L∞(Ω) a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity

Abstract

We consider a semilinear boundary value problem −Δu =f(x,u), in Ω, with Dirichlet boundary conditions, where Ω ⊂ RN with N > 2, is a bounded smooth domain, and f is a Carathéodory function, superlinear and subcritical at infinity. We provide L∞(Ω) a priori estimates for weak solutions in terms of their L2∗ (Ω)-norm, where 2*= 2N/N-2 is the critical Sobolev exponent. To establish our results, we do not assume any restrictions on the sign of the solutions, or on the non-linearity. Our approach is based on Gagliardo–Nirenberg and Caffarelli–Kohn–Nirenberg interpolation inequalities. Finally, we state sufficient conditions for having H01(Ω) uniform a priori bounds for non-negative solutions, so finally we provide suficient conditions for having L∞(Ω) uniform a priori bounds, which holds roughly speaking for superlinear and subcritical non-linearities.