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Existence and nonexistence of solutions for a quasilinear elliptic system. (English) Zbl 1326.35151

Summary: By a sub-super solution argument, we study the existence of positive solutions for the system \[ \begin{cases} -\varDelta_{p}u=a_{1}(x)F_{1}(x,u,v) &\text{ in}\;\varOmega,\\ -\varDelta_{q}v=a_{2}(x)F_{2}(x,u,v) &\text{ in}\;\varOmega,\\ u,v>0 &\text{ in}\;\varOmega,\\ u=v=0 &\text{ on}\;\partial\varOmega,\end{cases} \] 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.

MSC:

35J66 Nonlinear boundary value problems for nonlinear elliptic equations
35J50 Variational methods for elliptic systems
35B09 Positive solutions to PDEs
35B06 Symmetries, invariants, etc. in context of PDEs
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