WEAK SOLVABILITY OF NONLINEAR ELLIPTIC EQUATIONS INVOLVING VARIABLE EXPONENTS

We are concerned with the study of the existence and multiplicity of solutions for Dirichlet boundary value problems, involving the (p(m), q(m))− equation and the nonlinearity is superlinear but does not fulfil the Ambrossetti-Rabinowitz condition in the framework of Sobolev spaces with variable exponents in a complete manifold. The main results are proved using the mountain pass theorem and Fountain theorem with Cerami sequences. Moreover, an example of a (p(m), q(m)) equation that highlights the applicability of our theoretical results is also provided.