Existence results for quasilinear elliptic systems in $\Bbb R^N$

Stavrakakis, N.M. ; Zographopoulos, N.B.

Electronic Journal of Differential Equations, Tome 1999 (1999).

Résumé

Summary: We prove existence results for the quasilinear elliptic system $$ -\Delta_{p}u = \lambda a(x)|u|^{\gamma-2}u +\lambda b(x) |u|^{\alpha -1}|v|^{\beta +1}u, $ $ -\Delta_{q}v = \lambda d(x)|v|^{\delta-2}v +\lambda b(x)|u|^{\alpha +1}|v|^{\beta -1}v, where $\gamma$ and $\delta$ may reach the critical Sobolev exponents, and the coefficient functions a, b, and d may change sign. For the unperturbed system (a=0, b=0), we establish the existence and simplicity of a positive principal eigenvalue, under the assumption that $u(x), v(x)$ are positive, and $\lim_{|x| \to \infty} u(x) =\lim_{|x| \to \infty} u(x)=0$.$$

Détail

Classification : 35P30, 35J70, 35B45, 35B65
Keywords: p-Laplacian, nonlinear eigenvalue problems, homogeneous Sobolev spaces, maximum principle, palais-Smale condition