A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions

Building on the a priori estimates established in [ 3 ], we obtain a priori estimates for classical solutions to ellipticproblems with Dirichlet boundary conditions on regions with convex-starlike boundary. This includes ring-like regions. Arguments that go back to [ 4 ] are used to prove a priori bounds near the convex part of the boundary.Using that the boundary term in the Pohozaev identity on the boundary of a star-like region does not change sign, the proof isconcluded.

1. Introduction. In this paper we prove a priori bounds for the positive solutions to the boundary-value problem: in Ω, u = 0, on ∂Ω, (1.1) where Ω ⊂ R N , N ≥ 2, is a bounded C 2 domains with convex-starlike boundary, including ring-like regions, and f : R + → R + is a subcritical nonlinearity. We will say that a domain Ω has a convex-starlike boundary if ∂Ω = Γ 1 ∪ Γ 2 with Γ 1 ⊂ ∂Ω 1 , for some convex domain Ω 1 ⊂ R N , and n(x) · (x − y) < 0 for some y ∈ R N and for all x ∈ Γ 2 . Here n(x) denotes the outward normal to the boundary ∂Ω. A particular case appears when Ω = Ω 1 \ Ω 2 with Ω 2 ⊂ Ω 1 , where Ω 1 is convex, and Ω 2 star-like, that is n 2 (x) · (x − y) > 0, for some y ∈ R N , and for all x ∈ ∂Ω 2 .
Here n 2 (x) denotes the outward normal to the boundary ∂Ω 2 . In that case, we will say that Ω is a ring-like domain. Since (1.1) is invariant under translations, without loss of generality, we may assume y = 0, (in other words Ω 2 to be star-like with respect to zero).
Our main result is: If the nonlinearity f is locally Lipschitzian and satisfies: (H1) There exist contants C 0 > 0 and β 0 ∈ (0, 1) such that lim inf (H3) There exists a constant C 2 > 0 and a non-increasing function H : Then there exists a uniform constant C, depending only on Ω and f, such that for every u > 0, classical solution to (1.1), Unlike results in [4] or [3], we do not assume f (s) s N +2 N −2 to be nonincreasing. Moreover, Theorem 1 extends the results in [4] and [5] in that it allows a wider class of nonlinearities, such as f (s) = s N +2 N −2 / ln(s + 2) α , with α > 2/(N − 2), see [3,Corollary 2.2]. The reader is referred to Theorem 1.2 in [3] where a slightly more restrictive hypothesis (H1) is used for convex domains (β 0 = 1/2). In [2] one can see the role of regions with convex external boundaries in proving the uniqueness of positive solutions to semilinear elliptic boundary value problems.
2. Bounds near the convex boundary and Pohozaev identity. Due to n(x) · Due to the convexity of Ω 1 , arguing as in [4] (see also [3,Theorem A.3]) one sees that there exists a neighborhood of Γ 1 and a positive constant γ such that for every x in that neighborhood there exists a set K with |K | ≥ γ such that u is bounded below by u(x) in K . Here |K | denotes the Lebesgue measure of the set K . The main idea behind the proof of Theorem 2 is to use the convexity of Ω 1 and moving plane arguments. These lead to the following result whose proof may be found in Step 2 in the proof of [4, Theorem 1.1].

Theorem 2.
Let Ω ⊂ R N be a domain with convex-starlike boundary. Assume that the nonlinearity f is locally Lipschitzian and that satisfies (H4). If u ∈ C 2 (Ω) satisfies (1.1) and u > 0 in Ω, then there exists a constant δ > 0 depending only on Ω 1 and not on f or u, and a constant C depending only on Ω 1 and f but not on u, such that Any classical solutions to (1.1) satisfies the following identity known as Pohozaev identity, see [8], where n(x) is the outward normal vector to the boundary at x ∈ ∂Ω.
Since ∂Ω = Γ 1 ∩ Γ 2 is a convex-starlike boundary, for each x ∈ Γ 2 , we have and τ (x) is tangential to ∂Ω. In particular, (2.3) holds for any x ∈ Γ 2 . Moreover, since u vanishes on ∂Ω, for any tangential vector t(x) where ω t := {x ∈ Ω : d(x, Γ 1 ) < t}. From Schauder interior estimates, see [6, Theorem 6.2], Finally, combining L p estimates with Schauder boundary estimates, (see [1], [6]) Consequently, there exists two constants C, δ > 0 independent of u such that Next we prove that also Applying this inequality to any positive solution, and integrating on Ω we obtain that for some constant C independent on u. Applying this inequality to any positive solution, integrating on Ω, and using (3.6) we obtain that Consequently, since H is non-increasing, for any q > N/2. Therefore, from elliptic regularity, (see [6,Lemma 9.17]) (3.9) Let us restrict q ∈ (N/2, N ). From Sobolev embeddings, for 1/q * = 1/q − 1/N with q * > N we may write (3.10) From Morrey's Theorem, (see [1,Theorem 9.12 and Corollary 9.14]), there exists a constant C only dependent on Ω, q and N such that From now on, we shall argue by contradiction. Let {u k } k be a sequence of classical positive solutions to (1.1) and assume that and there exists y k ∈ ∂B(x k , R k ) such that u k (y k ) = β 0 u k . (3.14) Let us denote by Therefore Then, reasoning as in (3.8), we obtain . (3.16) From elliptic regularity, see (3.9), we deduce (3.17) Therefore, from Morrey's Theorem, see (3.12), for any x ∈ B(x k , R k ) (3.18) Taking x = y k in the above inequality and from (3.14) we obtain (3.20) or equivalently (3.21) This, (3.15), and the assumption that H is non-increasing imply where ω = ω N is the volume of the unit ball in R N .