Differential and Integral Equations

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

G. A. Afrouzi and K. J. Brown

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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

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

First available in Project Euclid: 21 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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


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

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