Bifurcation problems for superlinear elliptic indefinite equations

Abstract

In this paper, we are dealing with the following superlinear elliptic problem: $$\begin{cases} -\Delta u = \lambda u+h(x)u^p &\text{in }{\mathbb R}^N,\\ u\geq 0,\end{cases}\tag{P} $$ where $h$ is a $C^2$ function from ${\mathbb R}^N$ to ${\mathbb R}$ changing sign such that $\Omega^+ :=\{x\in {\mathbb R}^N\mid h(x)> 0\}$, $\Gamma :=\{x\in {\mathbb R}^N\mid h(x)=0 \}$ are bounded.

For $1< p< {(n+2)}/{(n-2)}$ we prove the existence of global and connected branches of solutions of (P) in ${\mathbb R}^-\times H^1({\mathbb R}^N)$ and in ${\mathbb R}\times L^{\infty}({\mathbb R}^N)$. The proof is based upon a local approach.