Classification of Stable Solutions to a Fractional Singular Elliptic Equation with Weight
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Classification of Stable Solutions to a Fractional Singular Elliptic Equation with Weight Anh Tuan Duong1,2 · Vu Trong Luong3 · Thi Quynh Nguyen4
Received: 11 March 2020 / Accepted: 25 June 2020 © Springer Nature B.V. 2020
Abstract Let p > 0 and (−)s is the fractional Laplacian with 0 < s < 1. The purpose of this paper is to establish a classification result for positive stable solutions to a fractional singular elliptic equation with weight (−)s u = −h(x)u−p in RN . Here N > 2s and h is a nonnegative, continuous function satisfying h(x) ≥ C|x|a , a ≥ 0, when |x| large. We prove the nonexistence of positive stable solutions of this equation under the condition 2(a + 2s) N < 2s + p + p2 + p p+1 or equivalently p > pc (N, s, a), where
pc (N, s, a) =
√
(N−2s)2 −2(N+a)(a+2s)+2 (a+2s)3 (2N−2s+a) (N−2s)(10s+4a−N)
+∞
if N < 10s + 4a . if N ≥ 10s + 4a
B A.T. Duong
[email protected] V.T. Luong [email protected]; [email protected] T.Q. Nguyen [email protected]
1
Division of Computational Mathematics and Engineering, Institute for Computational Science, Ton Duc Thang University, Ho Chi Minh City, Vietnam
2
Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam
3
VNU University of Education, Vietnam National University, Hanoi; 144 Xuan Thuy, Cau Giay, Hanoi, Vietnam
4
Faculty of Fundamental Science, Hanoi University of Industry, Ha Noi, Vietnam
A.T. Duong et al.
Keywords Liouville type theorems · Stable solutions · Fractional singular elliptic equations · Negative exponent nonlinearity Mathematics Subject Classification Primary 35B53 · 35J60 · Secondary 35B35
1 Introduction In this paper, we classify positive stable solutions of a fractional elliptic equation with singular nonlinearity (−)s u = −h(x)u−p in RN ,
(1.1)
where p > 0, 0 < s < 1, N > 2s and h is a nonnegative, continuous function satisfying the growth condition at infinity: there exist constants C > 0 and a ≥ 0 such that h(x) ≥ C|x|a , for |x| large.
(1.2)
Let us denote by Ls (RN ) the space of u ∈ L1loc (RN ) satisfying |u(x)| dx < ∞. N+2s RN (1 + |x|) It is well known that, see e.g., [37], the fractional Laplacian (−)s is well defined on C 2σ (RN ) ∩ Ls (RN ), for some σ > s, by u(x) − u(ξ ) s dξ, (−) u(x) = cN,s lim ε↓0 RN \B(x,ε) |x − ξ |N+2s where cN,s is the normalization constant and B(x, ε) = {ξ ∈ RN ; |ξ − x| < ε}. Throughout this paper, we always consider positive solutions of (1.1) in the space C 2σ (RN ) ∩ Ls (RN ) for some σ > s. Let us now give the definition of stable solutions, see e.g., [18, 32] for the local cases and [10, 23] for the nonlocal cases. Definition 1.1 A positive solution u of (1.1) is called stable if p RN
hu−p−1 φ 2 dx ≤
cN,s 2
RN
RN
(φ(x) − φ(y))2 dxdy, for all φ ∈ Cc1 (RN ). |x − y|N+2s
(1.3)
We next review some related results on this topic in literature. In recent years, the existence and nonexistence of stable solutions to elliptic equations/systems in local case s = 1 have been considerably studied, see [5–9, 13, 15–20,
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