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Abstract

We investigate the existence of solutions of the Dirichlet problem for the differential inclusion $0\in {\mit\Delta} x(y)+\partial _{x}G(y,x(y))$ for a.e. $y\in {\mit\Omega} ,$ which is a generalized Euler–Lagrange equation for the functional $J(x)=\int_{\mit\Omega}\{\frac{1}{2}% |\nabla x(y)|^{2}-G(y,x(y))\}\,dy.$ We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of $J$. We consider the case when $G$ is subquadratic at infinity.