On the superlinear Kirchhoff problem involving the double phase operator with variable exponents

The paper deals with the following Kirchhoff-double phase problem mL(u)D(u)=|u|p(x)-2u+b(x)|u|q(x)-2uinΩ,mL(u)|∇u|p(x)-2u+b(x)|∇u|q(x)-2u·ν=λg(x,u)on∂Ω,where L(u)=∫Ω(1p(x)|∇u|p(x)+b(x)q(x)|∇u|q(x))dxand Dis the double phase operator with variable exponents. The goal is to determine the precise positive interval of λfor which the above problem admits at least two nontrivial weak solutions without assuming the Ambrosetti–Rabinowitz condition. Next, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain Theorem with Cerami condition.