EXISTENCE AND REGULARITY FOR CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS

Description

EXISTENCE AND REGULARITY FOR CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS JEROME VETOIS Abstract. We establish existence and regularity results for doubly critical anisotropic equa- tions in domains of the Euclidean space. In particular, we answer a question posed by Fra- gala–Gazzola–Kawohl [24] when the maximum of the anisotropic configuration coincides with the critical Sobolev exponent. 1. Introduction In this paper, we investigate existence and regularity for doubly critical anisotropic equa- tions. In dimension n ≥ 2, we provide ourselves with an anisotropic configuration ??p = (p1, . . . , pn) with pi > 1 for all i = 1, . . . , n. We let D1, ??p (?) be the anisotropic Sobolev space defined as the completion of the vector space of all smooth functions with compact support in ? with respect to the norm ?u?D1,??p (?) = ∑n i=1 ?∂u/∂xi?Lpi (?). We are concerned with the following anisotropic problem of critical growth { ?∆??p u = ? |u| p??2 u in ? , u ? D1, ??p (?) , (1.1) on domains ? in the Euclidean space Rn, where ? is a positive real number, p? is the critical Sobolev exponent (see (1.3) below), and ∆??p is the anisotropic Laplace operator defined by ∆??p u = n∑ i=1 ∂ ∂xi ?pixiu , (1.2) where ?pixiu = |∂u/∂xi| pi?2 ∂u/∂xi for all i = 1, .

• configuration ??p


• nonnegative


• result


• ∂xi


• then there


• let n?


• critical anisotropic


• existence result

Sujets

• Sign (mathematics)
• Result

Informations

• Publié par: pefav
• Nombre de lectures: 10
• Langue: English

Extrait

EXISTENCE AND REGULARITY FOR CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS ´ ˆ ´ JEROME VETOIS

Abstract. We establish existence and regularity results for doubly critical anisotropic equa- tions in domains of the Euclidean space. In particular, we answer a question posed by Fra- gala`–Gazzola–Kawohl[24]whenthemaximumoftheanisotropicconfigurationcoincideswith the critical Sobolev exponent.

1. Introduction In this paper, we investigate existence and regularity for doubly critical anisotropic equa- tions. In dimension n ≥ 2, we provide ourselves with an anisotropic configuration −→ p = ( p 1 , . . . , p n ) with p i > 1 for all i 1 , . . . , n . We let D 1 , − p → ( Ω ) be the anisotropic Sobolev space = defined as the completion of the vector space of all smooth functions with compact support in Ω with respect to the norm k u k D 1 , −→ p ( Ω ) = P i n =1 k ∂u/∂x i k L p i ( Ω ) . We are concerned with the following anisotropic problem of critical growth λ p ∗ − 2 ( u − ∈ Δ D − p → 1 , u − p → = ( Ω ) | , u | u in Ω , (1.1) on domains Ω in the Euclidean space R n , where λ is a positive real number, p ∗ is the critical Sobolev exponent (see (1.3) below), and Δ −→ p is the anisotropic Laplace operator defined by Δ −→ p u = i = n X ∂ ∂ x i r x p i i u , (1.2) 1 where r p i u = | ∂u/∂x i | p i − 2 ∂u/∂x i for all i = 1 , . . . , n . As one can check, Δ −→ p involves di- x i rectional derivatives with distinct weights. Anisotropic operators appear in several places in the literature. Recent references can be found in physics [3, 7], in biology [11], and in image processing [46]. We consider in this paper the doubly critical situation p + = p ∗ , where p + = max ( p 1 , . . . , p n ) is the maximum value of the anisotropic configuration and p ∗ is the critical Sobolev exponent for the embeddings of the anisotropic Sobolev space D 1 , −→ p ( Ω ) into Lebesgue spaces. In this setting, not only the nonlinearity has critical growth, but the operator itself has critical growth in particular directions of the Euclidean space. As a remark, the notion of critical direction is a pure anisotropic notion which does not exist when dealing with the Laplace operator or the p -Laplace operator. Given i = 1 , . . . , n , the i -th direction is said to be critical if p i = p ∗ , resp. subcritical if p i < p ∗ . Critical directions induce a failure in the rescaling invariance rule associated with (1.1).

Date : July 16, 2010. Published in Advances in Differential Equations 16 (2011), no. 1/2, 61–83. 1