Existence and multiplicity results for double phase problem with nonlinear boundary condition

Abstract

In this paper, we consider the following double phase problem with a nonlinear boundary condition −div|∇u|p−2∇u+μ(x)|∇u|q−2∇u=f(x,u)−|u|p−2u−μ(x)|u|q−2uinΩ,|∇u|p−2∇u+μ(x)|∇u|q−2∇u⋅ν=g(x,u)on∂Ω.First of all, we prove the existence of a solution and infinitely many solutions for this problem with superlinear nonlinearity (without A–R condition). In addition, under the sublinear assumptions on f and g, we establish the existence of infinitely many solutions via Clark’s theorem for the aforementioned problem.