Nonlinear Boundary Value Problems via Minimization on Orlicz-Sobolev Spaces

Abstract

We develop arguments on convexity and minimization of energy functionals on Orlicz-Sobolev spaces to investigate existence of solution to the equation $\displaystyle -\mbox{div} (ϕ(|\nabla u|) \nabla u) = f(x,u) + h \mbox{in} Ω$ under Dirichlet boundary conditions, where $\Omega \subset {\mathbf{R}}^N$ is a bounded smooth domain, $\varphi :(0,\infty )⟶(0,\infty )$ is a suitable continuous function and $f:\Omega \times \mathbf{R}\to \mathbf{R}$ satisfies the Carathéodory conditions, while $h$ is a measure.

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