Branches of positive solutions for subcritical elliptic equations

Abstract

AbstractUsing \(L^{\infty }\) a-priori bounds for positive solutions to a class of subcritical elliptic problems in bounded C 2 domains, we prove the existence of a branch of positive solutions bifurcating from \((\lambda _{1},0)\), where \(\lambda _{1}\) is the first eigenvalue of the Dirichlet eigenvalue problem. We also provide sufficient conditions guarantying that either for any \(\lambda <\lambda _{1}\) there exists at least a positive solution, or for any continuum \((\lambda,u_{\lambda })\) of positive solution, there exists a \(\lambda ^{{\ast}} < 0\) such that \(\lambda ^{{\ast}} <\lambda <\lambda _{1}\) and the corresponding solutions are unbounded in the H 1(Ω)-norm as \(\lambda \rightarrow \lambda ^{{\ast}}\).