$$L^\infty (\Omega )$$ 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...
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Gespeichert in:
| Autor*in: |
Pardo, Rosa [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s) 2023 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of fixed point theory and applications - Springer International Publishing, 2007, 25(2023), 2 vom: 06. Feb. |
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| Übergeordnetes Werk: |
volume:25 ; year:2023 ; number:2 ; day:06 ; month:02 |
| Links: |
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| DOI / URN: |
10.1007/s11784-023-01048-w |
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| Katalog-ID: |
OLC213379784X |
|---|
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| 520 | |a 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. | ||
| 650 | 4 | |a A priori estimates | |
| 650 | 4 | |a a priori bounds | |
| 650 | 4 | |a singular weights | |
| 650 | 4 | |a subcritical problems | |
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10.1007/s11784-023-01048-w doi (DE-627)OLC213379784X (DE-He213)s11784-023-01048-w-p DE-627 ger DE-627 rakwb eng 510 VZ Pardo, Rosa verfasserin (orcid)0000-0003-1914-9203 aut $$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2023 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. A priori estimates a priori bounds singular weights subcritical problems Enthalten in Journal of fixed point theory and applications Springer International Publishing, 2007 25(2023), 2 vom: 06. Feb. (DE-627)548575800 (DE-600)2395316-0 (DE-576)340336587 1661-7738 nnns volume:25 year:2023 number:2 day:06 month:02 https://doi.org/10.1007/s11784-023-01048-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT AR 25 2023 2 06 02 |
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10.1007/s11784-023-01048-w doi (DE-627)OLC213379784X (DE-He213)s11784-023-01048-w-p DE-627 ger DE-627 rakwb eng 510 VZ Pardo, Rosa verfasserin (orcid)0000-0003-1914-9203 aut $$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2023 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. A priori estimates a priori bounds singular weights subcritical problems Enthalten in Journal of fixed point theory and applications Springer International Publishing, 2007 25(2023), 2 vom: 06. Feb. (DE-627)548575800 (DE-600)2395316-0 (DE-576)340336587 1661-7738 nnns volume:25 year:2023 number:2 day:06 month:02 https://doi.org/10.1007/s11784-023-01048-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT AR 25 2023 2 06 02 |
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10.1007/s11784-023-01048-w doi (DE-627)OLC213379784X (DE-He213)s11784-023-01048-w-p DE-627 ger DE-627 rakwb eng 510 VZ Pardo, Rosa verfasserin (orcid)0000-0003-1914-9203 aut $$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2023 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. A priori estimates a priori bounds singular weights subcritical problems Enthalten in Journal of fixed point theory and applications Springer International Publishing, 2007 25(2023), 2 vom: 06. Feb. (DE-627)548575800 (DE-600)2395316-0 (DE-576)340336587 1661-7738 nnns volume:25 year:2023 number:2 day:06 month:02 https://doi.org/10.1007/s11784-023-01048-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT AR 25 2023 2 06 02 |
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10.1007/s11784-023-01048-w doi (DE-627)OLC213379784X (DE-He213)s11784-023-01048-w-p DE-627 ger DE-627 rakwb eng 510 VZ Pardo, Rosa verfasserin (orcid)0000-0003-1914-9203 aut $$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2023 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. A priori estimates a priori bounds singular weights subcritical problems Enthalten in Journal of fixed point theory and applications Springer International Publishing, 2007 25(2023), 2 vom: 06. Feb. (DE-627)548575800 (DE-600)2395316-0 (DE-576)340336587 1661-7738 nnns volume:25 year:2023 number:2 day:06 month:02 https://doi.org/10.1007/s11784-023-01048-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT AR 25 2023 2 06 02 |
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10.1007/s11784-023-01048-w doi (DE-627)OLC213379784X (DE-He213)s11784-023-01048-w-p DE-627 ger DE-627 rakwb eng 510 VZ Pardo, Rosa verfasserin (orcid)0000-0003-1914-9203 aut $$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity 2023 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © The Author(s) 2023 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. A priori estimates a priori bounds singular weights subcritical problems Enthalten in Journal of fixed point theory and applications Springer International Publishing, 2007 25(2023), 2 vom: 06. Feb. (DE-627)548575800 (DE-600)2395316-0 (DE-576)340336587 1661-7738 nnns volume:25 year:2023 number:2 day:06 month:02 https://doi.org/10.1007/s11784-023-01048-w lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT AR 25 2023 2 06 02 |
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Enthalten in Journal of fixed point theory and applications 25(2023), 2 vom: 06. Feb. volume:25 year:2023 number:2 day:06 month:02 |
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| author |
Pardo, Rosa |
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510 VZ $$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity A priori estimates a priori bounds singular weights subcritical problems |
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ddc 510 misc A priori estimates misc a priori bounds misc singular weights misc subcritical problems |
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$$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity |
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$$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity |
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Journal of fixed point theory and applications |
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10.1007/s11784-023-01048-w |
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$$l^\infty (\omega )$$ a priori estimates for subcritical semilinear elliptic equations with a carathéodory non-linearity |
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$$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity |
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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. © The Author(s) 2023 |
| abstractGer |
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. © The Author(s) 2023 |
| abstract_unstemmed |
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. © The Author(s) 2023 |
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$$L^\infty (\Omega )$$ a priori estimates for subcritical semilinear elliptic equations with a Carathéodory non-linearity |
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