The Dirichlet problem with sublinear nonlinearities

Aleksandra Orpel¹

¹ Faculty of Mathematics University of /L/od/x Banacha 22 90-238 /L/od/x, Poland

Annales Polonici Mathematici, Tome 78 (2002) no. 2, pp. 131-140

Résumé

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.

Détail

DOI : 10.4064/ap78-2-4

Keywords: investigate existence solutions dirichlet problem differential inclusion mit delta partial x mit omega which generalized euler lagrange equation functional int mit omega frac nabla g develop duality theory formulate variational principle problem consequence duality derive variational principle minimizing sequences consider subquadratic infinity