Electron. J. Diff. Eqns., Vol. 2007(2007), No. 93, pp. 1-47.

Variational and topological methods for operator equations involving duality mappings on Orlicz-Sobolev spaces

George Dinca, Pavel Matei

Abstract:
Let $a:\mathbb{R}\to \mathbb{R}$ be a strictly increasing odd continuous function with $\lim_{t\to +\infty }a(t)=+\infty $ and $A(t)=\int_{0}^{t}a(s)\,ds$, $t\in \mathbb{R}$, the N-function generated by a. Let $\Omega $ be a bounded open subset of $\mathbb{R}^{N}$, $N\geq 2$, $T[u,u]$ a nonnegative quadratic form involving the only generalized derivatives of order m of the function $u\in W_{0}^{m}E_{A}(\Omega )$ and $g_{\alpha }:\Omega\times\mathbb{R}\to\mathbb{R}$, $| \alpha | <m$, be Caratheodory functions.
We study the problem
$$\displaylines{ J_{a}u=\sum_{| \alpha | <m}(-1)^{| \alpha | }D^{\alpha }g_{\alpha }(x,D^{\alpha }u) \quad\hbox{in }\Omega , \cr D^{\alpha }u=0\hbox{ on }\partial \Omega , | \alpha | \leq m-1, }$$
where $J_{a}$ is the duality mapping on $ \big(W_{0}^{m}E_{A}(\Omega ),\| \cdot \| _{m,A}\big) $, subordinated to the gauge function a (given by (1.5) and
$$ \| u\| _{m,A}=\| \sqrt{T[u,u]}\| _{(A)}, $$
$\| \cdot \| _{(A)}$ being the Luxemburg norm on $E_{A}(\Omega )$.
By using the Leray-Schauder topological degree and the mountain pass theorem of Ambrosetti and Rabinowitz, the existence of nontrivial solutions is established. The results of this paper generalize the existence results for Dirichlet problems with p-Laplacian given in [12] and [13].

Submitted June 4, 2007. Published June 21, 2007.
Math Subject Classifications: 35B38, 35B45, 47J30, 47H11.
Key Words: A priori estimate; critical points; Orlicz-Sobolev spaces; Leray-Schauder topological degree; Duality mapping; Nemytskij operator; Mountain Pass Theorem.

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George Dinca
Faculty of Mathematics and Computer Science, University of Bucharest
14, Academiei Str., 010014 Bucharest, Romania
email: dinca@fmi.unibuc.ro
Pavel Matei
Faculty of Mathematics and Computer Science, University of Bucharest
14, Academiei Str., 010014 Bucharest, Romania
email: pavel.matei@gmail.com

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