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

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Let us consider a quasi-linear boundary value problem -Δ ₚ u= f(x,u), - Δ p u = f ( x , u ) , in Ω , Ω , with Dirichlet boundary conditions, where Ω ⊂ RN Ω ⊂ R N , with $$p<N,$$ 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 Lp^* L p ∗ -norm, where p^*= Np/N-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|-μ\, f̃(s), | f ( x , s ) | ≤ | x | - μ f ~ ( s ) , where μ ∈ (0,p), μ ∈ ( 0 , p ) , and f̃(s)/|s|^pμ^*-1→ 0 f ~ ( s ) / | s | p μ ∗ - 1 → 0 as |s|→ ∞ | s | → ∞ , here p^*μ:=p(N-μ )/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|-μ\,|s|^p^*μ-2s [log (e+|s|) ]^α , f ( x , s ) = | x | - μ | s | p μ ∗ - 2 s [ log ( e + | s | ) ] α , with μ ∈ [1,p), μ ∈ [ 1 , p ) , then, for any ε >0 ε > 0 there exists a constant C_ε >0 C ε > 0 such that for any solution u∈ H¹₀(Ω ) u ∈ H 0 1 ( Ω ) , the following holds aligned [log (e+ u ∞ ) ]^α ≤ C_ε \, (1+ u p^* )^\, (p^*μ-p)(1+ε )\,, aligned [ log ( e + ‖ u ‖ ∞ ) ] α ≤ C ε ( 1 + ‖ u ‖ p ∗ ) ( p μ ∗ - p ) ( 1 + ε ) , where C_ε C ε is independent of the solution u .