$$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity

Abstract

AbstractWe consider a semilinear boundary value problem $$ -\Delta u= f(x,u),$$ - Δ u = f ( x , u ) , in $$\Omega ,$$ Ω , with Dirichlet boundary conditions, where $$\Omega \subset {\mathbb {R}}^N $$ Ω ⊂ R N with $$N> 2,$$ N > 2 , is a bounded smooth domain, and f is a Carathéodory function, superlinear and subcritical at infinity. We provide $$L^\infty (\Omega )$$ L ∞ ( Ω ) a priori estimates for weak solutions in terms of their $$L^{2^*}(\Omega )$$ L 2 ∗ ( Ω ) -norm, where $$2^*=\frac{2N}{N-2}\ $$ 2 ∗ = 2 N N - 2 is the critical Sobolev exponent. In particular, our results also apply to $$f(x,s)=a(x)\,\frac{|s|^{2^*_{N/r}-2}s}{\big [\log (e+|s|)\big ]^\beta }\,$$ f ( x , s ) = a ( x ) | s | 2 N / r ∗ - 2 s [ log ( e + | s | ) ] β , where $$a\in L^r(\Omega )$$ a ∈ L r ( Ω ) with $$N/2<r\le \infty $$ N / 2 < r ≤ ∞ , and $$2_{N/r}^*:=2^*\left( 1-\frac{1}{r}\right) $$ 2 N / r ∗ : = 2 ∗ 1 - 1 r . Assume $$N/2<r\le N$$ N / 2 < r ≤ N . We show that 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: $$\begin{aligned} \Big [\log \big (e+\Vert u\Vert _{\infty }\big )\Big ]^\beta \le C _\varepsilon \, \Big (1+\Vert u\Vert _{2^*}\Big )^{\, (2^*_{N/r}-2)(1+\varepsilon )}. \end{aligned}$$ [ log ( e + ‖ u ‖ ∞ ) ] β ≤ C ε ( 1 + ‖ u ‖ 2 ∗ ) ( 2 N / r ∗ - 2 ) ( 1 + ε ) . 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 $$H_0^1(\Omega )$$ H 0 1 ( Ω ) uniform a priori bounds for non-negative solutions, so finally we provide suficient conditions for having $$L^\infty (\Omega )$$ L ∞ ( Ω ) uniform a priori bounds, which holds roughly speaking for superlinear and subcritical non-linearities.