Electron. J. Diff. Eqns., Vol. 2002(2002), No. 89, pp. 1-21.

Elliptic equations with one-sided critical growth

Marta Calanchi & Bernhard Ruf

Abstract:
We consider elliptic equations in bounded domains $\Omega\subset \mathbb{R}^N $ with nonlinearities which have critical growth at $+\infty$ and linear growth $\lambda$ at $-\infty$, with $\lambda greater than \lambda_1$, the first eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms provided $N \ge 6$. In dimensions $N = 3,4,5$ an additional lower order growth term has to be added to the nonlinearity, similarly as in the famous result of Brezis-Nirenberg for equations with critical growth.

Submitted March 01, 2002. Published October 18, 2002.
Math Subject Classifications: 35J20.
Key Words: Nonlinear elliptic equation, critical growth, linking structure.

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Marta Calanchi
Dip. di Matematica, Universita degli Studi,
Via Saldini 50, 20133 Milano, Italy
calanchi@mat.unimi.it, ph. +39.02.50316144
Bernhard Ruf
Dip. di Matematica, Universita degli Studi,
Via Saldini 50, 20133 Milano, Italy
e-mail: ruf@mat.unimi.it, ph. +39.02.50316157

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