Single‐peak solutions for a subcritical Schrödinger equation with non‐power nonlinearity

Zhongyuan Liu, Ziying Liu and Wenhuan Xu

Mathematische Nachrichten, 2023, vol. 296, issue 8, 3459-3480

Abstract: This paper deals with the following slightly subcritical Schrödinger equation: −Δu+V(x)u=fε(u),u>0inRN,$$\begin{equation*} \hspace*{80pt}-\Delta u+V(x)u=f_\varepsilon (u),\quad u>0\quad \text{in}\quad \mathbb {R}^N, \end{equation*}$$where V(x)$V(x)$ is a nonnegative smooth function, fε(u)=up[ln(e+u)]ε$f_\varepsilon (u)=\frac{u^p}{[\ln (e+u)]^\varepsilon }$, p=N+2N−2$p=\frac{N+2}{N-2}$, ε>0$\varepsilon >0$, N≥7$N\ge 7$. Most of the previous works for the Schrödinger equations were mainly investigated for power‐type nonlinearity. In this paper, we will study the case when the nonlinearity fε(u)$f_\varepsilon (u)$ is a non‐power nonlinearity. We show that, for ε small enough, there exists a family of single‐peak solutions concentrating at the positive stable critical point of the potential V(x)$V(x)$.

Date: 2023
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