Lack of interior $L^q$ bounds for stable solutions to elliptic equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912976540270592 |
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| author | Villegas, Salvador |
| author_facet | Villegas, Salvador |
| contents | We consider stable solutions of semilinear elliptic equations of the form $-Δu=f(u)$ in a bounded domain $Ω\subset\mathbb{R}^N$. In a well-known paper \cite{cfrs}, Cabré, Figalli, Ros-Oton and Serra obtained interior estimates for the $W^{1,2}$-norm of $u$ in terms of the $L^1$-norm of $u$ and proved interior Hölder regularity for dimensions $N\leq 9$. All these results rely on the assumption that $f$ is nonnegative. We show that, for general nonlinearities $f\in C^\infty(\mathbb{R})$, it is impossible, in any dimension $N\geq 1$, to obtain an interior $L^q$ estimate in terms of the $L^p$-norm of $u$ whenever $1\leq p<q\leq \infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_20427 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Lack of interior $L^q$ bounds for stable solutions to elliptic equations Villegas, Salvador Analysis of PDEs 25J15, 35J61, 35B45 We consider stable solutions of semilinear elliptic equations of the form $-Δu=f(u)$ in a bounded domain $Ω\subset\mathbb{R}^N$. In a well-known paper \cite{cfrs}, Cabré, Figalli, Ros-Oton and Serra obtained interior estimates for the $W^{1,2}$-norm of $u$ in terms of the $L^1$-norm of $u$ and proved interior Hölder regularity for dimensions $N\leq 9$. All these results rely on the assumption that $f$ is nonnegative. We show that, for general nonlinearities $f\in C^\infty(\mathbb{R})$, it is impossible, in any dimension $N\geq 1$, to obtain an interior $L^q$ estimate in terms of the $L^p$-norm of $u$ whenever $1\leq p<q\leq \infty$. |
| title | Lack of interior $L^q$ bounds for stable solutions to elliptic equations |
| topic | Analysis of PDEs 25J15, 35J61, 35B45 |
| url | https://arxiv.org/abs/2603.20427 |