A Multiplicity Theorem for Superlinear Double Phase Problems

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.

Constant sign and nodal solutions for resonant double phase problems

“…In the past, most multiplicity results for double phase equations, assumed that the reaction is (p − 1)-superlinear. We mention the works of Deregowska-Gasinski-Papageorgiou [2], Gasinski-Papageorgiou [7], Gasinski-Winkert [8], Liu-Dai [12], Papageorgiou-Vetro-Vetro [21], Papageorgiou-Zhang [22]. Recently, Papageorgiou-Rădulescu-Zhang [20] and Papageorgiou-Pudelko-Rădulescu [17], developed the spectral properties of the weighted p-Lapiacian ∆ a p and proved multiplicity theorems for resonant problems.…”