Global $$C^{1,\alpha }$$ C 1 , α regularity and existence of multiple solutions for singular p(x)-Laplacian equations

Anisotropic singular double phase Dirichlet problems

“…We start by showing that every weak solution of a general anisotropic singular problem is essentially bounded. This property extends to the anisotropic singular framework results obtained by Fan and Zhao [16], Giacomoni, Schindler and Takač [25], Byun and Ko [11]. So, we consider the following anisotropic Dirichlet problem…”

“…An alternative approach, can be based on estimates of the Ladyzhenskaya-Uraltseva type (see [31]). This method was used by Papageorgiou and Rȃdulescu [39] (for problems driven by a general isotropic nonhomogeneous differential operator) and by Byun and Ko [11] (anisotropic problems driven by the p(z)-Laplacian with singular terms). We mention also the work of Acerbi and Mingione [1], who obtained local estimates for the gradient Du(•) (Calderon-Zygmund type estimates).…”

“…The starting point of our work is the recent paper of Byun and Ko [11], which deals with anisotropic equations driven by the p(z)-Laplacian and with a reaction of the form λu −η(z) + u q(z)−1 , where q ∈ C 1 (Ω) and p + < q(z) for all z ∈ Ω. The hypothesis on the exponent p(•) is stronger and it is required that p ∈ C 1 (Ω) and it satisfies a kind of directional monotonicity condition (see hypothesis (p M ) in [11]); this condition is motivated by the work of Fan, Zhang and Zhao [15]. To the best of our knowledge, the work of Byun and Ko [11] is the first one on anisotropic singular problems.…”

“…The hypothesis on the exponent p(•) is stronger and it is required that p ∈ C 1 (Ω) and it satisfies a kind of directional monotonicity condition (see hypothesis (p M ) in [11]); this condition is motivated by the work of Fan, Zhang and Zhao [15]. To the best of our knowledge, the work of Byun and Ko [11] is the first one on anisotropic singular problems.…”

Singular Anisotropic Problems with Competition Phenomena

“…In the past, anisotropic singular equations were studied without the presence of the concave term λu τ (z)−1 and with a superlinear perturbation which is positive. We refer to the works of Byun-Ko [2] and Saoudi-Ghanmi [21]. Both deal with equations driven by the anisotropic p-Laplacian only.…”