Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Sobolev exponents

Pages: 731 - 762, Volume 36, Issue 2, February 2016      doi:10.3934/dcds.2016.36.731

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Yi He - Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China (email)
Gongbao Li - Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China (email)

Abstract: We study the existence, concentration and multiplicity of weak solutions to the quasilinear Schrödinger equation with critical Sobolev growth \begin{equation*} \left\{ \begin{gathered} - {\varepsilon ^2}\Delta u + V(x)u - {\varepsilon ^2}\Delta (u^2)u = W(x){u^{q - 1}} + {u^{2\cdot{2^*} - 1}} {\text{ in }}{\mathbb{R}^N},\\ u > 0{\text{ in }}{\mathbb{R}^N},\\ \end{gathered} \right. \end{equation*} where $\varepsilon$ is a small positive parameter, $N \ge 3$, ${2^ * } = \frac{{2N}} {{N - 2}}$, $4 < q < 2 \cdot {2^ * }$, $\min V > 0$ and $\inf W > 0$. Under proper assumptions, we obtain the existence and concentration phenomena of soliton solutions of the above problem. With minimax theorems and Ljusternik-Schnirelmann theory, we also obtain multiple soliton solutions by employing the topology of the set where the potentials $V(x)$ attains its minimum and $W(x)$ attains its maximum.

Keywords:  Existence, concentration, multiplicity, quasilinear Schrödinger equation, critical growth.
Mathematics Subject Classification:  Primary: 35J20, 35J60, 35J92.

Received: March 2014;      Available Online: August 2015.

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