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Existence of solutions for singular critical semilinear elliptic equation
Applied Mathematics Letters 94 (2019) 217–223
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Existence of solutions for singular critical semilinear elliptic equation✩ Mengchao Wang, Qi Zhang ∗ School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China
article
info
Article history: Received 31 December 2018 Accepted 24 February 2019 Available online 6 March 2019 Keywords: Semilinear elliptic equation Hardy potential Hardy–Sobolev critical exponents Mountain pass lemma
abstract This paper is devoted to the existence of solutions for a singular critical semilinear elliptic equation. Some existence and multiplicity results are obtained by using mountain pass arguments and analysis techniques. The results of Ding and Tang (2007) and Kang (2007) and related are improved. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Consider the elliptic problem: ⎧ 2∗ (s)−2 ⎪ ⎨ − ∆u − µ u = |u| u + g(x, u), s 2 |x| |x| ⎪ ⎩ u = 0,
x ∈ Ω \{0}
(1.1)
x ∈ ∂Ω
where Ω is an open bounded domain in RN with smooth boundary ∂Ω and 0 ∈ Ω , N ≥ 3, 0 ≤ s < −s) 2, 0 ≤ µ < µ ¯ := ( N 2−2 )2 , 2∗ (s) = 2(N N −2 is the critical Sobolev–Hardy exponent with g ∈ C(Ω × R, R), ∫t G(x, t) = 0 g(x, s)ds. Throughout this paper, we make the following assumptions: (g1 ) g ∈ C(Ω × R, R) and there exist constants c1 , c2 > 0 and p ∈ (2, 2∗ ) such that |g(x, t)| ≤ p−1 c1 |t| + c2 |t| , ∀(x, t) ∈ Ω × R; (g2 ) There exists a constant K > 0 big enough such that G(x, t) ≥ Kt2 , ∀(x, t) ∈ Ω × R; (g3 ) There are constants ρ > 2 and ν > 0 such that ρG(x, t) ≤ g(x, t)t + νt2 , ∀(x, t) ∈ Ω × R. ✩ Supported by the innovative project of graduate students of Central South University, PR China (No.502211809) and NSFC, PR China 61873284 ∗ Corresponding author. E-mail addresses: [email protected] (M. Wang), [email protected] (Q. Zhang).
https://doi.org/10.1016/j.aml.2019.02.030 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
218
M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
For 0 ≤ µ < µ ¯, define the best constant
Aµ,s :=
inf
u∈H 1 (RN )\{0}
2
2
u (|∇u| − µ |x| 2 )dx , ∗ (s) 2 ) (∫ 2 |u| 2∗ (s) dx s |x| RN
∫
RN
(1.2)
which is attained by the functions yε (x) =
Cε √ √ ) N −2 ( γ (2−s) µ ¯ −µ/ µ ¯ 2−s |x| ε + |x|
(1.3)
√ √ √µ¯/(2−s) √ ¯− µ ¯ − µ, Cε = (2ε(¯ µ − µ)(N − s)/ µ ¯) . for all ε > 0, where γ = µ A lot of work has been focused on the study of semilinear elliptic equations with Hardy potential, both in bounded and unbounded domains. The main reason of interest in Hardy term relies on their criticality. Indeed, Hardy term has the same homogeneity as the Laplacian and does not belong to the Kato class, hence it cannot be regarded as a lower order perturbation term. In bounded or unbounded domains, the similar form of (1.1): ⎧ 2∗ (s)−2 ⎪ ⎨ − ∆u − µ u = |u| u + λf (x, u), s 2 |x| |x| ⎪ ⎩ u = 0,
in Ω
(1.4)
on ∂Ω
has received extensive attention. In [1], L. Ding et al. proved that, if f (x, u) satisfies some conditions and λ ∈ (0, λ∗ ) with λ∗ a constant, problem (1.4) possesses at least two positive solutions. In [2] and [3], D. Kang q−2 et al. investigated problem (1.4) with f (x, u) = |u| u, 2 ≤ q < 2∗ and f (x, u) = u respectively; obtained the existence of positive or nontrivial solutions. In [4], with f (x, u) = uq−1 , 0 < q < 1, problem (1.4) was also discussed by Yang et al. When λ = 1, L. Ding et al. [5] showed that if µ ∈ [0, µ ¯), f (x, u) satisfies some conditions then problem (1.4) admits at least two distinct nontrivial solutions. When f (x, u) = u, Cao et al. [6] showed that problem (1.4) possesses at least a pair of sign-changing solutions under some constraints on N, µ and λ. In [7], Cao et al. showed that problem (1.4) admits a nontrivial solution for all λ > 0, µ ∈ [0, µ ¯ − ( NN+2 )2 ). There are also quite a few excellent results in the special case of s = 0 even if we are not going to introduce them here. Readers who are interested can refer to papers such as [8–11] and the references therein. Inspired by [5] and [12], we continue to study the nontrivial solutions of problem (1.1) in this paper. The methods we employed here are the Mountain Pass Lemma and analysis techniques. The main results we obtained are presented in the following theorems. Theorem 1.1. Suppose that N ≥ 3, 0 ≤ µ < µ ¯ − 1, 0 ≤ s < 2, g(x, t) satisfies (g1 ), (g2 ) and (g3 ). Then problem (1.1) has at least a nonnegative solution. Theorem 1.2. Suppose that N ≥ 3, 0 ≤ µ < µ ¯ − 1, 0 ≤ s < 2, g(x, t) satisfies (g1 ), (g2 ) and (g3 ). Then problem (1.1) has at least two distinct nontrivial solutions. Remark 1.3. (i) Comparing to [5], we obtain the same results under weaker conditions by relaxing the restrictions on the nonlinearity g(x, u). (ii) Relative to [10], the nonlinearity we considered is more general. Hence, we make a substantial improvement on the works of [5,10].
