Singular Finsler double phase problems with nonlinear boundary condition

Abstract

In this paper we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have 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.

Combined effects of singular and superlinear nonlinearities in singular double phase problems in $\mathbb{R}^N$

“…in Ω, with u = 0 on ∂Ω is shown by applying variational tools, while (R N , F ) stands for a Minkowski space. This work was extended to singular Finsler problems with nonlinear boundary condition by Farkas-Fiscella-Winkert [17]. In all these works, the domain is bounded, only the existence of one weak solution is shown and the methods are different from ours.…”

Parametric superlinear double phase problems with singular term and critical growth on the boundary

“…Based on the new equivalent norm in W 1,H (Ω) we were able to suppose critical growth on the boundary of Ω. To the best of our knowledge there is only one paper concerning singular double phase problems with nonlinear boundary condition, namely the paper of Farkas-Fiscella-Winkert [13] who studied the problem − div(A(u)) + u p−1 + µ(x)u q−1 = u p * −1 + λ u γ−1 + g 1 (x, u)…”