Existence and multiplicity results for double phase problems with variable exponents

The double phase operator is a differential operator that finds applications in several fields, among the others elasticity theory, biophysics, quantum physics, chemical reactions. The aim of this thesis is to present existence and multiplicity results for nonlinear differential equations involving the double phase operator with variable exponents, under different boundary conditions. In addition, the problems considered are parameter-dependent and an interval of parameters for which every problem admits solutions is also provided. The investigation is based on variational methods and precisely critical point theory; indeed, the main tools are the Mountain Pass theorem due to A. Ambrosetti e P. Rabinowitz, a two critical points theorem established by G. Bonanno and G. D’Aguì and the Nehari manifold method.The first problem considered is a Dirichlet double phase problem where if the nonlinearity has a subcritical growth and verifies a superlinear condition, the existence of two nontrivial weak solutions with opposite energy sign is established. These results are contained in [6], in collaboration with G. Bonanno, G. D’Aguì and P. Winkert.Then, a Robin double phase problem with critical growth on the boundary is studied. In particular, if the nonlinear term has a subcritical growth and satisfies the classical Ambrosetti-Rabinowitz condition, the existence of two nontrivial weak solutions with opposite energy sign is guaranteed. These results are presented in [4], in collaboration with V. Morabito.Finally, the last part of the thesis is dedicated to the study of a nonlinear double phase problem with nonlinear Neumann boundary condition. Under very general assumptions on the nonlinearity, the existence of three nontrivial weak solutions is obtained. Specifically, a solution is nonnegative, another one is nonpositive and the third one is sign-changing with exactly two nodal domains. These results are obtained in [5], in collaboration with Á. Crespo-Blanco, P. Pucci and P. Winkert.