M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
219
2. Proof of main results We first define the equivalent norm and inner product in H01 (Ω ) for 0 ≤ µ < µ ¯ respectively: ∥u∥ :=
[∫ (
2
|∇u| − µ
Ω
u2 ) 2
|x|
] 12 ,
dx
(u, v) :=
∫ (
∇u∇v − µ
Ω
uv ) 2
|x|
∀u, v ∈ H01 (Ω ).
dx ,
By the Hardy inequality, this norm is equivalent to the usual norm in H01 (Ω ). In the following we give some definitions and lemmas. Lemma 2.1 (Hardy Inequality [13]). Assume that 1 < p < N and u ∈ W 1,p (RN ). Then ∫ ∫ p p |u| p p dx ≤ ( |∇u| dx. ) p N − p |x| N N R R Lemma 2.2 (Sobolev-Hardy Inequality [13]). Assume that 1 < p < N and p∗ (s) = there exists a constant C > 0, such that for any u ∈ W 1,p (RN ) (
∫ RN
∗
p (s) ) ∗p |u| p (s) ≤ C s dx |x|
∫
(N −s)p N −p , 0
(2.1) ≤ s ≤ p, Then
p
|∇u| dx.
(2.2)
RN
In order to study the existence of nonnegative solutions for (1.1), we shall firstly consider the existence of nontrivial solutions to the problem ∗
(u+ )2 (s)−1 − ∆u − µ 2 = + g + (x, u), s |x| |x| u
x ∈ Ω.
The energy functional corresponding to (2.3) is given by ∫ ∫ ∗ (u+ )2 (s) 1 1 2 J(u) = ∥u∥ − ∗ dx − G+ (x, u)dx, s 2 2 (s) Ω |x| Ω
(2.3)
u ∈ H01 (Ω ),
(2.4)
where u+ = max{u, 0},
G+ (x, t) =
{
t
∫
g + (x, s)ds,
g + (x, t) =
0
g(x, t), 0,
J(u) is well defined with J ∈ C 1 (H01 (Ω ), R) and ∫ ∫ ∗ (u+ )2 (s)−1 v ⟨J ′ (u), v⟩ = (u, v) − dx − g + (x, u)vdx, s |x| Ω Ω
t ≥ 0, t < 0.
u, v ∈ H01 (Ω ).
(2.5)
The critical points of the functional J are just weak solutions of problem (2.3). Let Uε (x) = yεC(x) , define a ε ∞ cut-off function φ ∈ C0 (Ω ) such that { 1, |x| ≤ R φ(x) = where B2R (0) ⊂ Ω , 0 ≤ φ(x) ≤ 1 f or R < |x| < 2R. (2.6) 0, |x| ≥ 2R, 1 ∫ ∫ 2∗ (s) −s 2∗ (s) −s Set uε (x) = φ(x)Uε (x), vε (x) = uε (x)/( Ω |uε | |x| dx) 2∗ (s) , hence Ω |vε | |x| dx = 1.
Lemma 2.3 ([2]). vε (x) satisfies the following estimates: N −2
∥vε ∥2 = Aµ,s + O(ε 2−s ),
(2.7)
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M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
⎧ √ µq ¯ ⎪ ⎪ ⎪O(ε 2−s ), ⎪ ⎪ ⎪ ⎪ ∫ ⎨ √ µq ¯ q |vε | dx = O(ε 2−s | ln ε|), ⎪ Ω ⎪ ⎪ √ √ ⎪ µ(N ¯ −q µ) ¯ ⎪ √ ⎪ ⎪ ¯ ⎩O(ε (2−s) µ−µ ),
1≤q< √
N √
, µ ¯+ µ ¯−µ N √ q=√ , µ ¯+ µ ¯−µ N √ √ < q < 2∗ . µ ¯+ µ ¯−µ
(2.8)
Lemma 2.4. Assume (g1 ) and (g3 ) hold, let {un } ⊂ H01 (Ω ) be a sequence such that J(un ) → c, J ′ (un ) → 0, N −s
2−s 2−s 1 ′ where c ∈ (0, 2(N −s) Aµ,s ). Then there exists u ∈ H0 (Ω ) such that un ⇀ u, up to a subsequence. J (u) = 0 and u is a nontrivial solution of problem (2.3).
