Singular Finsler Double Phase Problems with Nonlinear Boundary Condition

Csaba Farkas,Alessio Fiscella,Patrick Winkert

doi:10.1515/ans-2021-2143

Abstract

Abstract In this paper, we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques, we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces to the Euclidean norm, our work is the first one dealing with a singular double phase problem and nonlinear boundary condition.

Highlights

In this paper we consider singular Finsler double phase problems with nonlinear boundary condition
If F coincides with the Euclidean norm, that is, F (ξ) = Ni=1 |ξi|2 1/2 for ξ ∈ RN, (1.1) reduces to the usual double phase operator given by div |∇u|p−2∇u + μ(x)|∇u|q−2∇u
If μ ≡ 0 or infΩ μ > 0, (1.2) ( (1.1)) reduces to the (Finsler) p-Laplacian or the (Finsler) (q, p)-Laplacian, respectively. The study of such operators and corresponding energy functionals are motivated by physical phenomena, see, for example, the work of Zhikov [58] in order to describe models for strongly anisotropic materials

Summary

Introduction

In this paper we consider singular Finsler double phase problems with nonlinear boundary condition. The Finsler double phase operator is defined by div(A(u)) := div F p−1(∇u)∇F (∇u) + μ(x)F q−1(∇u)∇F (∇u) for u ∈ W 1,H(Ω) with W 1,H(Ω) being the Musielak-Orlicz Sobolev space and F is a positive homogeneous function such that F ∈ C∞(RN \{0}) and the Hessian matrix ∇2(F 2/2)(x) is positive definite for all x = 0. Anisotropic double phase operator, critical type exponent, existence results, Minkowski space, nonlinear boundary condition, singular problems.

Results

Conclusion