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Elliptic equations with one-sided critical growth. (English) Zbl 1022.35009
Authors’ summary: 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 \(\lambda > \lambda_1\), the first eigenvalue of the Laplacian. We prove that such equations have at least two solutions for certain forcing terms provided \(N \geq 6\). 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.
Reviewer: V.Mustonen (Oulu)

MSC:
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
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