We present some multiplicity results concerning semilinear elliptic Dirichlet problems with jumping nonlinearities where the jumping condition involves a set of high eigenvalues not including the first one. Using a variational method we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. Indeed, we prove that for every positive integer k there exists a positive integer n(k) such that, if the number of jumped eigenvalues is greater than n(k), then the problem has at least a solution which presents k peaks. Moreover, we describe the asymptotic behaviour of the solutions as the number of jumped eigenvalues tends to infinity; in particular, we analyse some concentration phenomena of the peaks (near points or suitable manifolds), we describe the asymptotic profile of the rescaled peaks, etc

Molle, R., Passaseo, D. (2014). Elliptic equations with jumping nonlinearities involving high eigenvalues. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 49(1-2), 861-907 [10.1007/s00526-013-0603-y].

Elliptic equations with jumping nonlinearities involving high eigenvalues

MOLLE, RICCARDO;
2014-01-01

Abstract

We present some multiplicity results concerning semilinear elliptic Dirichlet problems with jumping nonlinearities where the jumping condition involves a set of high eigenvalues not including the first one. Using a variational method we show that the number of solutions may be arbitrarily large provided the number of jumped eigenvalues is large enough. Indeed, we prove that for every positive integer k there exists a positive integer n(k) such that, if the number of jumped eigenvalues is greater than n(k), then the problem has at least a solution which presents k peaks. Moreover, we describe the asymptotic behaviour of the solutions as the number of jumped eigenvalues tends to infinity; in particular, we analyse some concentration phenomena of the peaks (near points or suitable manifolds), we describe the asymptotic profile of the rescaled peaks, etc
gen-2014
Pubblicato
Rilevanza internazionale
Articolo
Esperti anonimi
Settore MAT/05 - ANALISI MATEMATICA
English
Con Impact Factor ISI
Molle, R., Passaseo, D. (2014). Elliptic equations with jumping nonlinearities involving high eigenvalues. CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS, 49(1-2), 861-907 [10.1007/s00526-013-0603-y].
Molle, R; Passaseo, D
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/2108/90153
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