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.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
