Note on an Anisotropic P -laplacian Equation in R

In this paper, we study a kind of anisotropic p-Laplacian equations in R n. Nontrivial solutions are obtained using mountain pass theorem given by Ambrosetti-Rabinowitz [1].


Introduction
Consider the following anisotropic p-Laplacian problem in R n Each p i ∈ (1, ∞), a i = a i (x) are measurable real functions satisfying 0 < a 0 < a i (x) ∈ L ∞ (R n ) and we assume that the nonlinear function f satisfies the subcritical growth conditions for some r ∈ (p + − 1, p * − 1).Herein, p + = max(p 1 , . . ., p n ), 1 p = 1 n n i=1 1 p i and p * = np n − p with p < n and g : R n → R is a nonnegative function satisfying . (1.3) For (1.1) we assume that F ∈ C 1 (R n × R), where F (x, s) = s 0 f (x, t) dt and there exists θ > p + such that 0 < θF (x, s) ≤ sf (x, s), ∀x ∈ R n , ∀s ∈ R \ {0}. (1.4) In the isotropic case, we can refer the reader to the works by [3] and [7] where existence and regularity results are obtained.As to anisotropic equations with different orders of derivations in different directions involving critical exponents with unbounded nonlinearities, to our knowledge they were not intensively studied before, in passing, we mention the work [2].Let us mention also that in [4] the authors have studied another class of anisotropic elliptic equations.Via an adaptation of the concentration-compactness lemma of P.-L.Lions to anisotropic operators, they have obtained the existence of multiple nonnegative solutions.Let us point out that in the case of bounded domains, more work in this direction can be found in [5] where the authors proved existence and nonexistence results for some anisotropic quasilinear elliptic equations.The purpose of this paper is to obtain nontrivial weak solutions using mountain pass theorem (see e.g.[1]).
Our main result is the following.
for some r + 1 < p 1 < p * , then problem (1.1) has at least one nontrivial solution.
Let (W, • ) be the anisotropic Sobolev space defined by with the dual (W * , • * ) and the duality pairing •, • .W is a real reflexive Banach under the norm Let us recall that a weak solution of the equation for all v ∈ W .They coincide with the critical points of the C 1 -energy functional corresponding to problem (1.1) for all u ∈ W.
Remark 2.1.Let us remark that (1.3) implies 1 ω + r p * + 1 p * = 1 which guarantee that the integral given in right side of (2.1) is well defined.
To deal with the functional framework we apply the following mountain pass theorem [1].
Let us recall that the functional k=1 ⊂ E for which there exists M > 0 such that: I(u k ) ≤ M and I ′ (u k ) → 0 strongly in I ⋆ as n goes to infinity (called a (PS) sequence), has a convergent subsequence.
Let us define the functional J : , hence we see that J is well defined and continuously Gâteaux differentiable with Proof.Let u k be a sequence in W which converges weakly to u.On one hand, in view of Hölder's inequality and Sobolev embedding, we obtain for all 0 ≤ R ≤ +∞, |g| ω dx = 0.This implies with the fact that u k is a bounded sequence, for any ε, there exists On the other hand, since g ∈ L ωp 1 for all t ∈ R and a.e.
Using the compact imbedding W 1,p (B ε ) into L q (B ε ) for all q ∈ [1, p * ) and the continuity of the Nemytskii's operator N f associated with f for all v ∈ L p 1 (B ε ).Finally, in view of (2.4) and (2.5), we get for all v ∈ W. This completes the proof of Proposition 2.3 and consequently J ′ is compact.
Proof.We have Remark that F (x, tu) ≥ t θ F (x, u) for any t ≥ 1, this is due to the fact that the function F (x,tu) is increasing for all t > 0, we obtain Φ(t Proof.First we claim that the sequence u k is bounded.Indeed, Arguing by contradiction and consider a subsequence still denoted by u k such that ||u k || → ∞.We have in view of (1.4) Passing to limit in (3.2) as k → ∞, we obtain where M is the constant of Palais-Smale sequence.This gives a contradiction since p + < θ.Hence the sequence u k has a subsequence still denoted by u k which converges weakly to some u ∈ W. For any pair integer (n, m) we have EJQTDE, 2010 No. 73, p. 7 By Palais-Smale condition and Proposition 2.3, it is easy to see that the right side of (3.3) approaches zero.Finally, using the following algebraic relation with ρ = r if 1 < r ≤ 2 and ρ = 2 if 2 < r, the monotonicity of the anisotropic operator of problem (1.1) now gives the result.This concludes the proof of Lemma 3.3.
Therefore lemmas 3.1, 3.2 and 3.3 fit into conditions setting of Theorem 2.2 of section 2, this guarantees the existence of at least a nontrivial weak solution for (1.1).Remark 3.4.
1-One can prove that each solution u of problem (1.1) satisfies u ∈ L σ (R n ) with p * ≤ σ ≤ ∞.This regularity result is based on an iterative procedure given in the works [3] where similar results are obtained for the case of degenerate isotropic p-Laplacian problems.2-Let us also mention that since ω + ε < ωp 1 p 1 −1 , with 0 < ε < 1 small enough, the restrictive integrability condition g ∈ L ω+ε (R n ), suffices to proceed with the iterating method and to bound the maximal norm of the solution.
, . . ., p n ).We always have p − ≤ p ≤ p + .The Sobolev conjugate of p is denoted by p ⋆ , i.e., p ⋆ = np n−p .Anisotropic Sobolev spaces were introduced and studied by Nikol'skiȋ 1/p + u) = Let u k be a Palais-Smale sequence of Φ.Then u k possesses a subsequence converging strongly to some u ∈ W.