Lushun Wang | Search arXiv

arXiv2023

Normalized ground states solutions for nonautonomous Choquard equations

In this paper, we study normalized ground state solutions for the following nonautonomous Choquard equation: $$-Δu-λu=\left(\frac{1}{|x|^μ}\ast A|u|^{p}\right)A|u|^{p-2}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=c,\quad u\in H^1(\mathbb{R}^N,\mathbb{R}),$$ where $c>0$, $0<μ<N$, $λ\in\mathbb{R}$, $A\in C^1(\mathbb{R}^N,\mathbb{R})$. For $p\in(2_{*,μ}, \bar{p})$, we prove that the Choquard equation possesses ground state normalized solutions, and the set of ground states is orbitally stable. For $p\in (\bar{p},2^*_μ)$, we find a normalized solution, which is not a global minimizer. $2^*_μ$ and $2_{*,μ}$ are the upper and lower critical exponents due to the Hardy-Littlewood-Sobolev inequality, respectively. $\bar{p}$ is $L^2-$critical exponent. Our results generalize and extend some related results.

math.AP↗

arXiv2017

Solutions for biharmonic equations with steep potential wells

In this paper, we are concerned with the existence of least energy solutions for the following biharmonic equations: $$Δ^2 u+(λV(x)-δ)u=|u|^{p-2}u \quad in\quad \mathbb{R}^N$$ where $N\geq 5, 2 0$ is a parameter, $V(x)$ is a nonnegative potential function with nonempty zero sets $\mbox{int} V^{-1}(0)$, $0<δ<μ_0$ and $μ_0$ is the principle eigenvalue of $Δ^2$ in the zero sets $\mbox{int} V^{-1}(0)$ of $V(x)$. Here $\mbox{int} V^{-1}(0)$ denotes the interior part of the set $V^{-1}(0):=\{x\in \mathbb{R}^N: V(x)=0\}$. We prove that the above equation admits a least energy solution which is trapped near the zero sets $\mbox{int} V^{-1}(0)$ for $λ>0$ large.

math.AP↗