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In this paper we study quasilinear elliptic systems driven by variable exponent double phase operators involving fully coupled right-hand sides and nonlinear boundary conditions. The aim of our work is to establish an enclosure and existence result for such systems by means of trapping regions formed by pairs of sup- and supersolutions. Under very general assumptions on the data we then apply our result to get infinitely many solutions. Moreover, we also discuss the case when we have homogeneous Dirichlet boundary conditions and present some existence results for this kind of problem.

This issue of Discrete and Continuous Dynamical Systems-Series S focuses on the qualitative analysis of some concrete nonlinear problems, e.g., ordinary, partial differential equations, systems and inclusions. The ten contributions collected here give an overview on some very recent results on the existence, multiplicity and sign information of the solutions of a wide range of nonlinear differential problems involving different boundary value conditions and operators in divergence form. In our opinion, the synergy pointed out here between the classical nonlinear analysis methods, like the critical point theory, sub-super solutions methods, truncation and comparison techniques, Morse theory, set-valued analysis and so on, could represent a stimulating reason for reflection for all those researchers – pure or applied mathematicians, upper-level graduate students – who are interested in finding new perspectives of investigations on this field of research. The present volume is related to the topics of the Special Session 92: Variational, Topological and Set-Valued Methods for Nonlinear Problems, organized, together with Professor S. A. Marano, on the occasion of the last 11th AIMS Conference in Orlando, that we like to consider as a natural continuation of the special sessions held in the previous editions of AIMS Conferences, since 2010, by some of the authors involved in this project, as G. Bonanno, S. Carl and D. Motreanu. We wish to thank all the Colleagues who contributed to the realization of this issue of the DCDS-S and we acknowledge the important support of the past organizers to accomplish successfully this project. In particular, we wish to thank Prof. S. A. Marano and Prof. N. S. Papageorgiou to encourage us to realize this volume. Many thanks also to the AIMS, in the persons of Prof. Shouchuan Hu, for his work as member of the Scientific Commette of the AIMS Conferences, and of Prof. Alain Miranville, for his patience in assisting us during the editing process.

The existence of inffinitely many solutions for diffierential inclusions depending on two positive parameters and involving the p- biharmonic operator is established via variational methods.

We consider parametric Dirichlet problems driven by the sum of a Laplacian and a nonhomogeneous differential operator ( ( a , 2 ) -type equation) and with a reaction term which exhibits arbitrary polynomial growth and a nonlinear dependence on the parameter. We prove the existence of three distinct nontrivial smooth solutions for small values of the parameter, providing sign information for them: one is positive, one is negative and the third one is nodal.