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We consider nonlinear Neumann problems driven by the p(z)-Laplacian differential operator and with a p-superlinear reaction which does not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). In the process, we also prove two results of independent interest. The first is about the L ∞ -boundedness of the weak solutions. The second relates W 1, p(z) and C 1 local minimizers. Mathematics Subject Classification (2000)

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In this paper we consider quasilinear elliptic equations with double phase phenomena and a reaction term depending on the gradient. Under quite general assumptions on the convection term we prove the existence of a weak solution by applying the theory of pseudomonotone operators. Imposing some linear conditions on the gradient variable the uniqueness of the solution is obtained.2010 Mathematics Subject Classification. 35J15, 35J62, 35J92, 35P30.

We consider a double phase problems with unbalanced growth and a superlinear reaction, which need not satisfy the Ambrosetti–Rabinowitz condition. Using variational tools and the Nehari method, we show that the Dirichlet problem has at least three nontrivial solutions, a positive solution, a negative solution and a nodal solution. The nodal solution has exactly two nodal domains.

In this paper we study implicit obstacle problems driven by a nonhomogenous differential operator, called double phase operator, and a multivalued term which is described by Clarke’s generalized gradient. Based on a surjectivity theorem for multivalued mappings, Kluge’s fixed point principle and tools from nonsmooth analysis, we prove the existence of at least one solution.

We study parametric double phase problems involving superlinear nonlinearities without supposing any growth condition. Based on truncation and comparison methods the existence of two constant sign solutions is shown provided the parameter is larger than the first eigenvalue of the p-Laplacian. As a result of independent interest we prove a priori estimates for solutions for a general class of double phase problems with convection term.2010 Mathematics Subject Classification. 35J15, 35J62, 35J92, 35P30.

We consider nonlinear Neumann problems driven by p-Laplacian-type operators which are not homogeneous in general. We prove an existence and a multiplicity result for such problems. In the existence theorem, we assume that the right hand side nonlinearity is p-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. In the multiplicity result, when specialized to the case of the p-Laplacian, we allow strong resonance at infinity and resonance at 0.