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arXiv 2111.03097math.AP
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具有间断非线性项和临界增长项的薛定谔方程解的存在性

Existence of solution for Schrödinger equation with discontinuous nonlinearity and critical Growth

Geovany Fernandes Patricio

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摘要：

本文讨论了如下问题\begin{equation} \left\{\begin{aligned} -Δu + V(x)u & = γH_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}). \end{aligned} \right. \end{equation}非平凡解的存在性，其中$N\geq 3$，$γ\geq 0$，$H_{e}:\mathbb{R} \to \mathbb{R}$表示Heaviside函数，$a \geq 0$，$2<q<2^*$和$V:\mathbb{R^{N}} \to \mathbb{R}$是$\mathbb{Z}^{N}$ -周期的，且$β= 0$不属于$-Δ+V$的谱.

英文摘要：

This paper concerns with the existence of nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -Δu + V(x)u & = γH_{e}(|u|-a)|u|^{q-2}u+|u|^{2^{*}-2}u\;\;\mbox{ in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}). \end{aligned} \right. \end{equation} where, $N\geq 3$, $γ\geq 0$, $H_{e}:\mathbb{R} \to \mathbb{R}$ denotes the Heaviside function, $a \geq 0$, $2<q<2^*$ and $V:\mathbb{R^{N}} \to \mathbb{R}$ is $\mathbb{Z}^{N}$-periodic with $β= 0$ does not belong to the spectrum of $-Δ+V$.

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