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Nonhomogeneous boundary value problems in Orlicz-Sobolev spaces
Mihǎilescu, Mihai ; Rǎdulescu, Vicenţiu, 1958-
We study the nonlinear Dirichlet problem ▫$$-{\mathrm div}(log(1+|\nabla u|^q)|\nabla u|^{p-2} \nabla u) = -\lambda|u|^{p-2}u+|u|^{r-2}u\quad \text{in} \;\Omega,$$▫ ▫$$u = 0\quad \text{on} \; ...\partial\Omega,$$▫ where ▫$\Omega$▫ is a bounded domain in ▫$\Bbb R^N$▫ with smooth boundary, while ▫$p$▫, ▫$q$▫ and ▫$r$▫ are real numbers satisfying ▫$p, q>1$▫, ▫$p+q < \min\{N,r\}$▫, ▫$r < (Np-N+p)/(N-p)$▫. The main result of this note establishes that for any ▫$\lambda>0$▫ this boundary value problem has infinitely many solutions in the Orlicz-Sobolev space ▫$W^1_0L_\Phi(\Omega)$▫, where ▫$$\Phi(t) = \int^t_0s|s|^{p-2}\log(1+|s|^q)ds.$$▫
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