Existence of solutions for quasilinear p-biharmonic equations with critical Sobolev-Hardy terms

Abstract

In order to study a class of quasilinear p-biharmonic equations with Hardy terms and multi-critical Sobolev-Hardy exponents, the existence theorem of the solutions to the above problem is established by means of the Ekeland variational principle. Firstly, to guarantee the variational functional is bounded from below, it is restricted on a set Mη (usually called Nehari manifold). Secondly, the set Mη is divided into three parts M+η, M0η and M-η by using fibering maps. Moreover, the existence of minimum in M+η and M-η is proved by studying the properties of the two subsets. Finally, by using implicit function theorem, it is found that there exists a minimizing sequence {un} making the (PS)c conditions hold when the parameters satisfy certain conditions. Therefore, the existence of the solutions to the problem is proved. The conclusions provide a theoretical basis for judging the structure and properties of the solutions.

Keywords

• nonlinear functional analysis
• critical Sobolev-Hardy terms
• quasilinear p-biharmonic equations
• Ekeland's variational principle
• existence of the solution