Infinitely many sign-changing solutions for logarithmic Schrödinger equations via an $L^p$-perturbation approach

We study the logarithmic Schrödinger equation $$-\mathrm{\Delta }u+V(x)u=u\log u^2,x\in {\mathbb{R}}^N,N\ge 3.$$ Since the logarithmic energy is not $C^1$ on the natural space $H_V^1({\mathbb{R}}^N)$, direct invariant-set minimax arguments for sign-changing solutions are not available. We introduce an $L^p$-regularization perturbation, which restores a $C^1$ variational structure while preserving the logarithmic nonlinearity, and prove via a limiting argument that the original equation admits infinitely many sign-changing weak solutions.