Abstract
The main purpose of this paper is to establish the existence and multiplicity of nontrivial solutions for a quasilinear Neumann problem with critical exponent. It is shown, by the methods of the Lions concentration-compactness principle and the mountain pass lemma, that under certain conditions, the existence of nontrivial solutions are obtained.
Highlights
In this paper, we consider the following quasilinear elliptic problem with critical Sobolev exponent:⎧ ⎨–εp pu + V (x)|u|p–2u = Q(x)|u|p∗–2u + P(x)|u|q–2u, ⎩|∇ u|p–2 ∂u ∂ν = 0, x∈, x∈∂, (1.1)where ⊂ RN is a bounded domain with smooth boundary, pu = div(|∇u|p–2∇u), ε > 0, < p
Comte and Knaap [8] proved that there exists a nontrivial solution of problem (1.2) by variational method if Q(x) = 1 and λ = –μ
From Lemma 2.2 and the mountain pass theorem, we know that there exists at least one nontrivial solution to Problem (1.5)
Summary
Comte and Knaap [8] proved that there exists a nontrivial solution of problem (1.2) by variational method if Q(x) = 1 and λ = –μ. As for quasilinear elliptic problems with critical Sobolev exponent, the existence and multiplicity of solutions have been studied extensively. Motivated by the results of the above papers, we discuss the existence of nontrivial nonnegative solutions to Problem (1.1) by a variational method.
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