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

Castorina, Daniele; Sanchón, Manel

doi:10.4171/jems/576

Cited by 14 publications

(55 citation statements)

References 35 publications

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“…In this paper, in the spirit of the clever proof of [25] for the case p = 2, we extend some of the results of [10] for the degenerate p-Laplacian to the singular case 1 < p < 2. We obtain the boundedness of the extremal solution up to a critical dimension n p = p + 2 while we prove that it belongs to L 2n n−p−2 (Ω) if n > p + 2, for any smooth bounded domain Ω, under a standard (asymptotic) convexity assumption on the nonlinearity.…”

Section: Introduction and Main Resultsmentioning

confidence: 94%

“…if n > p + 2, where C 2 = C 2 (n, p, r). The borderline case n = p + 2 is slightly more involved, but we are still able to prove a Morrey type inequality as for the case n < p + 2 (see page 22 in [10] for the details). Lemma 2.1 is thus proved.…”

Section: Proof Of Theorem 11 Part (C)mentioning

confidence: 83%

“…The first estimate gives uniform L ∞ and L r bounds for u λ in terms of the W 1,p+2 0 norm in a neighborhood of the boundary, depending on the dimension. It can be directly derived from the a priori estimates for semistable solutions contained in Theorem 1.4 of [10], where Sanchón and the author extend to the case p > 2 the regularity results of [5,7] for u * in convex domains. Lemma 2.1.…”

Section: Proof Of Theorem 11mentioning

confidence: 99%

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Regularity of the extremal solution for singular p-Laplace equations

Castorina

2014

manuscripta math.

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Section: Introduction and Main Resultsmentioning

confidence: 94%

Section: Proof Of Theorem 11 Part (C)mentioning

confidence: 83%

Section: Proof Of Theorem 11mentioning

confidence: 99%

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Regularity of the extremal solution for singular p-Laplace equations

Castorina

2014

manuscripta math.

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“…Some years later, Castorina and Sanchón [51] obtained the boundedness of stable solutions in dimension…”

Section: Background and Known Resultsmentioning

confidence: 99%

Estimates and Rigidity for Stable Solutions to Some Nonlinear Elliptic Problems

Miraglio

2020

Bull. Aust. Math. Soc.

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La consulta d'aquesta tesi queda condicionada a l'acceptació de les següents condicions d'ús: La difusió d'aquesta tesi per mitjà del r e p o s i t o r i i n s t i t u c i o n a l UPCommons (http://upcommons.upc.edu/tesis) i el repositori cooperatiu TDX (h t t p : / / w w w. t d x. c a t /) ha estat autoritzada pels titulars dels drets de propietat intel•lectual únicament per a usos privats emmarcats en activitats d'investigació i docència. No s'autoritza la seva reproducció amb finalitats de lucre ni la seva difusió i posada a disposició des d'un lloc aliè al servei UPCommons o TDX. No s'autoritza la presentació del seu contingut en una finestra o marc aliè a UPCommons (framing). Aquesta reserva de drets afecta tant al resum de presentació de la tesi com als seus continguts. En la utilització o cita de parts de la tesi és obligat indicar el nom de la persona autora. ADVERTENCIA La consulta de esta tesis queda condicionada a la aceptación de las siguientes condiciones de uso: La difusión de esta tesis por medio del repositorio institucional UPCommons (http://upcommons.upc.edu/tesis) y el repositorio cooperativo TDR (http://www.tdx.cat/?locale-attribute=es) ha sido autorizada por los titulares de los derechos de propiedad intelectual únicamente para usos privados enmarcados en actividades de investigación y docencia. No se autoriza su reproducción con finalidades de lucro ni su difusión y puesta a disposición desde un sitio ajeno al servicio UPCommons No se autoriza la presentación de su contenido en una ventana o marco ajeno a UPCommons (framing). Esta reserva de derechos afecta tanto al resumen de presentación de la tesis como a sus contenidos. En la utilización o cita de partes de la tesis es obligado indicar el nombre de la persona autora. WARNING On having consulted this thesis you're accepting the following use conditions: Spreading this thesis by the i n s t i t u t i o n a l r e p o s i t o r y UPCommons (http://upcommons.upc.edu/tesis) and the cooperative repository TDX (http://www.tdx.cat/?locale-attribute=en) has been authorized by the titular of the intellectual property rights only for private uses placed in investigation and teaching activities. Reproduction with lucrative aims is not authorized neither its spreading nor availability from a site foreign to the UPCommons service. Introducing its content in a window or frame foreign to the UPCommons service is not authorized (framing). These rights affect to the presentation summary of the thesis as well as to its contents. In the using or citation of parts of the thesis it's obliged to indicate the name of the author.

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“…Among other results, they established pointwise, L q and W 1,q estimates which are optimal and do not depend on the non-linearity h(s). Furthermore, Castorina and Sanchón [9] obtain a priori estimates for semi-stable solutions of the reaction-diffusion problem −∆ p u = h(u) in Ω while the reaction term is driven by any positive C 1 non-linearity h and, as a main tool, they develop Morrey-type and Sobolev-type inequalities that involve the functional…”

Section: (12)mentioning

confidence: 99%

Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

Ó¹,

Clemente²

2019

Ann. Acad. Sci. Fenn. Math.

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We investigate the regularity of semi-stable, radially symmetric, and decreasing solutions for a class of quasilinear reaction-diffusion equations in the inhomogeneous context of Riemannian manifolds. We prove uniform boundedness, Lebesgue and Sobolev estimates for this class of solutions for equations involving the p-Laplace Beltrami operator and locally Lipschitz non-linearity. We emphasize that our results do not depend on the boundary conditions and the specific form of the non-linearities and metric. Moreover, as an application, we establish regularity of the extremal solutions for equations involving the p-Laplace Beltrami operator with zero Dirichlet boundary conditions. 1 \B δ F (u) dv g , where F (x, t) =ˆt 0 f (x, s) ds.

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

Cited by 14 publications

References 35 publications

Regularity of the extremal solution for singular p-Laplace equations

Regularity of the extremal solution for singular p-Laplace equations

Estimates and Rigidity for Stable Solutions to Some Nonlinear Elliptic Problems

Regularity of stable solutions to quasilinear elliptic equations on Riemannian models

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