Proof . First, we prove that {un } is bounded in H01 (Ω ). To prove the boundedness of {un }, arguing by contradiction, suppose that ∥un ∥ → ∞. Let vn = un /∥un ∥. Then ∥vn ∥ = 1 and ∥vn ∥q ≤ C for 1 ≤ q ≤ 2∗ . By (g3 ), we have 1 c + 1 + o(1)∥un ∥ ≥ J(un ) − ⟨J ′ (un ), un ⟩ θ ∫ (1 (1 1) 2∗ (s) 1 ) (u+ n) 2 − ∥un ∥ + − = dx s 2 θ θ 2∗ (s) Ω |x| ∫ ∫ 1 g + (x, un )un dx − G+ (x, un )dx + θ Ω Ω θ−2 ν ≥ ∥un ∥2 − ∥un ∥22 2θ θ ν 2 θ−2 − ∥vn ∥22 ), = ∥un ∥ ( 2θ θ
(2.9)
where θ = min{2∗ (s), ρ}, which implies 1≤
2ν lim inf ∥vn ∥22 . θ − 2 n→∞
(2.10)
Passing to a subsequence, we may assume that vn ⇀ v in H01 (Ω ), then vn → v in Lq (Ω ), 1 ≤ q < 2∗ , and vn → v a.e. on Ω . Hence, it follows from (2.10) that v ̸= 0. And c + o(1) J(un ) = lim n→∞ ∥un ∥2 ∥un ∥2 ∫ ∫ [1 2∗ (s) 1 G+ (x, un ) 2 ] (u+ n) = lim − ∗ dx − vn dx s 2 n→∞ 2 2 (s)∥un ∥ Ω u2n |x| Ω ∫ ] [1 −K vn2 dx ≤ lim n→∞ 2 Ω ∫ 1 2 = −K v dx 2 Ω <0
0 = lim
n→∞
(2.11)
when K big enough, which is a contradiction. Thus {un } is bounded in H01 (Ω ) and there exists u ∈ H01 (Ω ) such that un ⇀ u, up to a subsequence. Furthermore, J ′ (u) = 0 by the weak continuity of J ′ . If u = 0 in Ω , as the term g(x, u) is subcritical, from ⟨J ′ (un ), un ⟩ = o(1) we have ∥un ∥2 −
∫ Ω
∗
2 (s) (u+ n) dx = o(1). s |x|
(2.12)
M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
221
And, by the definition of Aµ,s , we have ∥un ∥2 ≥ Aµ,s
(∫ (u+ )2∗ (s) ) ∗2 2 (s) n dx . s |x| Ω
(2.13)
It follows from (2.12) and (2.13) that 2
o(1) ≥ ∥un ∥
(
2∗ (s)
1−
− Aµ,s 2
∥un ∥2
∗
(s)−2
)
.
(2.14)
If ∥un ∥ → 0, it contradicts c > 0. Therefore, N −s 2−s ∥un ∥2 ≥ Aµ,s + o(1).
(2.15)
By (2.8), (2.12) and (2.15), we get ∫ 2∗ (s) 1 (u+ 1 n) dx + o(1) ∥un ∥2 − ∗ s 2 2 (s) Ω |x| 2−s = ∥un ∥2 + o(1) 2(N − s) N −s 2−s 2−s Aµ,s ≥ + o(1), 2(N − s)
⟨J ′ (un ), un ⟩ =
(2.16)
N −s
which contradicts c <
2−s 2−s 2(N −s) Aµ,s
. Thus u ̸= 0 and it is a nontrivial solution of problem (2.3).
□
Lemma 2.5. Suppose that 0 ≤ µ < µ ¯ − 1, 0 ≤ s < 2, g(x,t) satisfies (g1 ) and (g2 ). Then there exists 1 u0 ∈ H0 (Ω )\{0} such that N −s 2−s 2−s Aµ,s . (2.17) sup J(tu0 ) < 2(N − s) t≥0 Proof . For t ≥ 0, we consider the function ∗
t2 t2 (s) h(t) := J(tvε ) = ∥vε ∥2 − ∗ − 2 2 (s)
∫
G+ (x, tvε )dx,
(2.18)
Ω
∗
2 2 (s) ¯ := t ∥vε ∥2 − t h(t) . 2 2∗ (s)
(2.19)
Note that limt→+∞ h(t) = −∞, h(0) = 0, and h(t) > 0 as t is small enough. Therefore, supt≥0 h(t) > 0 is attained for some tε > 0. Since ∫ ∗ 0 = h′ (tε ) = tε ∥vε ∥2 − t2ε (s)−1 − g + (x, tvε )vε dx, (2.20) RN
we have 2
∥vε ∥ =
∗ t2ε (s)−2
1 + tε
∫
Hence,
g + (x, tvε )vε dx ≥ t2ε
∗
(s)−2
.
(2.21)
Ω 2
tε ≤ ∥vε ∥ 2∗ (s)−2 ≜ t0ε .
(2.22)
By (g1 ), it is easy to verify that p−1
|g + (x, t)| ≤ c1 |t| + c2 |t| Hence, we obtain ∥vε ∥2 ≤ t2ε
∗
(s)−2
+
c1 2
∫
2
|vε | dx + Ω
.
c2 p−2 |tε | p
(2.23) ∫
p
|vε | dx. Ω
(2.24)
M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
222
By (2.7), (2.8) and (2.24), when ε small enough, we conclude that t2ε
∗
(s)−2
≥
1 Aµ,s . 2
(2.25) 2
¯ attains its maximum at t0 = ∥vε ∥ 2∗ (s)−2 and is increasing in the interval On the other hand, the function h(t) ε 0 [0, tε ]. Together with (2.7), (2.24) and G+ (x, t) ≥ Kt2 for t ≥ 0, we deduce that ∫ ¯ 0) − h(tε ) ≤ h(t G+ (x, tε vε )dx ε Ω ∫ 2(N −s) 2−s 2−s ∥vε ∥ − ≤ G+ (x, tε vε )dx 2(N − 2) Ω ∫ (2.26) N −s N −2 2−s 2 2−s Aµ,s + O(ε 2−s ) − K (tε )2 |vε | dx ≤ 2(N − s) Ω ∫ N −s 2 N −2 A 2−s µ,s 2∗ (s)−2 2 2−s Aµ,s + O(ε 2−s ) − K( ) |vε | dx. ≤ 2(N − s) 2 Ω From (2.8), there holds ∫
2
N −2 √ µ−µ ¯
|vε | dx = O(ε (2−s)
).
