A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces
Mihǎilescu, Mihai ; Rǎdulescu, Vicenţiu, 1958-
We study the nonlinear eigenvalue problem ▫$-{\rm div}(a(|\nabla u|)\nabla u) = \lambda|u|^{q(x)-2}u$▫ in ▫$\Omega$▫, ▫$u=0$▫ on ▫$\partial\Omega$▫, where ▫$\Omega$▫ is a bounded open set in ...▫$\mathbb R^N$▫ with smooth boundary, ▫$q$▫ is a continuous function, and ▫$a$▫ is a nonhomogeneous potential. We establish sufficient conditions on ▫$a$▫ and ▫$q$▫ such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases ▫$a(t) = t^{p-2}\log (1+t^r)$▫ and ▫$a(t) = t^{p-2} [\log (1+t)]^{-1}$▫.
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