Boundary value problems for a class of quasilinear differential systems with singular nonlinearities
Motivated by the study of positive radial solutions to some systems of partial differential equations on annular domains, the authors prove some results on existence and multiplicity of positive solutions to quasilinear differential systems of the form \[ (\varphi_{p}(u'))' + m(t)\varphi_{p}(u') + H_{v}(u,v) - h(t) = 0,\quad(\varphi_{q}(v'))' + n(t)\varphi_{q}(v') + H_{u}(u,v) - k(t) = 0, \] under periodic or Neumann boundary conditions on an interval \((a,b)\). Here, \(\varphi_{\beta}(s) = | s |^{\beta -2}s\), \(p,q > 1\), \(m,n \in L^{\infty}((a,b), \mathbb{R}^{+})\), \(h,k \in L^{1}(a,b)\) and \(H \in C^{1}((0,\infty)\times (0,\infty))\) has a singularity at \(u = 0\) or \(v = 0.\) The main tool used in the proofs is the Leray-Schauder degree theory.
scientific article; zbMATH DE number 1193538