(2.27)
Ω
And since
N −2 N −2 √ , < 2−s (2 − s) µ ¯−µ
choosing ε small enough, we have sup J(tvε ) = h(tε ) < t≥0
N −s 2−s 2−s . □ Aµ,s 2(N − s)
Proof of Theorem 1.1. From the Hardy and Hardy–Sobolev inequalities, we get: ∫ Ω
2∗ (s)
|u| 2∗ (s) , s dx ≤ C∥u∥ |x|
∥u∥qq ≤ C∥u∥q f or 1 ≤ q ≤ 2∗ ,
u ∈ H01 (Ω ).
(2.28)
It follows from (g1 ) that p−1
|g + (x, t)| ≤ c1 |t| + c2 |t| , 1 1 2 p |G+ (x, t)| ≤ c1 |t| + c2 |t| , 2 p
(2.29) (2.30)
for all t ∈ R and x ∈ Ω . By (2.28) and (2.30) we have, for c1 small enough ∫ ∫ ∗ 1 1 (u+ )2 (s) dx − G+ (x, u)dx J(u) = ∥u∥2 − ∗ s 2 2 (s) Ω |x| Ω 1 C Cc1 Cc2 2 2∗ (s) 2 ≥ ∥u∥ − ∗ ∥u∥ − ∥u∥2 − ∥u∥pp 2 2 (s) 2 p ∗ 1 − Cc1 C C ≥ ∥u∥2 − ∗ ∥u∥2 (s) − ∥u∥P . 2 2 (s) p So, there exists α > 0 such that J(u) ≥ α > 0 for u ∈ ∂Br (0) when r > 0 small enough. Moreover, by the nonnegativity of G+ (x, u), there holds ∫ ∫ ∗ 2∗ (s) t2 t2 (s) (u+ 0) J(tu0 ) = ∥u0 ∥2 − ∗ dx − G+ (x, tu0 )dx s 2 2 (s) Ω |x| Ω ∫ ∗ 2∗ (s) t2 t2 (s) (u+ 0) dx, ≤ ∥u0 ∥2 − ∗ s 2 2 (s) Ω |x|
M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
223
and lim J(tu0 ) → −∞,
as t → ∞.
t→∞
Hence we can choose t0 > 0 such that ∥t0 u0 ∥ > r and J(t0 u0 ) ≤ 0. Using the Mountain Pass Lemma, there is a sequence {un } ⊂ H01 (Ω ) satisfying J ′ (un ) → 0,
J(un ) → c ≥ α, where
c = inf max J(γ(t)), γ∈Γ t∈[0,1] { ( )⏐ } Γ = γ(t) ∈ C [0, 1], H01 ⏐γ(0) = 0, γ(1) = t0 u0 .
And, applying Lemmas 2.4 and 2.5, we get a (P S)c sequence {un } ⊂ H01 (Ω ), and u ∈ H01 (Ω ) such that J ′ (u) = 0. Thus u is a solution of problem (2.3). And then ⟨J ′ (u), u− ⟩ = 0, where u− = min{u, 0}. Hence u− = 0, that is, u ≥ 0. We get that u is a nonnegative solution of problem (1.1). □ Proof of Theorem 1.2. From Theorem 1.1, problem (2.3) has a nonnegative solution u1 . Set k(x, t) = −g(x, −t) for t ∈ R. It follows from Theorem 1.1 that the equation −∆u − µ
2∗ (s)−2
u 2
|x|
=
|u|
|x|
s
u + k(x, u),
has at least a nonnegative solution v. Let u2 = v, then u2 is a solution of −∆u − µ
u |x|
2
=
|u|
2∗ (s)−2 s
|x|
u + g(x, u).
Obviously u1 ≥ 0, u2 ≤ 0. So, Eq. (1.1) has at least two distinct nontrivial solutions. □ References [1] L. Ding, C.L. Tang, Existence and multiplicity of positive solutions for a class of semilinear elliptic equation involving Hardy term and Hardy-Sobolev critical exponents, J. Math. Anal. Appl. 339 (2) (2008) 1073–1083. [2] D.S. Kang, S.J. Peng, Positive solutions for singular critical elliptic problems, Appl. Math. Lett. 17 (2004) 411–416. [3] D.S. Kang, S.J. Peng, Solutions for semi-linear elliptic problems with critical HardyCSobolev exponents and Hardy potential, Appl. Math. Lett. 18 (10) (2005) 1094–1100. [4] H.T. Yang, J.H. Chen, A result on Hardy-Sobolev critical elliptic equations with boundary singularities, Commun. Pure Appl. Anal. 6 (2007) 191–201. [5] L. Ding, C.L. Tang, Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy-Sobolev critical exponents, Appl. Math. Lett. 20 (12) (2007) 1175–1183. [6] D.M. Cao, S.J. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Differential Equations 193 (2003) 424–434. [7] D.M. Cao, P.G. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations 205 (2004) 521–537. [8] A. Ferrero, F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177 (2001) 494–522. [9] D.S. Kang, S.J. Peng, Existence of solutions for a singular critical elliptic equation, J. Math. Anal. Appl. 284 (2003) 724–732. [10] D.S. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in RN , Nonlinear Anal. 66 (2007) 241–252. [11] E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations 156 (1999) 407–426. [12] X.H. Tang, Infinitely many solutions for semilinear Schr?dinger equations with sign-changing potential andnonlinearity, J. Math. Anal. Appl. 401 (2013) 407–415. [13] N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (12) (2000) 5703–5743.
