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

By Bin Ge, Dejing Lv, Jian-Fang Lu. Nonlinear Analysis, 2019

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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.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/c75a2f429a3230af8ba7e28b44a18fda14a1092f