## Differential and Integral Equations

- Differential Integral Equations
- Volume 14, Number 1 (2001), 37-50.

### Unbounded principal eigenfunctions for problems on all ${\bf R}^N$

#### Abstract

We investigate the existence of principal eigenvalues, i.e., values of $\lambda$ for which the corresponding eigenfunction is positive, for the problem $ - \Delta u(x) = \lambda g(x) u(x)$ for $x \in \mathbb R ^N$ where $g$ is a smooth function which may change sign. Unlike most previous studies the eigenfunction is not required to $\to 0$ as $|x| \to \infty$. It is shown that there may exist a closed interval of principal eigenvalues $[\lambda_* , \lambda^*]$ and sufficient conditions are given to ensure that principal eigenfunctions $\to 0$ as $|x| \to \infty$ if and only if $\lambda = \lambda^*$ or $\lambda_*$.

#### Article information

**Source**

Differential Integral Equations, Volume 14, Number 1 (2001), 37-50.

**Dates**

First available in Project Euclid: 21 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.die/1356123373

**Mathematical Reviews number (MathSciNet)**

MR1797930

**Zentralblatt MATH identifier**

1161.35359

**Subjects**

Primary: 35P05: General topics in linear spectral theory

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B40: Asymptotic behavior of solutions 35J25: Boundary value problems for second-order elliptic equations

#### Citation

Afrouzi, G. A.; Brown, K. J. Unbounded principal eigenfunctions for problems on all ${\bf R}^N$. Differential Integral Equations 14 (2001), no. 1, 37--50. https://projecteuclid.org/euclid.die/1356123373