# Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

Daniele Castorina; Manel Sanchón

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 11, page 2949-2975
- ISSN: 1435-9855

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topCastorina, Daniele, and Sanchón, Manel. "Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities." Journal of the European Mathematical Society 017.11 (2015): 2949-2975. <http://eudml.org/doc/277353>.

@article{Castorina2015,

abstract = {We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of $–\Delta _p u= g(u)$ in a smooth bounded domain $\Omega \subset \mathbb \{R\}^n$. In particular, we obtain new $L^r$ and $W^\{1,r\}$ bounds for the extremal solution $u^\star $ when the domain is strictly convex. More precisely, we prove that $u^\star \in L^\{\infty \}(\Omega )$ if $n\le p+2$ and $u^\star \in L^\{\frac\{np\}\{n-p-2\}\} (\Omega ) \cap W^\{1,p\}_0 (\Omega )$ if $n > p + 2$.},

author = {Castorina, Daniele, Sanchón, Manel},

journal = {Journal of the European Mathematical Society},

keywords = {geometric inequalities; mean curvature of level sets; Schwarz symmetrization; $p$-Laplace equations; regularity of stable solutions; geometric inequalities; mean curvature of level sets; Schwarz symmetrization; -Laplace equations; regularity of stable solutions},

language = {eng},

number = {11},

pages = {2949-2975},

publisher = {European Mathematical Society Publishing House},

title = {Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities},

url = {http://eudml.org/doc/277353},

volume = {017},

year = {2015},

}

TY - JOUR

AU - Castorina, Daniele

AU - Sanchón, Manel

TI - Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities

JO - Journal of the European Mathematical Society

PY - 2015

PB - European Mathematical Society Publishing House

VL - 017

IS - 11

SP - 2949

EP - 2975

AB - We prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish a priori estimates for semistable solutions of $–\Delta _p u= g(u)$ in a smooth bounded domain $\Omega \subset \mathbb {R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\star $ when the domain is strictly convex. More precisely, we prove that $u^\star \in L^{\infty }(\Omega )$ if $n\le p+2$ and $u^\star \in L^{\frac{np}{n-p-2}} (\Omega ) \cap W^{1,p}_0 (\Omega )$ if $n > p + 2$.

LA - eng

KW - geometric inequalities; mean curvature of level sets; Schwarz symmetrization; $p$-Laplace equations; regularity of stable solutions; geometric inequalities; mean curvature of level sets; Schwarz symmetrization; -Laplace equations; regularity of stable solutions

UR - http://eudml.org/doc/277353

ER -

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