A Contribution to the Theory of Quasilinear Elliptic Equations and Application to the Minimization of Integral Functionals

Abstract

The aim of this work is to study the existence of solutions of quasilinear elliptic problems of the type

$$\left\{\begin{array}{ll}-{\rm div}([a(x) + |u|^q] \nabla u) + b(x)u|u|^{p-1}|\nabla u|^2 = f(x), & {\rm in}\,\Omega;\\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \; u = 0, & \,{\rm on}\,\partial\Omega. \end{array}\right.$$

Then we study the minimization of integral functional of the type

$$J(v) = \frac{1}{2} \int\limits_\Omega [a(x)+|v|^r]|\nabla v|^2 - \int\limits_\Omega fv,\qquad \qquad \qquad\qquad\qquad\qquad\qquad (0.1)$$

with ${{f \in {L^m}(\Omega)}}$ . Since we can have ${{m < \frac {2N}{N+2}}}$ , the study of minimization with nonregular data f (i.e. ${f \notin W^{-1,2}(\Omega)}$) will be possible.