C3 λ F uμ dx 2.39 F uμ dx q RN μ λ , where C1 λ > 0, C2 λ > 0, and C3 λ > are constants Hence μ2 ≥ μ C1 λ λ p RN ∇uμ dx C2 λ q RN ∇uμ dx − C3 λ , 2.40 12 Boundary Value Problems that is, RN |∇uμ |p dx ≤ C4 λ , RN |∇uμ |p dx ≤ C5 λ for some constant C4 λ , C5 λ independent of μ We see that there exists ε0 > and a sequence μn → such that p RN If RN q ∇uμn dx ≥ ε0 , RN ∇uμn dx ≥ ε0 > 2.41 |∇uμn |p dx ≥ ε0 then RN |∇uμn |q dx ≥ η ≥ Then, using the fact that RN F uμn dx ≤ C for some constant C > 0, we get ∞ Iμn μn − N−p /N −p/N λ μn ε0 p μn ≥ − N−q /N −q/N C λ μn η − −→ ∞ q λ ∞ as μn → and this contradicts the fact that limμ → Iμ /μ p lim μ → RN when RN ∇uμ dx c ≤ Therefore ∞ lim Iμ 0, 2.42 μ→0 0, 2.43 |∇uμn |p dx ≥ ε0 > we can use the same method to obtain that limμ → RN |∇uμ |q dx So lim p μ → RN this implies that limμ → ∇uμ dx RN F uμ dx ∞ Iμ ∞ This shows that limμ → Iμ /μ which obtain 2.32 ∇uμ dx 0, ∞ lim Iμ μ→0 0, 2.44 and consequently μ≥− μ q lim μ → RN λ RN F uμ dx −→ ∞ Finally, we observe that limμ → Iμ /μ 2.45 ∞ > Iλ /λ Proof of Main Theorems ∞ Theorem 3.1 Suppose that Iλ < for some λ > 0, then there exists < α0 ≤ λ such that problem ∞ ∞ Iα0 has a minimizer Moreover each minimizing sequence for Iα0 up to a translation is relatively 1,p 1,q N compact in W ∩ W R Proof According to Lemma 2.6 the set α1 ; ∞ I∞ Iα > λ for each α ∈ 0, α1 α λ 3.1 Boundary Value Problems 13 is nonempty We define sup α1 ; α0 ∞ I∞ Iα > λ for each α ∈ 0, α1 α λ 3.2 ∞ It follows from the continuity of Iλ that < α0 ≤ λ , ∞ Iα0 ∞ Iα α0 ∞ I < 0, λ λ α ∞ > Iλ , λ 3.3 for all < α < α0 This yields ∞ Iα0 α0 ∞ I λ λ α0 − α ∞ Iλ λ α ∞ ∞ I < Iα0 −α λ λ ∞ Iα 3.4 for each α ∈ 0, α0 ∞ Let {um } ⊂ W 1,p ∩ W 1,q RN be a minimizing sequence for Iα0 Since {um } is bounded 1,p 1,q N u in W ∩ W R , um → u a.e on RN First we consider the we may assume that um case u ≡ In this case by Lemma 2.2 um → for q < α < p∗ or Corollary there exists a v / in W 1,p ∩ W 1,q RN sequence {um } ⊂ RN such that um · ym In the first case limm → ∞ RN F um dx and consequently ∞ Iα0 lim I ∞ um m→∞ lim m→∞ p RN q |∇um |p dx RN |∇um |q dx − RN F um dx ≥ 0, 3.5 v / in W 1,p ∩ W 1,q RN holds and letting vm x which is impossible Hence um · ym um x ym from Brezis-Lieb lemma see 21, Lemma 1.32 we have RN a x |um |p b x |um |q dx RN RN a x ym |vm |p a x ym |v|p RN a x b x b x b x ym |vm |q dx ym |v|q dx 3.6 ym |vm − v|p ym |vm − v|q dx o1 We now show that RN a x ym |v|p b x ym |v|q dx α0 3.7 14 Boundary Value Problems In the contrary case from Lemma 2.1 we have 0< RN a x ym |v|p ym |v|q dx < α0 b x 3.8 By 3.21 we have lim m → ∞ RN ym |vm − v|p a x ym |vm − v|q dx −→ α0 − λ, b x 3.9 λ RN p ym |v| a x q ym |v| dx b x On the other hand, by Lemmas 2.1 and 2.4 we have RN RN |∇vm |p F vm dx |∇vm |q dx ≥ RN RN |∇v|p F v dx RN |∇v|q dx F vm − v dx o1, |∇ vm − v |p |∇ vm − v |q dx RN o1 , 3.10 and this implies that ∞ Iα0 ≥ I ∞ v I ∞ vm − v ∞ o ≥ Iλ ∞ Iα0 −λ0 o1 3.11 ∞ ∞ ∞ Letting m → ∞ we get Iα0 ≥ Iλ Iα0 −λ0 which contradicts 3.4 Therefore RN a x ym |v|p b x ym |v|q dx α0 It then follows from 3.6 that vm → v in Lp ∩ Lq RN By the Gagliardo∞ Nirenberg inequality vm → v in Ls RN , q ≤ s < ∞ Obviously this implies that Iα0 I ∞ v ∞ p q I v ·−ym and RN a x |v ·−ym | b