Existence of semiclassical states for a quasilinear Schrödinger equation on ℝ N with exponential critical growth

2016 ◽  
Vol 32 (11) ◽  
pp. 1279-1296 ◽  
Author(s):  
Shao Jun Li ◽  
Carlos A. Santos ◽  
Min Bo Yang
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Critical Growth ◽  
Quasilinear Schrödinger Equation ◽  
Exponential Critical Growth ◽  
Semiclassical States

Multiplicity and Concentration Results for a Magnetic Schrödinger Equation With Exponential Critical Growth in ℝ2

2020 ◽  
Author(s):  
Pietro d’Avenia ◽  
Chao Ji
Keyword(s):  
Magnetic Field ◽  
Schrödinger Equation ◽  
Variational Methods ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Critical Growth ◽  
Concentration Of Solutions ◽  
Exponential Critical Growth ◽  
Penalization Technique

Abstract In this paper we study the following nonlinear Schrödinger equation with magnetic field $$\begin{align*} \left(\frac{\varepsilon}{i}\nabla-A(x)\right)^{2}u+V(x)u=f(| u|^{2})u,\quad x\in\mathbb{R}^{2}, \end{align*}$$where $\varepsilon>0$ is a parameter, $V:\mathbb{R}^{2}\rightarrow \mathbb{R}$ and $A: \mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ are continuous potentials, and $f:\mathbb{R}\rightarrow \mathbb{R}$ has exponential critical growth. Under a local assumption on the potential $V$, by variational methods, penalization technique, and Ljusternik–Schnirelmann theory, we prove multiplicity and concentration of solutions for $\varepsilon $ small.

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