Existence and multiplicity of solutions for elliptic equations with jumping nonlinearities |
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Authors: | Riccardo Molle Donato Passaseo |
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Affiliation: | a Dipartimento di Matematica, Università di Roma “Tor Vergata”, Via della Ricerca Scientifica n. 1, 00133 Roma, Italy b Dipartimento di Matematica “E. De Giorgi”, Università di Lecce, P.O. Box 193, 73100 Lecce, Italy |
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Abstract: | In this paper we are concerned with a semilinear elliptic Dirichlet problem with jumping nonlinearity and, using a completely variational method, we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. In order to prove this fact, we show that for every positive integer k, when a suitable parameter is large enough, there exists a solution which presents k peaks. Under the assumptions we consider in this paper, new (unexpected) phenomena are observed in the study of this problem and new methods are required to construct the k-peaks solutions and describe their asymptotic behavior (weak limits of the rescaled solutions, localization of the concentration points of the peaks, asymptotic profile of the rescaled peaks, etc.). |
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Keywords: | Jumping nonlinearities Multiplicity of solutions Variational methods |
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