Existence and concentration of semiclassical states for nonlinear Schrodinger equations

zbMATH Open1259.35076arXiv1201.2215MaRDI QIDQ3165037

Shaowei Chen

Publication date: 25 October 2012

Abstract: In this paper, we study the following semilinear Schr"odinger equation ${\displaystyle }$ -epsilon^2 riangle u+ u+ V(x)u=f(u), uin H^{1}(mathbb{R}^{N}), ${\displaystyle }$ where $Ngeq2$ and $epsilon>0$ is a small parameter. The function $V$ is bounded in $mathbb{R}^N$, $in{f}_{mathbb{R}^N}(1+V(x))>0$ and it has a possibly degenerate isolated critical point. Under some conditions on $f,$ we prove that as $epsilonightarrow0,$ this equation has a solution which concentrates at the critical point of $V$.}

Full work available at URL: https://arxiv.org/abs/1201.2215

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zbMATH Keywords

semilinear Schrödinger equationvariational reduction method

Mathematics Subject Classification ID

Degenerate elliptic equations (35J70) Schrödinger operator, Schrödinger equation (35J10) Variational methods for second-order elliptic equations (35J20)

Related Items (4)

Semiclassical non-concentration near hyperbolic orbits ⋮ Existence of semiclassical states for a coupled Schrödinger system with potentials and nonlocal nonlinearities ⋮ Multiple semiclassical standing waves for fractional nonlinear Schrödinger equations ⋮ Localized concentration of semi-classical states for nonlinear Dirac equations

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