x |v ·−ym | dx α0 To complete the proof we show that vm → v in W 1,p ∩ W 1,q RN Indeed, we have p ∞ Iα0 ≥ p − RN RN RN q |∇vm |p dx |∇v|p dx F v dx q RN RN RN |∇vm |q dx − |∇v|q dx p F v − F vm RN RN F vm dx o1 |∇vm − v|p dx dx q RN |∇vm − v|q dx 3.12 o1 Since limm → ∞ RN F v −F vm dx 0, we deduce from 3.12 that ∇vm → ∇v in Lp ∩Lq RN and hence vm → v in W 1,p ∩ W 1,q RN ∞ If u / 0, we repeat the previous argument to show that Iα0 is attained Theorem 3.2 Suppose that F x, t ≥ F t on RN × R and that Iλ < for some λ > 0, then the infimum Iλ0 is attained for some < λ0 ≤ λ Boundary Value Problems 15 ∞ Proof Since F x, t ≥ F t on RN × R we have Iμ ≤ Iμ for μ ≥ We distinguish two cases: i ∞ ∞ Iλ Iλ < 0, ii Iλ < Iλ Case i By Theorem 3.1 there exists λ0 ∈ 0, λ such that ∞ Iλ0 I∞ u , RN a x |u|p b x |u|q dx λ0 for some u ∈ W 1,p ∩ W 1,q RN 3.13 Thus p Iλ ≤ I u ≤ p RN p RN |∇u|p dx |∇u| dx q q RN |∇u|q dx − q RN |∇u| dx − RN RN F x, u dx F u dx 3.14 I ∞ u ∞ Iλ0 ∞ If Iλ0 Iλ0 , then I also attains its infimum Iλ0 at u Therefore it remains to consider the case ∞ Iλ0 < Iλ0 Consequently we need to prove the following claim ∞ If Iλ < Iλ for some λ > 0, then there exists α0 ∈ 0, λ such that problem Iα0 has a solution This obviously completes the proof of case i and also provides the proof of case ii ∞ ∞ If By virtue of Lemma 2.5, Iβ Iλ−β is continuous for β ∈ 0, λ and also I0 I0 ∞ Iλ < Iλ for some λ > 0, then there exists γ > such that Iλ < I β ∞ Iλ−β 3.15 for all β ∈ 0, γ Let α0 sup γ; Iλ < Iβ ∞ Iλ−β , for ≤ β < γ 3.16 Then we have Iλ Iα0 Iλ < Iα ∞ Iλ−α0 , ∞ Iλ−α 3.17 for ≤ α < α0 This implies that ∞ Iλ < Iλ ≤ 0, 3.18 ∞ ∞ ∞ Iα0 < Iλ − Iλ−α0 ≤ Iα0 ≤ 0, 3.19 Iα0 ∞ Iλ−α0 and hence we show that Iα0 is attained by a u ∈ W 1,p ∩ W 1,q RN and every minimizing sequence for Iα0 is relatively compact in W 1,p ∩ W 1,q RN Let {um } be a minimizing sequence for Iα0 Since 16 Boundary Value Problems u in W 1,p ∩ W 1,q RN , um {um } is bounded, we may assume that um N Arguing indirectly we assume that u ≡ on R We see that |F x, um |dx lim m → ∞ B 0,R F um dx lim m → ∞ B 0,R u a.e on RN 3.20 for each R > We now write I um p I ∞ RN |∇um |p dx um q RN |∇um |q dx − F um − F x, um RN RN F x, um dx 3.21 dx We show that lim m → ∞ RN F um − F x, um dx 3.22 Towards this end we write RN F um − F x, um dx ≤ |F x, um |dx F um dx B 0,R 3.23 B 0,R |x|≥R,|um |≤δ |x|≥R,δ≤|um |≤1/δ F um − F x, um |x|≥R,|um |>1/δ We now define the following quantities: δ F t − F x, t sup |t|p 0 for some ζ ∈ R, then problem Iλ has a minimizer for some λ > Proof The condition F ζ > for some ζ > implies that u x/σ , σ > 0, we have W 1,p ∩ W 1,q RN Letting v x I∞ v σN pσ p RN |∇u|p dx qσ q RN |∇u|q dx − RN RN F u x dx > for some u ∈ F u x dx sufficiently large Hence there exists λ > such that Iλ ≤ Iλ < and the result follows from Theorem 3.2 Remark 3.4 It is a standard argument that minimizers of Iμ correspond to weak solutions of problem 1.1 with λ appearing as a Lagrange multiplier Such a λ is then called the principal eigenvalue for problem 1.1 Remark 3.5 If a ∈ LN/p RN , b ∈ LN/q RN , a, b < 0, we can use the similar method to study inf{I u ; u ∈ W 1,p ∩ W 1,q RN , RN a− x |u|p b− x |u|q dx λ}, λ 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