On the regularity theory for mixed anisotropic and nonlocal p-Laplace equations and its applications to singular problems

Abstract

We establish existence results for a class of mixed anisotropic and nonlocal p-Laplace equations with singular nonlinearities. We consider both constant and variable singular exponents. Our argument is based on an approximation method. To this end, we also discuss the necessary regularity properties of weak solutions of the associated non-singular problems. More precisely, we obtain local boundedness of subsolutions, the Harnack inequality for solutions and the weak Harnack inequality for supersolutions.

Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems

“…In the anisotropic case, such results can be found in [49], where Picone identity played a crucial role and in the nonlocal setting, we refer to [26]. In the mixed case, see [39,40]. Here, we show that subsolution estimates can be derived using the Picone identity from Theorem 1.13.…”

“…When g ≡ 0 and ǫ = 1, higher Hólder regularity is obtained in [25,43] assuming that H(x) = |x|. Further, when g ≡ 0 and ǫ = 1, weak Harnack inequality for (3.13) is proved in [40] for H(x) = |x| and extended to any Finsler-Minkowski norm H in [39] for any 1 < p < ∞. Further, recently, to study a related singular critical problem, when p = 2, ǫ ∈ (0, 1], 0 < s < 1 and H(x) = |x|, weak Harnack inequality is established in [16,Proposition 3.3].…”

“…Below, we prove a linear relationship between the solutions of the system of equations. For the study of mixed singular equations, see [5,39,44] and the references therein.…”

“…To the best of our knowledge, most of our results are valid even for p = 2, that is for the operator −∆u + (−∆) s . Regarding the anisotropic mixed equation, as far we are aware, some regularity results are recently developed in [42,39,56,57] and also related eigenvalue and extremal problems are studied, see [38,45]. We also refer to [4,40] and the references therein.…”