An existence result for a class of superlinear \(p\)-Laplacian semipositone systems. (English) Zbl 1046.34043

The authors consider the following system (*) \(-( r^{N-1}\phi (u^{\prime })) ^{\prime }=\lambda r^{N-1}f(v);\) \(a<r<b;\) \(-( r^{N-1}\phi (v^{\prime })) ^{\prime }=\lambda r^{N-1}g(u);\) \(a<r<b;\) \(u(a)=0=u(b);\) \(v(a)=0=v(b),\) where \(\phi (s)=\alpha (s^{2})s\) is an odd, increasing homeomorphism of the real line, \(\lambda >0.\) They consider the following conditions; (A1) For each \(c>0\) there exists a constant \(A_{c}>0\) such that \(\phi ^{-1}(cx)\geq A_{c}\phi ^{-1}(x)\) \(\forall x\geq 0\) and \(A_{c}\to \infty \) as \(c\to \infty .\) (A2) \(f,g:[ 0,\infty ) \to \mathbb{R}\) are continuous, and there exists \(M>0\) such that \(f(z)\geq -M/2\) and \(g(z)\geq -M/2\) for \(z\in [ 0,\infty ) .\) (A3) \(\lim_{z\to \infty }h^{*}(z)/\phi (z)=\infty \) and \(\lim_{z\to \infty }A_{h^{*}(z)}/z=\infty \), where \(h^{*}(z)=\inf_{w\geq z}\{ \min (f(w),\) \(g(w))\} \) and \(A_{c}\)is defined as in (A1). Using Leray-Schauder degree theory, they prove that if the conditions (A1), (A2) and (A3) hold, than there exists \(\lambda ^{*}>0\) such that for \(0<\lambda <\lambda ^{*},\) system (*) has at least one positive solution. An interesting example is given.


34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations