Poho\v{z}aev identity and the existence of normalized ground state solutions for variable exponent problems

TL;DR

This paper studies normalized solutions for nonlinear variable exponent problems, introducing a new variational approach, establishing existence and regularity of solutions, and deriving a Pohozaev identity specific to this setting.

Contribution

It introduces the first study of normalized solutions in variable exponent problems, overcoming the lack of scaling invariance with a constrained variational method.

Findings

Existence of a ground state solution for variable exponent problems.

Solutions are locally $C^{1,eta}$ regular.

A Pohozaev-type identity specific to variable exponents is established.

Abstract

In this article, we investigate normalized solutions for nonlinear problems involving variable exponents. To the best of our knowledge, normalized solutions have not been previously studied in this setting, and our results appear to be new. A key difficulty is that the standard scaling argument, which is important in the classical normalized solution approach, is no longer available in the variable exponent setup. To address this, we work with a constrained variational framework and establish the existence of a ground state solution. We further show that these solutions are $C^{1,\alpha}_{loc}(\mathbb{R}^N)$. Finally, we derive a Poho\v zaev-type identity adapted to the variable exponent structure in $\mathbb{R}^N$, which is used to prove that the solution is a ground state.