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Existence and concentration result for a quasilinear Schrödinger equation with critical growth

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Abstract

This paper is concerned with the concentration of positive ground states solutions for a modified Schrödinger equation

$$\begin{aligned} -\varepsilon ^{2}\triangle u+V(x)u-\varepsilon ^{2}\triangle (u^{2})u=K(x)|u|^{p-2}u+|u|^{22^{*}-2}u,\quad \text{ in } \; \mathbb {R}^{N}, \end{aligned}$$

where \(4<p<22^{*}, \varepsilon >0\) is a parameter and \(2^{*}:=\frac{2N}{N-2}(N\ge 3)\) is the critical Sobolev exponent. We prove the existence of a positive ground state solution \(v_{\varepsilon }\) and \(\varepsilon \) sufficiently small under some suitable conditions on the nonnegative functions V(x) and K(x). Moreover, \(v_{\varepsilon }\) concentrates around a global minimum point of V as \(\varepsilon \rightarrow 0\). The proof of the main result is based on minimax theorems and concentration compact theory.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Liuyang Shao & Haibo Chen

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  1. Liuyang Shao
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  2. Haibo Chen
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Correspondence to Haibo Chen.

Additional information

This work is partially supported by Natural Science Foundation of China 11671403. By the Fundamental Research Funds for the Central Universities of Central South University 2017zzts056 and by the Mathematics and Interdisciplinary Sciences project of CSU.

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Shao, L., Chen, H. Existence and concentration result for a quasilinear Schrödinger equation with critical growth. Z. Angew. Math. Phys. 68, 126 (2017). https://doi.org/10.1007/s00033-017-0869-6

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  • DOI: https://doi.org/10.1007/s00033-017-0869-6

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