Eigenvalue problems associated with nonhomogeneous differential operators in Orlicz-Sobolev spaces
Mihǎilescu, Mihai ; Rǎdulescu, Vicenţiu, 1958-
The nonlinear eigenvalue problem ▫$$-{\rm div}[(a_1(|\nabla u|) + a_2(|\nabla u|))\nabla u] = \lambda|u|^{q(x)-2}u\quad \text{in } \Omega,$$▫ ▫$$u=0 \quad \text{on } \partial\Omega,$$▫ is studied ...here in the framework of Orlicz-Sobolev spaces, where ▫$\Omega$▫ is a bounded domain in ▫$\Bbb R^N, a_i\:(0,\infty) \to \Bbb R$▫, ▫$i=1,2$▫, are such that ▫$\phi_i(t)$▫ defined as ▫$a_i(t)t$▫ if ▫$t\ne0$▫ and 0 if ▫$t=0$▫ are odd, increasing homeomorphisms from ▫$\Bbb R$▫ onto ▫$\Bbb R$▫ and ▫$q\:\Omega \to (0,\infty)$▫ is continuous. Problems of this kind, with variable exponents, have been treated mainly for the particular case of the ▫$p$▫-Laplacian ▫$-\Delta_p$▫ by different authors. Here it is proved under suitable assumptions on ▫$a_i$▫ and ▫$q$▫ that there is ▫$\lambda_1 > 0$▫, given by some kind of (non-homogeneous) Rayleigh quotient, such that any ▫$\lambda > \lambda_1$▫ is an eigenvalue (in the sense that there is a nontrivial weak solution in the Orlicz-Sobolev space). Moreover, for some ▫$\lambda_0 < \lambda_1$▫, there is no solution on ▫$(0,\lambda_0)$▫. It is anopen question if ▫$\lambda_1 = \lambda_0$▫ or not.
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