On a class of logarithmic Schrödinger equations via perturbation method
Chen Huang, Zhipeng Yang
TL;DR
The paper addresses the existence and multiplicity of weak solutions to the logarithmic Schrödinger equation $-\Delta u+V(x)u=u\log u^{2}$ in $\mathbb{R}^N$, under a potential $V$ with $\inf V>0$ and $V(x)\to\infty$ as $|x|\to\infty$. It introduces a novel perturbative variational approach by adding a small $p$-Laplacian term, defining the $C^1$ functional $I_{\lambda}$ on the intersection space $X=W^{1,p}(\mathbb{R}^N)\cap H^1_V(\mathbb{R}^N)$, and proving the Palais–Smale condition and mountain-pass geometry uniformly in $\lambda$. By establishing $\lambda$-independent bounds and passing to the limit $\lambda\to0^{+}$, the authors obtain a nontrivial weak solution of the original problem. They further develop a symmetric, genus-based minimax framework to produce infinitely many weak solutions, with their energies tending to infinity as the index grows, again by passing to the limit $\lambda\to0^{+}$. Overall, the work provides a robust, adaptable perturbative method for logarithmic-type nonlinearities that remains in the natural Hilbert space in the limit.
Abstract
In this paper, we consider the following logarithmic Schrödinger equation \[ -Δu + V(x)u = u \log u^{2} \quad \text{in }\ \mathbb{R}^{N}. \] Assuming that \(V(x)\in C(\mathbb{R}^{N})\) and \(V(x)\to+\infty\) as \(|x|\to\infty\), we develop a new perturbative variational approach to overcome the lack of \(C^{1}\)-smoothness of the associated functional and prove the existence and multiplicity of nontrivial weak solutions.