### Bifurcation problems for superlinear elliptic indefinite equations

DOI: http://dx.doi.org/10.12775/TMNA.2000.028

#### Abstract

In this paper, we are dealing with the following superlinear elliptic problem:

$$\cases

-\Delta u = \lambda u+h(x)u^p &\text{in }{\mathbb R}^N,\\

u\geq 0,\endcases\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.

$$\cases

-\Delta u = \lambda u+h(x)u^p &\text{in }{\mathbb R}^N,\\

u\geq 0,\endcases\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.

#### Keywords

Global bifurcation; uniform a priori bounds; moving plane; Kelvin transform; blow up analysis

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