Contents lists available at ScienceDirect
Applied Mathematics Letters www.elsevier.com/locate/aml
Existence of solutions for singular critical semilinear elliptic equation✩ Mengchao Wang, Qi Zhang ∗ School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, PR China
article
info
Article history: Received 31 December 2018 Accepted 24 February 2019 Available online 6 March 2019 Keywords: Semilinear elliptic equation Hardy potential Hardy–Sobolev critical exponents Mountain pass lemma
abstract This paper is devoted to the existence of solutions for a singular critical semilinear elliptic equation. Some existence and multiplicity results are obtained by using mountain pass arguments and analysis techniques. The results of Ding and Tang (2007) and Kang (2007) and related are improved. © 2019 Elsevier Ltd. All rights reserved.
1. Introduction Consider the elliptic problem: ⎧ 2∗ (s)−2 ⎪ ⎨ − ∆u − µ u = |u| u + g(x, u), s 2 |x| |x| ⎪ ⎩ u = 0,
x ∈ Ω \{0}
(1.1)
x ∈ ∂Ω
where Ω is an open bounded domain in RN with smooth boundary ∂Ω and 0 ∈ Ω , N ≥ 3, 0 ≤ s < −s) 2, 0 ≤ µ < µ ¯ := ( N 2−2 )2 , 2∗ (s) = 2(N N −2 is the critical Sobolev–Hardy exponent with g ∈ C(Ω × R, R), ∫t G(x, t) = 0 g(x, s)ds. Throughout this paper, we make the following assumptions: (g1 ) g ∈ C(Ω × R, R) and there exist constants c1 , c2 > 0 and p ∈ (2, 2∗ ) such that |g(x, t)| ≤ p−1 c1 |t| + c2 |t| , ∀(x, t) ∈ Ω × R; (g2 ) There exists a constant K > 0 big enough such that G(x, t) ≥ Kt2 , ∀(x, t) ∈ Ω × R; (g3 ) There are constants ρ > 2 and ν > 0 such that ρG(x, t) ≤ g(x, t)t + νt2 , ∀(x, t) ∈ Ω × R. ✩ Supported by the innovative project of graduate students of Central South University, PR China (No.502211809) and NSFC, PR China 61873284 ∗ Corresponding author. E-mail addresses: [email protected] (M. Wang), [email protected] (Q. Zhang).
https://doi.org/10.1016/j.aml.2019.02.030 0893-9659/© 2019 Elsevier Ltd. All rights reserved.
218
M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
For 0 ≤ µ < µ ¯, define the best constant
Aµ,s :=
inf
u∈H 1 (RN )\{0}
2
2
u (|∇u| − µ |x| 2 )dx , ∗ (s) 2 ) (∫ 2 |u| 2∗ (s) dx s |x| RN
∫
RN
(1.2)
which is attained by the functions yε (x) =
Cε √ √ ) N −2 ( γ (2−s) µ ¯ −µ/ µ ¯ 2−s |x| ε + |x|
(1.3)
√ √ √µ¯/(2−s) √ ¯− µ ¯ − µ, Cε = (2ε(¯ µ − µ)(N − s)/ µ ¯) . for all ε > 0, where γ = µ A lot of work has been focused on the study of semilinear elliptic equations with Hardy potential, both in bounded and unbounded domains. The main reason of interest in Hardy term relies on their criticality. Indeed, Hardy term has the same homogeneity as the Laplacian and does not belong to the Kato class, hence it cannot be regarded as a lower order perturbation term. In bounded or unbounded domains, the similar form of (1.1): ⎧ 2∗ (s)−2 ⎪ ⎨ − ∆u − µ u = |u| u + λf (x, u), s 2 |x| |x| ⎪ ⎩ u = 0,
in Ω
(1.4)
on ∂Ω
has received extensive attention. In [1], L. Ding et al. proved that, if f (x, u) satisfies some conditions and λ ∈ (0, λ∗ ) with λ∗ a constant, problem (1.4) possesses at least two positive solutions. In [2] and [3], D. Kang q−2 et al. investigated problem (1.4) with f (x, u) = |u| u, 2 ≤ q < 2∗ and f (x, u) = u respectively; obtained the existence of positive or nontrivial solutions. In [4], with f (x, u) = uq−1 , 0 < q < 1, problem (1.4) was also discussed by Yang et al. When λ = 1, L. Ding et al. [5] showed that if µ ∈ [0, µ ¯), f (x, u) satisfies some conditions then problem (1.4) admits at least two distinct nontrivial solutions. When f (x, u) = u, Cao et al. [6] showed that problem (1.4) possesses at least a pair of sign-changing solutions under some constraints on N, µ and λ. In [7], Cao et al. showed that problem (1.4) admits a nontrivial solution for all λ > 0, µ ∈ [0, µ ¯ − ( NN+2 )2 ). There are also quite a few excellent results in the special case of s = 0 even if we are not going to introduce them here. Readers who are interested can refer to papers such as [8–11] and the references therein. Inspired by [5] and [12], we continue to study the nontrivial solutions of problem (1.1) in this paper. The methods we employed here are the Mountain Pass Lemma and analysis techniques. The main results we obtained are presented in the following theorems. Theorem 1.1. Suppose that N ≥ 3, 0 ≤ µ < µ ¯ − 1, 0 ≤ s < 2, g(x, t) satisfies (g1 ), (g2 ) and (g3 ). Then problem (1.1) has at least a nonnegative solution. Theorem 1.2. Suppose that N ≥ 3, 0 ≤ µ < µ ¯ − 1, 0 ≤ s < 2, g(x, t) satisfies (g1 ), (g2 ) and (g3 ). Then problem (1.1) has at least two distinct nontrivial solutions. Remark 1.3. (i) Comparing to [5], we obtain the same results under weaker conditions by relaxing the restrictions on the nonlinearity g(x, u). (ii) Relative to [10], the nonlinearity we considered is more general. Hence, we make a substantial improvement on the works of [5,10].
