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An eigenvalue problem for generalized Laplacian in Orlicz-Sobolev spaces. (English) Zbl 0926.46030

Summary: Let \(m:[0,\infty)\to [0,\infty)\) be an increasing continuous function with \(m(t)= 0\) if and only if \(t= 0\), \(m(t)\to\infty\) as \(t\to\infty\) and \(\Omega\subset \mathbb{R}^N\) a bounded domain. In this note, we show that for every \(r>0\) there exists a function \(u_r\) solving the minimization problem \[ \inf\Biggl\{ \int_\Omega M(|\nabla u|) \Biggl| u\in W^1_0L_M(\Omega),\;\int_\Omega M(u)= r\Biggr\}, \] where \(M(t)= \int^{| t|}_0 m(s)ds\). Moreover, the function \(u_r\) is a weak solution to the corresponding Euler-Lagrange equation \[ -\text{div}\Biggl( {m(|\nabla u|)\over|\nabla u|} \nabla u\Biggr)= \lambda{m(| u|)\over| u|} u\quad \text{in }\Omega,\qquad u= 0\quad\text{on }\partial\Omega \] for some \(\lambda>0\). We emphasize that no \(\Delta_2\)-condition is needed for \(M\) or \(\overline M\); so the associated functionals are not continuously differentiable, in general.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47F05 General theory of partial differential operators
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