Multiplicity of solutions to class of nonlocal elliptic problems with critical exponents

Abstract

In this paper, we establish existence of infinitely many weak solutions for a class of quasilinear stationary Kirchhoff‐type equations, which involves a general variable exponent elliptic operator with critical growth. Precisely, we study the following nonlocal problem: where Ω is a bounded smooth domain of , with homogeneous Dirichlet boundary conditions on ∂Ω, the nonlinearity is a continuous function, is a function of the class is a continuous function, whose properties will be introduced later, λ is a positive parameter and . We assume that , where is the critical Sobolev exponent. We will prove that the problem has infinitely many solutions and also we obtain the asymptotic behavior of the solution as λ → 0+. Furthermore, we emphasize that a difference with previous researches is that the conditions on a(·) are general overall enough to incorporate some interesting differential operators. Our work covers a feature of the Kirchhoff's problems, that is, the fact that the Kirchhoff's function M in zero is different from zero, it also covers a wide class of nonlocal problems for p(x) > 1, for all . The main tool to find critical points of the Euler–Lagrange functional associated with this problem is through a suitable truncation argument, concentration‐compactness principle for variable exponent found in Bonder and Silva (2010), and the genus theory introduced by Krasnoselskii. The result of this paper extends or complements, or else completes recent papers and is new in several directions for the stationary Kirchhoff equations involving the p(x)‐Laplacian type operators.