Mountain pass solutions and an indefinite superlinear elliptic problem on $\mathbb R^{\mathbb N}$

Abstract

We consider the elliptic problem $$ -\Delta u-\lambda u=a(x) g(u), $$ with $a(x)$ sign-changing and $g(u)$ behaving like $u^p$, $p> 1$. Under suitable conditions on $g(u)$ and $a(x)$, we extend the multiplicity, existence and nonexistence results known to hold for this equation on a bounded domain (with standard homogeneous boundary conditions) to the case that the bounded domain is replaced by the entire space $\mathbb R^N$. More precisely, we show that there exists $\Lambda> 0$ such that this equation on $\mathbb R^N$ has no positive solution for $\lambda> \Lambda$, at least two positive solutions for $\lambda\in (0,\Lambda)$, and at least one positive solution for $\lambda\in (-\infty,0]\cup\{\Lambda\}$.

Our approach is based on some descriptions of mountain pass solutions of semilinear elliptic problems on bounded domains obtained by a special version of the mountain pass theorem. These results are of independent interests.