This paper establishes interior Morrey regularity for stable solutions of the $ p $-Laplace equation $ -\Delta_p u = h(u) $ in $ B_1\subset\mathbb{R}^d $, where $ h $ belongs to $ C^1(\mathbb{R}) $. The results cover the endpoint case left unresolved by Cabré et al. Moreover, for positive nonlinearities $ h $, a global Morrey estimate was obtained for stable solutions of a Dirichlet problem on any smooth, bounded, strictly convex domain.
Citation: Meng Qu, Zhi Wang. Endpoint Morrey interior regularity of stable solutions to the $ p $-Laplace equation[J]. Electronic Research Archive, 2026, 34(7): 4765-4776. doi: 10.3934/era.2026210
This paper establishes interior Morrey regularity for stable solutions of the $ p $-Laplace equation $ -\Delta_p u = h(u) $ in $ B_1\subset\mathbb{R}^d $, where $ h $ belongs to $ C^1(\mathbb{R}) $. The results cover the endpoint case left unresolved by Cabré et al. Moreover, for positive nonlinearities $ h $, a global Morrey estimate was obtained for stable solutions of a Dirichlet problem on any smooth, bounded, strictly convex domain.
| [1] |
D. Castorina, M. Sanchón, Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities, J. Eur. Math. Soc., 17 (2015), 2949–2975. https://doi.org/10.4171/JEMS/576 doi: 10.4171/JEMS/576
|
| [2] |
A. Farina, B. Sciunzi, E. Valdinoci, On a Poincaré type formula for solutions of singular and degenerate elliptic equations, Manuscripta Math., 132 (2010), 335–342. https://doi.org/10.1007/s00229-010-0349-1 doi: 10.1007/s00229-010-0349-1
|
| [3] | L. Dupaigne, Stable Solutions of Elliptic Partial Differential Equations, $1^{st}$ edition, Chapman and Hall/CRC, 2011. https://doi.org/10.1201/b10802 |
| [4] |
X. Cabré, Boundedness of stable solutions to semilinear elliptic equations: A survey, Adv. Nonlinear Stud., 17 (2017), 355–368. https://doi.org/10.1515/ans-2017-0008 doi: 10.1515/ans-2017-0008
|
| [5] |
K. Wang, Recent progress on stable and finite Morse index solutions of semilinear elliptic equations, Electron. Res. Arch., 29 (2021), 3805–3816. https://doi.org/10.3934/era.2021062 doi: 10.3934/era.2021062
|
| [6] |
X. Cabré, A. Figalli, X. Ros-Oton, J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math., 224 (2020), 187–252. https://doi.org/10.4310/acta.2020.v224.n2.a1 doi: 10.4310/acta.2020.v224.n2.a1
|
| [7] |
D. D. Joseph, T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Ration. Mech. Anal., 49 (1973), 241–269. https://doi.org/10.1007/BF00250508 doi: 10.1007/BF00250508
|
| [8] |
F. Peng, Y. R. Zhang, Y. Zhou, Optimal regularity and the Liouville property for stable solutions to semilinear elliptic equations in $\mathbb{R}^n$ with $n\geq10$, Anal. PDE, 17 (2024), 3335–3353. https://doi.org/10.2140/apde.2024.17.3335 doi: 10.2140/apde.2024.17.3335
|
| [9] |
X. Cabré, P. Miraglio, M. Sanchón, Optimal regularity of stable solutions to nonlinear equations involving the $p$-Laplacian, Adv. Calculus Var., 15 (2022), 749–785. https://doi.org/10.1515/acv-2020-0055 doi: 10.1515/acv-2020-0055
|
| [10] |
X. Cabré, F. Charro, The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space, Adv. Math., 380 (2021), 107592. https://doi.org/10.1016/j.aim.2021.107592 doi: 10.1016/j.aim.2021.107592
|
Meng Qu, Zhi Wang. Endpoint Morrey interior regularity of stable solutions to the $ p $-Laplace equation[J]. Electronic Research Archive, 2026, 34(7): 4765-4776. doi: 10.3934/era.2026210
DownLoad: