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multiplicity of solutions; weakly nonlinear elliptic equations
In this paper existence and multiplicity of solutions of the elliptic problem $\Cal L u + \lambda_1u+\mu u^+vu^-+g(x,u)=f$ in $\Omega$ $Bu=0$ on $\partial\Omega$, are discussed provided the parameters $\mu$ and $v$ are close to the first eigenvalue $\lamda_1$. The sufficient conditions presented here are more general than those in given by S. Fučík in his aerlier paper.
[1] A. Ambrosetti G. Mancini: Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance. The case of the simple eigenvalue. Journal of Diff. Eq., vol. 28, (1978), 220-245. DOI 10.1016/0022-0396(78)90068-2 | MR 0492839
[2] S. Fučík: Remarks on a result by A. Ambrosetti and G. Prodi. U.M.I., (4), 11 (1975), 259-267. MR 0382849
[3] M. A. Krasnoselskij: Topological methods in the theory of nonlinear integral equations. Pergamon Press, London, 1964.
[4] J. Minty: Monotone operators in Hilbert space. Duke Math. Journal 29 (1962), 341 - 346. DOI 10.1215/S0012-7094-62-02933-2 | MR 0169064
[5] A. N. Kolmogorov S. V. Fomin: Элементы теории функций и функционального анализа. Nauka, Moskva, 1972.
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