Espectro de Fuík e equações elípticas com não linearidade de salto / Fucik Spectrum and elliptic equations with jumping nonlinearities
AUTOR(ES)
Rafael Antonio Rossato
DATA DE PUBLICAÇÃO
2010
RESUMO
We study the Fucik Spectrum for the Laplacian operator, that is, the set SIGMAof the couples (mü, nü) ARE THIS ESTA CONTAINEDR POT. 2, for which the problem { - DELTAu(x) = mü nüPOT. + (x); EPSILONOMEGA, Bu = 0; x EPSILONPARTIALOMEGA, admits a nontrivial solution, where OMEGAEPSILONR POT. nis a bounded domain, u POT. + (x) = max {0, u(x)}, u POT. -(x) = {0, - u(x)} and B represents some boundary condition. We first show abstract results about the Fucik Spectrum and then we compute it explicitly in the one dimensional case for the Dirichlet and Neumann problems. These results one applied at the study of the solvability of the problem. { - DELTAu(x) = f (x, u(x)), x EPSILONOMEGA, Bu = 0; x EPSILONPARTIALOMEGA, whe3n the nonlinearity f is a suitable pertubation of müu POT. + - nüu+ - nüu POT. n; we describe different behaviors depending on the parameters (mü, nü). Finally, we consider the Fucik Spectrum in higher dimension. In this case it is not possible to compute it explicitly, so we will show a variational characterization of the first nontrivial curve. This characterization will allow to obtain some information on the properties of this curve and also further results on the solvability of (2)
ASSUNTO(S)
equações elípticas espectro de fucik fucik spectrum elliptic equations