M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
219
2. Proof of main results We first define the equivalent norm and inner product in H01 (Ω ) for 0 ≤ µ < µ ¯ respectively: ∥u∥ :=
[∫ (
2
|∇u| − µ
Ω
u2 ) 2
|x|
] 12 ,
dx
(u, v) :=
∫ (
∇u∇v − µ
Ω
uv ) 2
|x|
∀u, v ∈ H01 (Ω ).
dx ,
By the Hardy inequality, this norm is equivalent to the usual norm in H01 (Ω ). In the following we give some definitions and lemmas. Lemma 2.1 (Hardy Inequality [13]). Assume that 1 < p < N and u ∈ W 1,p (RN ). Then ∫ ∫ p p |u| p p dx ≤ ( |∇u| dx. ) p N − p |x| N N R R Lemma 2.2 (Sobolev-Hardy Inequality [13]). Assume that 1 < p < N and p∗ (s) = there exists a constant C > 0, such that for any u ∈ W 1,p (RN ) (
∫ RN
∗
p (s) ) ∗p |u| p (s) ≤ C s dx |x|
∫
(N −s)p N −p , 0
(2.1) ≤ s ≤ p, Then
p
|∇u| dx.
(2.2)
RN
In order to study the existence of nonnegative solutions for (1.1), we shall firstly consider the existence of nontrivial solutions to the problem ∗
(u+ )2 (s)−1 − ∆u − µ 2 = + g + (x, u), s |x| |x| u
x ∈ Ω.
The energy functional corresponding to (2.3) is given by ∫ ∫ ∗ (u+ )2 (s) 1 1 2 J(u) = ∥u∥ − ∗ dx − G+ (x, u)dx, s 2 2 (s) Ω |x| Ω
(2.3)
u ∈ H01 (Ω ),
(2.4)
where u+ = max{u, 0},
G+ (x, t) =
{
t
∫
g + (x, s)ds,
g + (x, t) =
0
g(x, t), 0,
J(u) is well defined with J ∈ C 1 (H01 (Ω ), R) and ∫ ∫ ∗ (u+ )2 (s)−1 v ⟨J ′ (u), v⟩ = (u, v) − dx − g + (x, u)vdx, s |x| Ω Ω
t ≥ 0, t < 0.
u, v ∈ H01 (Ω ).
(2.5)
The critical points of the functional J are just weak solutions of problem (2.3). Let Uε (x) = yεC(x) , define a ε ∞ cut-off function φ ∈ C0 (Ω ) such that { 1, |x| ≤ R φ(x) = where B2R (0) ⊂ Ω , 0 ≤ φ(x) ≤ 1 f or R < |x| < 2R. (2.6) 0, |x| ≥ 2R, 1 ∫ ∫ 2∗ (s) −s 2∗ (s) −s Set uε (x) = φ(x)Uε (x), vε (x) = uε (x)/( Ω |uε | |x| dx) 2∗ (s) , hence Ω |vε | |x| dx = 1.
Lemma 2.3 ([2]). vε (x) satisfies the following estimates: N −2
∥vε ∥2 = Aµ,s + O(ε 2−s ),
(2.7)
220
M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
⎧ √ µq ¯ ⎪ ⎪ ⎪O(ε 2−s ), ⎪ ⎪ ⎪ ⎪ ∫ ⎨ √ µq ¯ q |vε | dx = O(ε 2−s | ln ε|), ⎪ Ω ⎪ ⎪ √ √ ⎪ µ(N ¯ −q µ) ¯ ⎪ √ ⎪ ⎪ ¯ ⎩O(ε (2−s) µ−µ ),
1≤q< √
N √
, µ ¯+ µ ¯−µ N √ q=√ , µ ¯+ µ ¯−µ N √ √ < q < 2∗ . µ ¯+ µ ¯−µ
(2.8)
Lemma 2.4. Assume (g1 ) and (g3 ) hold, let {un } ⊂ H01 (Ω ) be a sequence such that J(un ) → c, J ′ (un ) → 0, N −s
2−s 2−s 1 ′ where c ∈ (0, 2(N −s) Aµ,s ). Then there exists u ∈ H0 (Ω ) such that un ⇀ u, up to a subsequence. J (u) = 0 and u is a nontrivial solution of problem (2.3).
