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

DOI: http://dx.doi.org/10.12775/TMNA.2003.028

#### 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.

$$

-\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.

#### Keywords

Mountain pass solution; Morse index; a priori estimates

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