Monatsh Math (2015) 177:325–327 DOI 10.1007/s00605-015-0759-y
Addendum to: Positive solutions of elliptic problems involving both critical Sobolev nonlinearities on exterior regions Qi Han1
Received: 21 January 2015 / Accepted: 10 March 2015 / Published online: 9 April 2015 © Springer-Verlag Wien 2015
Keywords
First exterior p-harmonic Steklov eigenvalue · Moser’s iteration method
Mathematics Subject Classification
Primary 35J92 · 46E35
First, it seems to me the space M q, p () described in Han [3] was originally defined 1 (), yet I was not aware of in Section 5.1.1 of Maz’ya [6] by the notation W p, q this fact then. Interested reader may also find Gol’dshtein et al. [2, Theorem 2.11] helpful. On the other hand, the proof of [3, Appendix B] can be simplified using the same strategy of Bandle and Reichel [1, Lemma 3.1] via Moser’s iteration method, which is more straightforward than the measure-theoretic method of Lindqvist [5]. We reprove that part for the sake of future reference for the reader as the following appendant result to [3]. Theorem 1.1 Assume U is an exterior region with a Lipschitz boundary ∂U as defined in [3], p ∈ (1, N ) and s ∈ E 1, p (U ) is a weak solution of the following problem − p s = 0 in U, subject to |∇s| p−2
∂s = δ |s|q−2 s on ∂U. ∂ν
(1)
Communicated by A. Jüngel. To my little angel Jacquelyn and her mother, my dear wife, Jingbo.
B 1
Qi Han
[email protected] Department of Mathematical Sciences, Worcester Polytechnic Institute (WPI), Worcester, MA 01609-2280, USA
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Q. Han
Then, one has s∞,U + s∞,∂U ≤ C s p,∂U , where 1 < q ≤ p, δ > 0 is a constant and C > 0 is an absolute constant that depends only on p, N , U and δ. Proof Write, as a common practice, s + := max {s, 0} and s − := max {−s, 0}. Below, we only consider the function s + and assume without loss of generality that s + ≥ 1. Let k > 0 be a constant (fixed temporarily), and accordingly define ϕ := + pt p t + + and ω := s min s + , k . Then, we have ϕ, ω > 0 in ,k s min s E 1, p (U ) as well as ⎧
p pt pt ⎪ ⎪ ⎨ |∇s| p−2 ∇s · ∇ϕ = ∇s + min s + , k p + pt s + χ(s + )t ≤k , t t ⎪ ⎪ ⎩ ∇ω = ∇s + min s + , k + t s + χ s + t ≤k . ( ) By virtue of (x + y) p ≤ 2 p−1 (x p + y p ) when x, y ≥ 0 and p ≥ 1, one has |∇ω| p ≤ [2 (1 + t)] p−1 |∇s| p−2 ∇s · ∇ϕ,
since it holds max t p−1 , 1 = [max {t, 1}] p−1 ≤ (1 + t) p−1 in view of t ≥ 0. Using the weak form of (1), and noticing [3, Estimate (3.6)] and 1 < q ≤ p, we have ∂U
p(N −1) N −2
ω
N −2
N −1
dσ
|∇ω| d x ≤ C1 [2 (1 + t)] |∇s| p−2 ∇s · ∇ϕ d x U s q−1 ϕ dσ ≤ C1 (1 + t) p−1 ω p dσ. = C1 [2 (1 + t)] p−1 δ
≤ C1
p
p−1
U
∂U
∂U
Here, C1 , C1 > 0 are constants that depend on p, U and δ. Letting k → ∞ yields ∂U
s+
(t+1) p(N −1) N −2
dσ
1 N −2 p(N −1) t+1
≤
C1 (1 + t) p−1
∂U
+ p(t+1) dσ s
1 p(t+1)
,
(2) provided we have a priori fact s + ∈ L p(t+1) (∂U, dσ ) that is true when t = t0 = 0. N −1 + ∈ Next, we define tl := (tl−1 + 1) N −2 − 1 inductively and observe that s p(t +1) L l (∂U, dσ ) for every integer l ≥ 1. Moreover, a routine calculation using (2) easily leads to + s ≤ p(t +1),∂U l
l−1 1 + p(t +1) m s C1 (1 + tm ) p−1 . p,∂U m=0
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Thus, we are done by taking l → ∞ in view of the following elementary estimate ∞
∞ 1 ln C (1 + tm ) p−1 1 p−1 p(tm +1) C1 (1 + tm ) = exp p (tm + 1) m=0 m=0 ∞ p − 1 C2 + ln (1 + tm ) ≤ exp p tm + 1 m=0 ∞ N −2 m . ≤ exp C2 N −1 m=0
Here, C2 , C2 > 0 are constants that depend on p, U and δ. So, s + ∈ L ∞ (∂U, dσ ). Similarly, one has s − ∈ L ∞ (∂U, dσ ). Hence, s = s + − s − ∈ L ∞ (∂U, dσ )
follows, and by the maximum principle we have s ∈ L ∞ (U ) as well. As a final remark, one notices that we can take δ(z) ≥ 0 to be a function in L r (∂U, dσ ) with r ∈ (N − 1, ∞] and have the same result when we define tl := N −1 (tl−1 + 1) r −1 r N −2 − 1. Moreover, when ∂U is assumed to be more regular such as of class C 1,α , then s ∈ C 1,β (U¯ ∩ B R ) for some α, β > 0 that depends on ∂U, R(> Rb ) by virtue of standard elliptic interior regularity results and Theorem 2 of Lieberman [4]. Notice that s → 0 as |x| → ∞ when s > 0.
References 1. Bandle, C., Reichel, W.: A linear parabolic problem with non-dissipative dynamical boundary conditions. In: Recent Advances in Elliptic and Parabolic Problems (Zurich, 2004), pp. 45–77. World Scientific Publishing Co., Pte. Ltd., Hackensack (2006) 2. Gol’dshtein, V., Motreanu, V., Ukhlov, A.: Embeddings of weighted Sobolev spaces and degenerate Dirichlet problems involving the weighted p-Laplacian. Complex Var. Elliptic Equ. 56, 905–930 (2011) 3. Han, Q.: Positive solutions of elliptic problems involving both critical Sobolev nonlinearities on exterior regions. Monatsh. Math. 176, 107–141 (2015) 4. Lieberman, G.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12, 1203–1219 (1988) 5. Lindqvist, P.: Addendum: on the equation div |∇u| p−2 ∇u + λu p−2 u = 0. Proc. Am. Math. Soc. 116, 583–584 (1992) 6. Maz’ya, V.G.: Sobolev Spaces. Springer, Heidelberg (2011)
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