Infinitely many solutions to Kirchhoff double phase problems with variable exponents

TL;DR

This paper proves the existence of infinitely many solutions to a class of elliptic equations involving variable exponents and double phase operators, with solutions' norms approaching zero, under mild assumptions.

Contribution

It introduces new existence results for Kirchhoff double phase problems with variable exponents using critical point theory and a priori bounds, expanding the understanding of such nonlinear elliptic equations.

Findings

Existence of infinitely many solutions with norms converging to zero.

Application of abstract critical point theory to variable exponent problems.

Development of a priori bounds for generalized double phase problems.

Abstract

In this work we deal with elliptic equations driven by the variable exponent double phase operator with a Kirchhoff term and a right-hand side that is just locally defined in terms of very mild assumptions. Based on an abstract critical point result of Kajikiya (2005) and recent a priori bounds for generalized double phase problems by the authors (2022), we prove the existence of a sequence of nontrivial solutions whose $L^\infty$-norms converge to zero.