Abstract
We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. Using the Nehari manifold, we show that the problem has at least three nontrivial bounded solutions: nodal, positive and by the symmetry of the behaviour at +∞ and −∞ also negative.
Highlights
Let Ω ⊆ RN be a bounded domain with a smooth boundary ∂Ω
We study the following double phase problem (P)
If a ∈ L∞(Ω) \ {0}, a(z) 0 for almost all z ∈ Ω and 1 < r < +∞, by ∆ra we denote the weighted r-Laplacian differential operator defined by ∆rau = div (a(z)|Du|r−2Du)
Summary
Let Ω ⊆ RN be a bounded domain (that is, a bounded connected set in the Ndimensional Euclidean space) with a smooth boundary ∂Ω. The unbalanced growth of θ(z, ·) requires the use of Musielak–Orlicz spaces for the treatment of problem (P) The interest for this kind of problems, was revived recently with the work of Mingione and coworkers who produced interesting interior regularity results for local minimizers of such functionals. We mention the very recent work of Ragusa–Tachikawa [7] who extended the interior regularity results to anisotropic double phase functionals. The use of the Nehari manifold helps us overcome the difficulties that originate from the fact that for double phase problems we have no global regularity theory and so many of the tools and techniques of “balanced” problems cannot be used. We mention the two recent informative survey articles by Mingione–Radulescu [1] and Radulescu [22]
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
