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

Abstract

AbstractLet us consider a quasi-linear boundary value problem $$ -\Delta _p u= f(x,u),$$ - Δ p u = f ( x , u ) , in $$\Omega ,$$ Ω , with Dirichlet boundary conditions, where $$\Omega \subset \mathbb {R}^N $$ 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 $$L^\infty $$ L ∞ a priori estimates for weak solutions, in terms of their $$L^{p^*}$$ L p ∗ -norm, where $$p^*= \frac{Np}{N-p}\ $$ 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, $$|f(x,s)|\le |x|^{-\mu }\, \tilde{f}(s),$$ | f ( x , s ) | ≤ | x | - μ f ~ ( s ) , where $$\mu \in (0,p),$$ μ ∈ ( 0 , p ) , and $$\tilde{f}(s)/|s|^{p_{\mu }^*-1}\rightarrow 0$$ f ~ ( s ) / | s | p μ ∗ - 1 → 0 as $$|s|\rightarrow \infty $$ | s | → ∞ , here $$p^*_{\mu }:=\frac{p(N-\mu )}{N-p}$$ 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 $$f(x,s)=|x|^{-\mu }\,\frac{|s|^{p^*_{\mu }-2}s}{\big [\log (e+|s|)\big ]^\alpha },$$ f ( x , s ) = | x | - μ | s | p μ ∗ - 2 s [ log ( e + | s | ) ] α , with $$\mu \in [1,p),$$ μ ∈ [ 1 , p ) , then, for any $$\varepsilon >0$$ ε > 0 there exists a constant $$C_\varepsilon >0$$ C ε > 0 such that for any solution $$u\in H^1_0(\Omega )$$ u ∈ H 0 1 ( Ω ) , the following holds $$\begin{aligned} \Big [\log \big (e+\Vert u\Vert _{\infty }\big )\Big ]^\alpha \le C_\varepsilon \, \Big (1+\Vert u\Vert _{p^*}\Big )^{\, (p^*_{\mu }-p)(1+\varepsilon )}\,, \end{aligned}$$ [ log ( e + ‖ u ‖ ∞ ) ] α ≤ C ε ( 1 + ‖ u ‖ p ∗ ) ( p μ ∗ - p ) ( 1 + ε ) , where $$C_\varepsilon $$ C ε is independent of the solution u.