Electron. J. Diff. Eqns., Vol. 1999(1999), No. 39, pp. 1-15.

Existence results for quasilinear elliptic systems in RN

N. M. Stavrakakis & N. B. Zographopoulos

Abstract:
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 $\lim_{|x| \to \infty} u(x) =\lim_{|x| \to \infty} u(x)=0$.

An addendum was attached on November 4, 2003. The results concerning the critical Sobolev exponent are false; other results still hold. See last page of this article.

Submitted July 16, 1999. Published October 4, 1999.
Math Subject Classifications: 35P30, 35J70, 35B45, 35B65.
Key Words: p-Laplacian, nonlinear eigenvalue problems, homogeneous Sobolev spaces, maximum principle, Palais-Smale Condition.

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N. M. Stavrakakis (e-mail: nikolas@central.ntua.gr)
N. B. Zographopoulos (e-mail: nz@math.ntua.gr)
Department of Mathematics
National Technical University
Zografou Campus
157 80 Athens, Greece
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