Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity

Yi  He; Gongbao  Li

doi:10.3934/mcrf.2016016

Article Contents

2016, Volume 6, Issue 4: 551-593. Doi: 10.3934/mcrf.2016016

This issue Previous Article An optimal control model of carbon reduction and trading Next Article Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions

Concentrating solitary waves for a class of singularly perturbed quasilinear Schrödinger equations with a general nonlinearity

Yi He^1, and
Gongbao Li^2,

1.
School of Mathematics and Statistics, South-Central University For Nationalities, Wuhan, 430074
2.
Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079

Received: December 31, 2015

Revised: February 29, 2016

Published: November 2016

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Abstract

We are concerned with a class of singularly perturbed quasilinear Schrödinger equations of the following form: \[ - {\varepsilon ^2}\Delta u - {\varepsilon ^2}\Delta ({u^2})u + V(x)u = h(u),{\text{ }}u > 0{\text{ in }}{\mathbb{R}^N}, \] where $\varepsilon $ is a small positive parameter, $N \ge 3$ and the nonlinearity $h$ is of critical growth. We construct a family of positive solutions ${u_\varepsilon } \in {H^1}({\mathbb{R}^N})$ of the above problem which concentrates around local minima of $V$ as $\varepsilon \to 0$ under certain assumptions on $h$. Our result especially solves the above problem in the case where $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(2 < q \le 4,{\text{ }}\lambda > 0)$ and completes the study made in some recent works in the sense that, in those papers only the case where $h(u) \sim \lambda {u^{q - 1}} + {u^{2 \cdot {2^ * } - 1}}{\text{ }}(4 < q < 2 \cdot {2^ * },{\text{ }}\lambda > 0)$ was considered. Moreover, our main results extend also the arguments used in Byeon and Jeanjean [14], which deal with Schrödinger equations with subcritical nonlinearities, to the quasilinear Schrödinger equations with critical nonlinearities.

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Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 35J92.

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