Proof . First, we prove that {un } is bounded in H01 (Ω ). To prove the boundedness of {un }, arguing by contradiction, suppose that ∥un ∥ → ∞. Let vn = un /∥un ∥. Then ∥vn ∥ = 1 and ∥vn ∥q ≤ C for 1 ≤ q ≤ 2∗ . By (g3 ), we have 1 c + 1 + o(1)∥un ∥ ≥ J(un ) − ⟨J ′ (un ), un ⟩ θ ∫ (1 (1 1) 2∗ (s) 1 ) (u+ n) 2 − ∥un ∥ + − = dx s 2 θ θ 2∗ (s) Ω |x| ∫ ∫ 1 g + (x, un )un dx − G+ (x, un )dx + θ Ω Ω θ−2 ν ≥ ∥un ∥2 − ∥un ∥22 2θ θ ν 2 θ−2 − ∥vn ∥22 ), = ∥un ∥ ( 2θ θ
(2.9)
where θ = min{2∗ (s), ρ}, which implies 1≤
2ν lim inf ∥vn ∥22 . θ − 2 n→∞
(2.10)
Passing to a subsequence, we may assume that vn ⇀ v in H01 (Ω ), then vn → v in Lq (Ω ), 1 ≤ q < 2∗ , and vn → v a.e. on Ω . Hence, it follows from (2.10) that v ̸= 0. And c + o(1) J(un ) = lim n→∞ ∥un ∥2 ∥un ∥2 ∫ ∫ [1 2∗ (s) 1 G+ (x, un ) 2 ] (u+ n) = lim − ∗ dx − vn dx s 2 n→∞ 2 2 (s)∥un ∥ Ω u2n |x| Ω ∫ ] [1 −K vn2 dx ≤ lim n→∞ 2 Ω ∫ 1 2 = −K v dx 2 Ω <0
0 = lim
n→∞
(2.11)
when K big enough, which is a contradiction. Thus {un } is bounded in H01 (Ω ) and there exists u ∈ H01 (Ω ) such that un ⇀ u, up to a subsequence. Furthermore, J ′ (u) = 0 by the weak continuity of J ′ . If u = 0 in Ω , as the term g(x, u) is subcritical, from ⟨J ′ (un ), un ⟩ = o(1) we have ∥un ∥2 −
∫ Ω
∗
2 (s) (u+ n) dx = o(1). s |x|
(2.12)
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221
And, by the definition of Aµ,s , we have ∥un ∥2 ≥ Aµ,s
(∫ (u+ )2∗ (s) ) ∗2 2 (s) n dx . s |x| Ω
(2.13)
It follows from (2.12) and (2.13) that 2
o(1) ≥ ∥un ∥
(
2∗ (s)
1−
− Aµ,s 2
∥un ∥2
∗
(s)−2
)
.
(2.14)
If ∥un ∥ → 0, it contradicts c > 0. Therefore, N −s 2−s ∥un ∥2 ≥ Aµ,s + o(1).
(2.15)
By (2.8), (2.12) and (2.15), we get ∫ 2∗ (s) 1 (u+ 1 n) dx + o(1) ∥un ∥2 − ∗ s 2 2 (s) Ω |x| 2−s = ∥un ∥2 + o(1) 2(N − s) N −s 2−s 2−s Aµ,s ≥ + o(1), 2(N − s)
⟨J ′ (un ), un ⟩ =
(2.16)
N −s
which contradicts c <
2−s 2−s 2(N −s) Aµ,s
. Thus u ̸= 0 and it is a nontrivial solution of problem (2.3).
□
Lemma 2.5. Suppose that 0 ≤ µ < µ ¯ − 1, 0 ≤ s < 2, g(x,t) satisfies (g1 ) and (g2 ). Then there exists 1 u0 ∈ H0 (Ω )\{0} such that N −s 2−s 2−s Aµ,s . (2.17) sup J(tu0 ) < 2(N − s) t≥0 Proof . For t ≥ 0, we consider the function ∗
t2 t2 (s) h(t) := J(tvε ) = ∥vε ∥2 − ∗ − 2 2 (s)
∫
G+ (x, tvε )dx,
(2.18)
Ω
∗
2 2 (s) ¯ := t ∥vε ∥2 − t h(t) . 2 2∗ (s)
(2.19)
Note that limt→+∞ h(t) = −∞, h(0) = 0, and h(t) > 0 as t is small enough. Therefore, supt≥0 h(t) > 0 is attained for some tε > 0. Since ∫ ∗ 0 = h′ (tε ) = tε ∥vε ∥2 − t2ε (s)−1 − g + (x, tvε )vε dx, (2.20) RN
we have 2
∥vε ∥ =
∗ t2ε (s)−2
1 + tε
∫
Hence,
g + (x, tvε )vε dx ≥ t2ε
∗
(s)−2
.
(2.21)
Ω 2
tε ≤ ∥vε ∥ 2∗ (s)−2 ≜ t0ε .
(2.22)
By (g1 ), it is easy to verify that p−1
|g + (x, t)| ≤ c1 |t| + c2 |t| Hence, we obtain ∥vε ∥2 ≤ t2ε
∗
(s)−2
+
c1 2
∫
2
|vε | dx + Ω
.
c2 p−2 |tε | p
(2.23) ∫
p
|vε | dx. Ω
(2.24)
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By (2.7), (2.8) and (2.24), when ε small enough, we conclude that t2ε
∗
(s)−2
≥
1 Aµ,s . 2
(2.25) 2
¯ attains its maximum at t0 = ∥vε ∥ 2∗ (s)−2 and is increasing in the interval On the other hand, the function h(t) ε 0 [0, tε ]. Together with (2.7), (2.24) and G+ (x, t) ≥ Kt2 for t ≥ 0, we deduce that ∫ ¯ 0) − h(tε ) ≤ h(t G+ (x, tε vε )dx ε Ω ∫ 2(N −s) 2−s 2−s ∥vε ∥ − ≤ G+ (x, tε vε )dx 2(N − 2) Ω ∫ (2.26) N −s N −2 2−s 2 2−s Aµ,s + O(ε 2−s ) − K (tε )2 |vε | dx ≤ 2(N − s) Ω ∫ N −s 2 N −2 A 2−s µ,s 2∗ (s)−2 2 2−s Aµ,s + O(ε 2−s ) − K( ) |vε | dx. ≤ 2(N − s) 2 Ω From (2.8), there holds ∫
2
N −2 √ µ−µ ¯
|vε | dx = O(ε (2−s)
).
