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

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.