Lishan Lin | Searcharxiv

arXiv2012

Existence and concentration of semiclassical states for nonlinear Schrodinger equations

In this paper, we study the following semilinear Schrödinger equation $$ -ε^2\triangle u+ u+ V(x)u=f(u),\ u\in H^{1}(\mathbb{R}^{N}), $$ where $N\geq 2$ and $ε>0$ is a small parameter. The function $V$ is bounded in $\mathbb{R}^N$, $\inf_{\mathbb{R}^N}(1+V(x))>0$ and it has a possibly degenerate isolated critical point. Under some conditions on $f,$ we prove that as $ε\rightarrow 0,$ this equation has a solution which concentrates at the critical point of $V$.}

math.AP↗

arXiv2009

Results on entire solutions for a degenerate critical elliptic equation with anisotropic coefficients

In this paper, we study the following degenerate critical elliptic equations with anisotropic coefficients $$ -div(|x_{N}|^{2\alpha}\nabla u)=K(x)|x_{N}|^{\alpha\cdot 2^{*}(s)-s}|u|^{2^{*}(s)-2}u {in} \mathbb{R}^{N} $$ where $x=(x_{1},...,x_{N})\in\mathbb{R}^{N},$ $N\geq 3,$ $\alpha>1/2,$ $0\leq s\leq 2$ and $2^{*}(s)=2(N-s)/(N-2).$ Some basic properties of the degenerate elliptic operator $-div(|x_{N}|^{2\alpha}\nabla u)$ are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions of this equation. As a consequence, we obtain some nonexistence results for solutions of this equation.

math.AP↗