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

Author(s)

Hurtado, Elard Juarez ^[1] ; Miyagaki, Olimpio H. ^[2] ; Rodrigues, Rodrigo S. ^[2]

Affiliation

^[1] Univ Brasilia, Dept Math, BR-70910900 Brasilia, DF - Brazil

^[2] Univ Fed Sao Carlos, Dept Math, Sao Carlos - Brazil

• Document type: Journal article
• Source: MATHEMATICAL METHODS IN THE APPLIED SCIENCES; DEC 2021.
• Web of Science Citations: 0

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: [-M(A(u))div(a(vertical bar del u vertical bar(p(x)))vertical bar del u vertical bar(p(x)-2) del u) = lambda f(x,u) + vertical bar u vertical bar(s(x)-2)u in Omega, u = 0 on partial derivative Omega, where Omega is a bounded smooth domain of Double-struck capital R-N, with homogeneous Dirichlet boundary conditions on partial differential partial derivative Omega, the nonlinearity f: (Omega) over bar x R -> R is a continuous function, a : R+ -> R+ is a function of the class C-1, M : R-0(+) -> R+ is a continuous function, whose properties will be introduced later, lambda is a positive parameter and p, s is an element of C((Omega) over bar). We assume that e = [x is an element of Omega : s(x) = gamma{*}(x)] not equal (empty set) , where gamma{*}(x) = N gamma(x)/(N - gamma(x)) 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 lambda -> 0(+). Furthermore, we emphasize that a difference with previous researches is that the conditions on a(center dot) 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 x is an element of(Omega) over bar. 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. (AU)

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