An existence result for singular Finsler double phase problems

Abstract

In this paper, we study a class of singular double phase problems defined on Minkowski spaces in terms of Finsler manifolds and with right-hand sides that allow a certain type of critical growth for such problems. Under very general assumptions and based on variational tools we prove the existence of at least one nontrivial weak solution for such a problem. This is the first work dealing with a Finsler double phase operator even in the nonsingular case.

On an anisotropic p-Laplace equation with variable singular exponent

“…As far as we are aware, Finsler p-Laplace equations with singular nonlinearities have been very less understood. We refer to the recent articles by Biset-Mebrate-Mohammed [11], Farkas-Winkert [25] and Farkas-Fiscella-Winkert [24].…”

Singular Finsler double phase problems with nonlinear boundary condition

“…has been shown by the first and the third author in [26]. The current paper can be seen as a nontrivial extension of the one in [26] to the case of a nonlinear boundary condition including critical growth.…”

“…has been shown by the first and the third author in [26]. The current paper can be seen as a nontrivial extension of the one in [26] to the case of a nonlinear boundary condition including critical growth. In particular, we are able to cover the situation when 1 < p < 2 and/or 1 < q < 2, which has not been considered in [26] where 2 ≤ p < q.…”

On an anisotropic double phase problem with singular and sign changing nonlinearity

“…Although it is worth mentioning that anisotropic singular problem is very less understood. When a(x) = 0, singular anisotropic problems is studied in Biset-Mebrate-Mohammed [3], Farkas-Winkert [11] and Farkas-Fiscella-Winkert [10], Bal-Garain-Mukherjee [1]. To the best of our knowledge, in the double phase context, anisotropic singular problems has been first discussed in Farkas-Winkert [12], where the authors proved existence of one weak solution in the critical case.…”