L^ a-priori estimates for subcritical p-laplacian equations with a Carathéodory non-linearity

Abstract Let us consider a quasi-linear boundary value problem - ₚ u= f (x, u), - Δ p u = f (x, u), in, Ω, with Dirichlet boundary conditions, where RN Ω ⊂ R N, with p p N, is a bounded smooth domain strictly convex, and the non-linearity f is a Carathéodory function p -super-linear and subcritical. We provide L^ L ∞ a priori estimates for weak solutions, in terms of their L^p^* L p ∗ -norm, where p^*= NpN-p\ p ∗ = Np N - p is the critical Sobolev exponent. No hypotheses on the sign of the solutions, neither of the non-linearities are required. This method is based in elliptic regularity for the p -Laplacian combined either with Gagliardo–Nirenberg or with Caffarelli–Kohn–Nirenberg interpolation inequalities. By a subcritical non-linearity we mean, for instance, |f (x, s) | |x|^- \, f (s), | f (x, s) | ≤ | x | - μ f ~ (s), where (0, p), μ ∈ (0, p), and f (s) /|s|^p ^*-1 0 f ~ (s) / | s | p μ ∗ - 1 → 0 as |s| | s | → ∞, here p^*: =p (N-) N-p p μ ∗: = p (N - μ) N - p is the critical Hardy–Sobolev exponent. Our non-linearities includes non-power non-linearities. In particular we prove that when f (x, s) =|x|^- \, |s|^p^* -₂s (e+|s|) ^, f (x, s) = | x | - μ | s | p μ ∗ - 2 s log (e + | s |) α, with 1, p), μ ∈ [ 1, p), then, for any >0 ε > 0 there exists a constant C_ >0 C ε > 0 such that for any solution u H¹₀ () u ∈ H 0 1 (Ω), the following holds aligned [ (e+ u) ^ C_ \, (1+ u ₏^*) ^\, (p^* -p) (1+) \, , aligned log (e + ‖ u ‖ ∞) α ≤ C ε (1 + ‖ u ‖ p ∗) (p μ ∗ - p) (1 + ε), where C_ C ε is independent of the solution u.