Existence and nonexistence of solutions for a quasilinear elliptic system

Qin Li¹ ; Zuodong Yang²

¹ Institute of Mathematics School of Mathematical Sciences Nanjing Normal University Jiangsu Nanjing 210023, China
² Institute of Mathematics School of Mathematical Sciences Nanjing Normal University Jiangsu Nanjing 210023, China and School of Teacher Education Nanjing Normal University Jiangsu Nanjing 210097, China

Annales Polonici Mathematici, Tome 113 (2015) no. 2, pp. 155-164.

Résumé

By a sub-super solution argument, we study the existence of positive solutions for the system $$\left\{\begin{array}{l@{\quad}l} -\varDelta_{p}u=a_{1}(x)F_{1}(x,u,v) {\rm in}\ \varOmega,\\ -\varDelta_{q}v=a_{2}(x)F_{2}(x,u,v) {\rm in}\ \varOmega,\\ u,v>0 {\rm in}\ \varOmega,\\ u=v=0 {\rm on}\ \partial\varOmega,\end{array}\right. $$ where $\varOmega$ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary or $\varOmega=\mathbb{R}^{N}$. A nonexistence result is obtained for radially symmetric solutions.

Détail

DOI : 10.4064/ap113-2-3

Keywords: sub super solution argument study existence positive solutions system begin array quad vardelta varomega vardelta varomega varomega partial varomega end array right where varomega bounded domain mathbb smooth boundary varomega mathbb nonexistence result obtained radially symmetric solutions