Topological Methods in Nonlinear Analysis

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

Yihong Du and Yuxia Guo

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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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 22, Number 1 (2003), 69-92.

Dates
First available in Project Euclid: 30 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1475266318

Mathematical Reviews number (MathSciNet)
MR2037267

Zentralblatt MATH identifier
1254.35066

Citation

Du, Yihong; Guo, Yuxia. Mountain pass solutions and an indefinite superlinear elliptic problem on $\mathbb R^{\mathbb N}$. Topol. Methods Nonlinear Anal. 22 (2003), no. 1, 69--92. https://projecteuclid.org/euclid.tmna/1475266318


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