Acta mathematica scientia,Series B ›› 2020, Vol. 40 ›› Issue (5): 1391-1404.doi: 10.1007/s10473-020-0513-y

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THE PERTURBATION PROBLEM OF AN ELLIPTIC SYSTEM WITH SOBOLEV CRITICAL GROWTH

Qi LI   

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
  • Received:2019-01-16 Revised:2019-09-03 Online:2020-10-25 Published:2020-11-04
  • Supported by:
    Q. Li was supported by the excellent doctorial dissertation cultivation grant (2018YBZZ067 and 2019YBZZ057) from Central China Normal University.

Abstract: In this paper, we study the following perturbation problem with Sobolev critical exponent: \begin{equation}\label{eqs0.1} \left\{ \begin{array}{ll} -\Delta u=(1+\varepsilon K(x)){{u}^{{{2}^{*}}-1}}+\frac{\alpha }{{{2}^{*}}}{{u}^{\alpha -1}}{{v}^{\beta }}+\varepsilon h(x){{u}^{p}},\ \ &x\in \mathbb{R}^N,\\[2.5mm] -\Delta v=(1+\varepsilon Q(x)){{v}^{{{2}^{*}}-1}}+\frac{\beta }{{{2}^{*}}}{{u}^{\alpha }}{{v}^{\beta -1}}+\varepsilon l(x){{v}^{q}},\ \ &x\in \mathbb{R}^N,\\[2mm] u> 0,\,v> 0,\ \ &x\in \mathbb{R}^N, \end{array} \right. \end{equation} where $0 < p,\,q < 1$, $\alpha +\beta ={{2}^{*}}:=\frac{2N}{N-2}$, $\alpha,\,\beta\geq 2$ and $N=3, 4$. Using a perturbation argument and a finite dimensional reduction method, we get the existence of positive solutions to problem (0.1) and the asymptotic property of the solutions.

Key words: perturbation argument, finite dimensional reduction method, critical exponent

CLC Number: 

  • 35J47
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