×

Maximum principle and existence of positive solutions for nonlinear systems involving degenerate \(p\)-Laplacian operators. (English) Zbl 1137.35322

Summary: We study the maximum principle and existence of positive solutions for the nonlinear system
\[ \begin{aligned} -\Delta _{p,_{P}}u&= a(x)|u|^{p-2}u+b(x)|u|^{\alpha }|v|^{\beta }v+f \quad \text{in } \Omega , \\ -\Delta _{Q,q}v&= c(x)|u|^{\alpha }|v|^{\beta }u+d(x)|v|^{q-2}v+g \quad \text{in } \Omega , \\ u&=v=0 \quad \text{on }\partial \Omega , \end{aligned} \] where the degenerate \(p\)-Laplacian defined as \(\Delta _{p,_{P}}u=\text{div}[P(x)|\nabla u|^{p-2}\nabla u]\). We give necessary and sufficient conditions for having the maximum principle for this system and then we prove the existence of positive solutions for the same system by using an approximation method.

MSC:

35B50 Maximum principles in context of PDEs
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
PDFBibTeX XMLCite
Full Text: EuDML EMIS