Branches of positive solutions for subcritical elliptic equations

Abstract

Using $L^{\infty }$ 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 $\lambda \rightarrow \lambda ^{{\ast}}$.