The Perturbation Problem of an Elliptic System with Sobolev Critical Growth

Abstract

In this paper, we study the following perturbation problem with Sobolev critical exponent:

$$\left\{ {\begin{array}{*{20}{c}} { - \Delta u = (1 + \varepsilon K(x)){u^{2* - 1}} + \frac{\alpha }{{{2^*}}}{u^{\alpha - 1}}{v^\beta } + \varepsilon h(x){u^p},}&{x \in {\mathbb{R}^N},} \\ { - \Delta v = (1 + \varepsilon Q(x)){v^{2* - 1}} + \frac{\beta }{{{2^*}}}{u^\alpha }{v^{\beta - 1}} + \varepsilon l(x){v^q},}&{x \in {\mathbb{R}^N},} \\ {u > 0,v > 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}&{x \in {\mathbb{R}^N},} \end{array}} \right.$$

((0.1))

where $0 < p,q < 1,\alpha + \beta = {2^*}: = \tfrac{{2N}}{{N - 2}},\alpha ,\beta \geqslant 2$ 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.