## Advances in Differential Equations

- Adv. Differential Equations
- Volume 2, Number 6 (1997), 981-1003.

### Principal eigenvalues for some quasilinear elliptic equations on $\mathbb{R}^N$

J. Fleckinger, R. F. Manásevich, N. M. Stavrakakis, and F. de Thélin

#### Abstract

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.

#### Article information

**Source**

Adv. Differential Equations Volume 2, Number 6 (1997), 981-1003.

**Dates**

First available in Project Euclid: 22 April 2013

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1366638680

**Mathematical Reviews number (MathSciNet)**

MR1606355

**Zentralblatt MATH identifier**

1023.35505

**Subjects**

Primary: 35J65: Nonlinear boundary value problems for linear elliptic equations

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35P15: Estimation of eigenvalues, upper and lower bounds

#### Citation

Fleckinger, J.; Manásevich, R. F.; Stavrakakis, N. M.; de Thélin, F. Principal eigenvalues for some quasilinear elliptic equations on $\mathbb{R}^N$. Adv. Differential Equations 2 (1997), no. 6, 981--1003. https://projecteuclid.org/euclid.ade/1366638680