Electron. J. Differential Equations, Vol. 2017 (2017), No. 54, pp. 1-11.

Existence of positive solutions to perturbed nonlinear Dirichlet problems involving critical growth

Huixing Zhang, Ran Zhang

Abstract:
We consider the following perturbed nonlinear elliptic problem with critical growth
$$\displaylines{ -\varepsilon^2\Delta u+V(x)u=f(x)|u|^{p-2}u +\frac{\alpha}{\alpha+\beta}K(x)|u|^{\alpha-2}u|v|^\beta,\quad x\in \mathbb{R}^N,\cr -\varepsilon^2\Delta v+V(x)v=g(x)|v|^{p-2}v +\frac{\beta}{\alpha+\beta}K(x)|u|^\alpha|v|^{\beta-2}v,\quad x\in \mathbb{R}^N,\cr u(x),\quad v(x)\to 0 \quad \text{as } |x|\to\infty. }$$
Using variational methods, we prove the existence of positive solutions.

Submitted April 20, 2016. Published February 21, 2017.
Math Subject Classifications: 35B33, 35J60, 35J65.
Key Words: Perturbed nonlinear Dirichlet problem; critical growth; Palais-Smale condition; variational methods.

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Huixing Zhang
Department of Mathematics
China University of Mining and Technology
Xuzhou, Jiangsu 221116, China
email: huixingzhangcumt@163.com
Ran Zhang
Department of Mathematics
China University of Mining and Technology
Xuzhou, Jiangsu 221116, China
email: ranzhang_zhangran@163.com

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