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

Abstract We consider a semilinear boundary value problem %$ -\Delta u= f(x,u),%$ in %$\Omega ,%$ with Dirichlet boundary conditions, where %$\Omega \subset {\mathbb {R}}^N %$ with %$N> 2,%$ is a bounded smooth domain, and f is a Carathéodory function, superlinear and subcritical at infinity. We p...
Ausführliche Beschreibung

Abstract We consider a semilinear boundary value problem %$ -\Delta u= f(x,u),%$ in %$\Omega ,%$ with Dirichlet boundary conditions, where %$\Omega \subset {\mathbb {R}}^N %$ with %$N> 2,%$ is a bounded smooth domain, and f is a Carathéodory function, superlinear and subcritical at infinity. We provide %$L^\infty (\Omega )%$ a priori estimates for weak solutions in terms of their %$L^{2^*}(\Omega )%$-norm, where %$2^*=\frac{2N}{N-2}\ %$ is the critical Sobolev exponent. In particular, our results also apply to %$f(x,s)=a(x)\,\frac{|s|^{2^*_{N/r}-2}s}{\big [\log (e+|s|)\big ]^\beta }\,%$, where %$a\in L^r(\Omega )%$ with %$N/2<r\le \infty %$, and %$2_{N/r}^*:=2^*\left( 1-\frac{1}{r}\right) %$. Assume %$N/2<r\le N%$. We show that for any %$\varepsilon >0%$ there exists a constant %$C_\varepsilon >0%$ such that for any solution %$u\in H^1_0(\Omega )%$, the following holds: [log(e+‖u‖∞)]β≤Cε(1+‖u‖2∗)(2N/r∗-2)(1+ε).%$\begin{aligned} \Big [\log \big (e+\Vert u\Vert _{\infty }\big )\Big ]^\beta \le C _\varepsilon \, \Big (1+\Vert u\Vert _{2^*}\Big )^{\, (2^*_{N/r}-2)(1+\varepsilon )}. \end{aligned}%$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 %$H_0^1(\Omega )%$ uniform a priori bounds for non-negative solutions, so finally we provide suficient conditions for having %$L^\infty (\Omega )%$ uniform a priori bounds, which holds roughly speaking for superlinear and subcritical non-linearities. Ausführliche Beschreibung

Autor*in:	Pardo, Rosa [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2023

Schlagwörter:	A priori estimates a priori bounds singular weights subcritical problems

Anmerkung:	© The Author(s) 2023

Übergeordnetes Werk:	Enthalten in: Journal of fixed point theory and applications - Cham (ZG) : Springer International Publishing AG, 2007, 25(2023), 2 vom: 06. Feb.
Übergeordnetes Werk:	volume:25 ; year:2023 ; number:2 ; day:06 ; month:02

Links:	Volltext

DOI / URN:	10.1007/s11784-023-01048-w

Katalog-ID:	SPR049260669