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Positive solutions for a class of concave-convex semilinear elliptic systems with double critical exponents

Authors :
Han-Ming Zhang
Jia-Feng Liao
Source :
Electronic Journal of Qualitative Theory of Differential Equations, Vol 2023, Iss 20, Pp 1-24 (2023)
Publication Year :
2023
Publisher :
University of Szeged, 2023.

Abstract

In this paper, we consider the following concave-convex semilinear elliptic system with double critical exponents:~ \begin{equation} \begin{cases} -\Delta u=|u|^{2^{*}-2}u+\frac{\alpha}{2^{*}}|u|^{\alpha-2}|v|^{\beta}u+\lambda|u|^{q-2}u,~~&\mbox{in}~\Omega,\\ -\Delta v=|v|^{2^{*}-2}v+\frac{\beta}{2^{*}}|u|^{\alpha}|v|^{\beta-2}v+\mu|v|^{q-2}v,~~&\mbox{in}~\Omega,\\ u,~v>0,~~&\mbox{in}~\Omega,\\ u=v=0,~~&\mbox{on}~\partial\Omega, \end{cases} \nonumber \end{equation} where $\Omega\subset\mathbb{R}^{N}(N\geq3)$ is a bounded domain with smooth boundary,~$\lambda,~\mu>0,~11,~\beta>1,~\alpha+\beta=2^{*}=\frac{2N}{N-2}$. By the Nehari manifold method and variational method, we obtain two positive solutions which improves the recent results in the literature.

Details

Language :
English
ISSN :
14173875
Volume :
2023
Issue :
20
Database :
Directory of Open Access Journals
Journal :
Electronic Journal of Qualitative Theory of Differential Equations
Publication Type :
Academic Journal
Accession number :
edsdoj.55b7a9c20844d59c8643f01ecee271
Document Type :
article
Full Text :
https://doi.org/10.14232/ejqtde.2023.1.20