Existence and Asymptotic Behaviour for a Kirchhoff Type Equation With Variable Critical Growth Exponent

Author(s)

Hurtado, Elard Juarez ; Miyagaki, Olimpio Hiroshi ; Rodrigues, Rodrigo da Silva

• Document type: Journal article
• Source: Milan Journal of Mathematics; v. 85, n. 1, p. 71-102, JUN 2017.
• Web of Science Citations: 2

Abstract

In this paper, we establish existence and asymptotic behaviour of nontrivial weak solution of a class of quasilinear stationary Kirchhoff type equations involving the variable exponent spaces with critical growth, namely where is a bounded smooth domain of , with homogeneous Dirichlet boundary conditions on , the nonlinearities is a continuous function, is a function of the class , is a continuous function whose properties will be introduced later, and is a positive parameter. We assume that , where is the critical Sobolev exponent. We show that the problem has at least one solution, which it converges to zero, in the norm of the space X as . Our result extends, complement and complete in several ways some of the recent works. We want to emphasize that a difference of some previous research is that the conditions on are general enough to incorporate some differential operators of great interest. In particular, we can cover a general class of nonlocal operators for , for all . The main tools used are the Mountain Pass Theorem without the Palais-Smale condition given in {[}11] and the Concentration Compactness Principle for variable exponent found in {[}9]. We remark that it will be necessary a suitable truncation argument in the Euler- Lagrange operator associated. (AU)

• FAPESP's process: 15/11912-6 - Solutions for elliptic problems
• Grantee: Rodrigo da Silva Rodrigues
• Support Opportunities: Regular Research Grants