Multiple positive solutions for a class of \(p\)-\(q\)-Laplacian systems with multiple parameters and combined nonlinear effects. (English) Zbl 1240.35153

From the abstract: Consider the system \(-\Delta _{p}u=\lambda _{1}f(v)+\mu _{1}h(u),\, -\Delta _{q}v=\lambda _{2}g(u)+\mu _{2}\gamma (v)\) in \(\Omega \) with the Dirichlet boundary conditions, where \(\Delta _{s}z=\operatorname {div}(| \nabla z| ^{s-2}\nabla z)\), \(s>1\), \(\lambda _{1},\lambda _{2}>0\) and \(\mu _1, \mu _2\geq 0\) 0 are parameters and \(\Omega \) is a bounded domain in \(\mathbb {R}^{N}\) with smooth boundary. For some classes of non-negative monotone functions \(f,g,h\) and \(\gamma \) , we discuss a multiplicity result for positive solutions for a certain range of parameters \(\lambda _{1}, \mu _1, \lambda _{2}\) and \(\mu _2\). We use the method of sub- and super-solutions to establish our results.


35J57 Boundary value problems for second-order elliptic systems
35J70 Degenerate elliptic equations
35J60 Nonlinear elliptic equations
35J62 Quasilinear elliptic equations
47J15 Abstract bifurcation theory involving nonlinear operators