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1997 Principal eigenvalues for some quasilinear elliptic equations on $\mathbb{R}^N$
J. Fleckinger, R. F. Manásevich, N. M. Stavrakakis, F. de Thélin
Adv. Differential Equations 2(6): 981-1003 (1997).


We improve some previous results for the principal eigenvalue of the $p$-Laplacian defined on $\mathbb{R}^N,$ study regularity of the corresponding eigenfunctions and give an existence result of the type below the first eigenvalue (or between the first eigenvalues) for a certain perturbed problem. Based on our approach for the equation we deduce existence, uniqueness and simplicity of positive principal eigenvalues for the $p$-Laplacian system $$ \begin{align} &{-\Delta}_{p} u = \lambda a(x) |u|^{p-2}u + \lambda b(x) |u|^{\alpha - 1} u |v|^{\beta +1}, \quad x \in \mathbb{R}^N, \\ &{-\Delta}_{q} v = \lambda b(x) |u|^{\alpha + 1} |v|^{\beta -1}v + \lambda d(x) |v|^{q - 2}v, \quad x \in \mathbb{R}^N, \\ & 0 <u, 0<v, \text{ in } \quad \mathbb{R}^N, \lim_{|x| \rightarrow +\infty} u(x) = \lim_{|x| \rightarrow +\infty} v(x) = 0. \end{align} $$ We also establish the regularity of the corresponding eigenfunctions.


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J. Fleckinger. R. F. Manásevich. N. M. Stavrakakis. F. de Thélin. "Principal eigenvalues for some quasilinear elliptic equations on $\mathbb{R}^N$." Adv. Differential Equations 2 (6) 981 - 1003, 1997.


Published: 1997
First available in Project Euclid: 22 April 2013

zbMATH: 1023.35505
MathSciNet: MR1606355

Primary: 35J65
Secondary: 35B05, 35P15

Rights: Copyright © 1997 Khayyam Publishing, Inc.


Vol.2 • No. 6 • 1997
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