Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 93, pp. 1-47.

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

Authors: George Dinca (Univ. of Bucharest, Romania)
         Pavel Matei  (Univ. of Bucharest, Romania)

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 Carath\'{e}odory functions.

 We  study the problem
 \begin{gather*}
 J_{a}u=\sum_{| \alpha | <m}(-1)^{| \alpha
 | }D^{\alpha  }g_{\alpha  }(x,D^{\alpha  }u)
 \quad\text{in }\Omega , \\
 D^{\alpha  }u=0\text{  on }\partial \Omega , | \alpha | \leq m-1,
 \end{gather*}
 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
 \begin{equation*}
 \| u\| _{m,A}=\| \sqrt{T[u,u]}\| _{(A)},
 \end{equation*}
 $\| \cdot \| _{(A)}$ 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.