(2.27)
Ω
And since
N −2 N −2 √ , < 2−s (2 − s) µ ¯−µ
choosing ε small enough, we have sup J(tvε ) = h(tε ) < t≥0
N −s 2−s 2−s . □ Aµ,s 2(N − s)
Proof of Theorem 1.1. From the Hardy and Hardy–Sobolev inequalities, we get: ∫ Ω
2∗ (s)
|u| 2∗ (s) , s dx ≤ C∥u∥ |x|
∥u∥qq ≤ C∥u∥q f or 1 ≤ q ≤ 2∗ ,
u ∈ H01 (Ω ).
(2.28)
It follows from (g1 ) that p−1
|g + (x, t)| ≤ c1 |t| + c2 |t| , 1 1 2 p |G+ (x, t)| ≤ c1 |t| + c2 |t| , 2 p
(2.29) (2.30)
for all t ∈ R and x ∈ Ω . By (2.28) and (2.30) we have, for c1 small enough ∫ ∫ ∗ 1 1 (u+ )2 (s) dx − G+ (x, u)dx J(u) = ∥u∥2 − ∗ s 2 2 (s) Ω |x| Ω 1 C Cc1 Cc2 2 2∗ (s) 2 ≥ ∥u∥ − ∗ ∥u∥ − ∥u∥2 − ∥u∥pp 2 2 (s) 2 p ∗ 1 − Cc1 C C ≥ ∥u∥2 − ∗ ∥u∥2 (s) − ∥u∥P . 2 2 (s) p So, there exists α > 0 such that J(u) ≥ α > 0 for u ∈ ∂Br (0) when r > 0 small enough. Moreover, by the nonnegativity of G+ (x, u), there holds ∫ ∫ ∗ 2∗ (s) t2 t2 (s) (u+ 0) J(tu0 ) = ∥u0 ∥2 − ∗ dx − G+ (x, tu0 )dx s 2 2 (s) Ω |x| Ω ∫ ∗ 2∗ (s) t2 t2 (s) (u+ 0) dx, ≤ ∥u0 ∥2 − ∗ s 2 2 (s) Ω |x|
M. Wang and Q. Zhang / Applied Mathematics Letters 94 (2019) 217–223
223
and lim J(tu0 ) → −∞,
as t → ∞.
t→∞
Hence we can choose t0 > 0 such that ∥t0 u0 ∥ > r and J(t0 u0 ) ≤ 0. Using the Mountain Pass Lemma, there is a sequence {un } ⊂ H01 (Ω ) satisfying J ′ (un ) → 0,
J(un ) → c ≥ α, where
c = inf max J(γ(t)), γ∈Γ t∈[0,1] { ( )⏐ } Γ = γ(t) ∈ C [0, 1], H01 ⏐γ(0) = 0, γ(1) = t0 u0 .
And, applying Lemmas 2.4 and 2.5, we get a (P S)c sequence {un } ⊂ H01 (Ω ), and u ∈ H01 (Ω ) such that J ′ (u) = 0. Thus u is a solution of problem (2.3). And then ⟨J ′ (u), u− ⟩ = 0, where u− = min{u, 0}. Hence u− = 0, that is, u ≥ 0. We get that u is a nonnegative solution of problem (1.1). □ Proof of Theorem 1.2. From Theorem 1.1, problem (2.3) has a nonnegative solution u1 . Set k(x, t) = −g(x, −t) for t ∈ R. It follows from Theorem 1.1 that the equation −∆u − µ
2∗ (s)−2
u 2
|x|
=
|u|
|x|
s
u + k(x, u),
has at least a nonnegative solution v. Let u2 = v, then u2 is a solution of −∆u − µ
u |x|
2
=
|u|
2∗ (s)−2 s
|x|
u + g(x, u).
Obviously u1 ≥ 0, u2 ≤ 0. So, Eq. (1.1) has at least two distinct nontrivial solutions. □ References [1] L. Ding, C.L. Tang, Existence and multiplicity of positive solutions for a class of semilinear elliptic equation involving Hardy term and Hardy-Sobolev critical exponents, J. Math. Anal. Appl. 339 (2) (2008) 1073–1083. [2] D.S. Kang, S.J. Peng, Positive solutions for singular critical elliptic problems, Appl. Math. Lett. 17 (2004) 411–416. [3] D.S. Kang, S.J. Peng, Solutions for semi-linear elliptic problems with critical HardyCSobolev exponents and Hardy potential, Appl. Math. Lett. 18 (10) (2005) 1094–1100. [4] H.T. Yang, J.H. Chen, A result on Hardy-Sobolev critical elliptic equations with boundary singularities, Commun. Pure Appl. Anal. 6 (2007) 191–201. [5] L. Ding, C.L. Tang, Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy-Sobolev critical exponents, Appl. Math. Lett. 20 (12) (2007) 1175–1183. [6] D.M. Cao, S.J. Peng, A note on the sign-changing solutions to elliptic problems with critical Sobolev and Hardy terms, J. Differential Equations 193 (2003) 424–434. [7] D.M. Cao, P.G. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations 205 (2004) 521–537. [8] A. Ferrero, F. Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177 (2001) 494–522. [9] D.S. Kang, S.J. Peng, Existence of solutions for a singular critical elliptic equation, J. Math. Anal. Appl. 284 (2003) 724–732. [10] D.S. Kang, Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents in RN , Nonlinear Anal. 66 (2007) 241–252. [11] E. Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations 156 (1999) 407–426. [12] X.H. Tang, Infinitely many solutions for semilinear Schr?dinger equations with sign-changing potential andnonlinearity, J. Math. Anal. Appl. 401 (2013) 407–415. [13] N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (12) (2000) 5703–5743.