EXISTENCE AND REGULARITY FOR CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS

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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, .

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critical anisotropic

existence result

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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 in =1 k ∂u/∂x i k L pi ( 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 xp ii 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

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