Multi-bump solutions for logarithmic Schr\"odinger equations

TL;DR

This paper proves the existence of infinitely many multi-bump solutions for spatially periodic logarithmic Schrödinger equations using a periodic approximation approach.

Contribution

It introduces a novel method employing large-period problems to establish multiple solutions for the logarithmic Schrödinger equation.

Findings

Existence of infinitely many multi-bump solutions.

Solutions are distinct under c^N-action.

Method applicable to periodic Schrd6dinger equations.

Abstract

We study spatially periodic logarithmic Schr\"odinger equations: \begin{equation}\tag{LS} -\Delta u + V(x)u=Q(x)u\log u^2, \quad u>0\quad \text{in}\ \mathbb{R}^N, \end{equation} where $N\geq 1$ and $V(x)$, $Q(x)$ are spatially $1$-periodic functions of class $C^1$. We take an approach using spatially $2L$-periodic problems ($L\gg 1$) and we show the existence of infinitely many multi-bump solutions of $(LS)$ which are distinct under $\mathbb{Z}^N$-action.