Mixed double phase equations with local and nonlocal operators

Abstract

In this paper, we study a new class of mixed double phase problems that combine local and nonlocal operators. We consider two different models. The first model is driven by the fractional $p$-Laplacian together with a local double phase operator, while the second model involves the local $p$-Laplacian coupled with a fractional double phase operator. In order to describe the interaction between local and nonlocal effects within the double phase framework, we introduce an appropriate variational setting based on classical and fractional Musielak-Orlicz Sobolev spaces. Within this setting, we establish several existence and multiplicity results for weak solutions by means of variational and topological techniques. In particular, for the problem driven by the fractional $p$-Laplacian and a local double phase operator, we prove the existence of a nonnegative solution using the Nehari manifold method in the presence of concave-convex nonlinearities. We also investigate the associated Brezis-Nirenberg type problem and obtain the existence of infinitely many solutions via genus theory. For the problem governed by the local $p$-Laplacian and a fractional double phase operator, we show the existence of at least two nontrivial constant sign solutions by exploiting the variational structure of the associated energy functional. Furthermore, in the subcritical case, we prove the existence of a least energy sign-changing solution by combining the Poincar\'{e}-Miranda existence theorem with the quantitative deformation lemma.