Multiple solutions for a class of double phase problem without the Ambrosetti–Rabinowitz conditions

TLDR: In this paper, the existence and multiplicity of weak solutions for a class of the double phase problem − div ( | ∇ u | p − 2 ∆ ∆ u + a (x, u)) = λ f ( x, u ), in Ω, u = 0, on ∂ Ω, where N ≥ 2 and 1 p q N.

Abstract: In the present paper, in view of the variational approach, we consider the existence and multiplicity of weak solutions for a class of the double phase problem − div ( | ∇ u | p − 2 ∇ u + a ( x ) | ∇ u | q − 2 ∇ u ) = λ f ( x , u ) , in Ω , u = 0 , on ∂ Ω , where N ≥ 2 and 1 p q N . Firstly, by the Fountain and Dual Theorem with Cerami condition, we obtain some existence of infinitely many solutions for the above problem under some weaker assumptions on f . Secondly, we prove that this problem has at least one nontrivial solution for any parameter λ > 0 small enough, and also that the solution blows up, in the Sobolev norm, as λ → 0 + . Finally, by imposing additional assumptions on f , we establish the existence of infinitely many solutions by using Krasnoselskii’s genus theory for the above equation.