On a Dirichlet Problem with Generalizedp(x)-Laplacian and Some Applications
Marek Galewski · Numerical Functional Analysis and Optimization · 2007
We consider the existence of solutions to the Dirichlet problem in case the nonlinearity satisfies local growth conditions, where denotes the pseudo-Laplacian with p(x)-growth. We introduce a dual variational method and provide duality relations for both action and dual action functionals that later results in the necessary and sufficient conditions for the existence of a solution. Based on these results, we obtain a sufficient condition for stability of solutions. Finally, we show that certain optimal control problem with state constraints admits a solution.