ON THE NONLINEAR DIRICHLET PROBLEM WITH p(i)-LAPLACIAN

for A; = 0 ,1 , . . . and with suitable assumptions on V which are valid for the p(x)Laplacian operator. Following some ideas from [11] we construct a dual variational method which applies to more general type of nonlinearities than those that are subject to a Palais-Smale type condition. We relate critical values and critical points to the action functional for which (1.1) is the Euler-Lagrange equation and the dual action functional (introduced in the paper and different from the Clarke dual functional) on specially constructed subsets of their domains. A variational approach concerning existence results for a class of problems involving critical and subcritical growth is shown in [2]. Critical point theory in certain Sobolev spaces is used to obtain existence and multiplicity results in sublinear and superlinear cases as well [5], where the problem is written as