The minimizing problem involving p--Laplacian and Hardy--Littlewood--Sobolev upper critical exponent
aa r X i v : . [ m a t h . A P ] M a y The minimizing problem involving p–Laplacian andHardy–Littlewood–Sobolev upper critical exponent ∗ Yu Su † Haibo Chen ‡ School of Mathematics and Statistics, Central South University,Changsha, 410083 Hunan, P.R.China.
Abstract:
In this paper, we study the minimizing problem: S p, ,α,µ := inf u ∈ W ,p ( R N ) \{ } ´ R N |∇ u | p d x − µ ´ R N | u | p | x | p d x (cid:16) ´ R N ´ R N | u ( x ) | p ∗ α | u ( y ) | p ∗ α | x − y | α d x d y (cid:17) p · p ∗ α , where N > p ∈ (1 , N ), µ ∈ h , (cid:16) N − pp (cid:17) p (cid:17) , α ∈ (0 , N ) and p ∗ α = p (cid:16) N − αN − p (cid:17) is theHardy–Littlewood–Sobolev upper critical exponent. Firstly, by using refinement of Hardy-Littlewood-Sobolev inequality, we prove that S p, ,α,µ is achieved in R N by a radially sym-metric, nonincreasing and nonnegative function. Secondly, we give a estimation of extremalfunction. Keywords:
Refinement of Hardy–Littlewood–Sobolev inequality; Hardy–Littlewood–Sobolev upper critical exponent; Minimizing.
MSC (2010) Classifications:
In this paper, we consider the minimizing problem: S p, ,α,µ := inf u ∈ W ,p ( R N ) \{ } ´ R N |∇ u | p d x − µ ´ R N | u | p | x | p d x (cid:16) ´ R N ´ R N | u ( x ) | p ∗ α | u ( y ) | p ∗ α | x − y | α d x d y (cid:17) p · p ∗ α , ( P )where N > p ∈ (1 , N ), µ ∈ h , (cid:16) N − pp (cid:17) p (cid:17) , α ∈ (0 , N ) and p ∗ α = p (cid:16) N − αN − p (cid:17) is the Hardy–Littlewood–Sobolev upper critical exponent.The paper was motivated by some works appeared in recent years. For p = 2, problem ( P ) isclosely related to the nonlinear Choquard equation as follows: − ∆ u + V ( x ) u = ( | x | α ∗ | u | q ) | u | q − u, in R N , (1.1) ∗ This research was supported by National Natural Science Foundation of China 11671403. † E-mail: [email protected] (Y. Su). ‡ Corresponding author: E-mail: math [email protected] (H. Chen). α ∈ (0 , N ) and N − αN q N − αN − . For q = 2 and α = 1, the equation (1.1) goes back to thedescription of the quantum theory of a polaron at rest by Pekar in 1954 [16] and the modeling ofan electron trapped in its own hole in 1976 in the work of Choquard, as a certain approximation toHartree–Fock theory of one–component plasma [18]. For q = N − N − and α = 1, by using the Greenfunction, it is obvious that equation (1 .
1) can be regarded as a generalized version of Schr¨odinger–Newton system: − ∆ u + V ( x ) u = | u | N +1 N − φ, in R N , − ∆ φ = | u | N +1 N − , in R N . The existence and qualitative properties of solutions of Choquard type equations (1.1) have beenwidely studied in the last decades (see [13]). Moroz and Van Schaftingen [12] considered equation(1.1) with lower critical exponent N − αN if the potential 1 − V ( x ) should not decay to zero at infinityfaster than the inverse of | x | . In [2], the authors studied the equation (1.1) with critical growth inthe sense of Trudinger–Moser inequality and studied the existence and concentration of the groundstates. In 2016, Gao and Yang [8] firstly investigated the following critical Choquard equation: − ∆ u = (cid:18) ˆ R N | u | ∗ α | x − y | α d y (cid:19) | u | ∗ α − u + λu, in Ω , (1.2)where Ω is a bounded domain of R N , with lipschitz boundary, N > α ∈ (0 , N ) and λ > − ∆) s u = ˆ R N | u | ∗ α,s | x − y | α d y ! | u | ∗ α,s − u + λu, in Ω , (1.3)where Ω is a bounded domain of R N with C , boundary, s ∈ (0 , N > s , α ∈ (0 , N ) and λ >
0, 2 ∗ α,s = N − αN − s is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality.By using variational methods, they established the existence, multiplicity and nonexistence ofnontrivial solutions to problem (1.3).For p = 2, in 2017, Pucci, Xiang and Zhang [20] studied the Schr¨odinger–Choquard–Kirchhoffequations involving the fractional p–Laplacian as follows:( a + b k u k p ( θ − s )[( − ∆) sp u + V ( x ) | u | p − u ] = λf ( x, u ) + ˆ R N | u | p ∗ α,s | x − y | α d y ! | u | p ∗ α,s − u in R N , (1.4)where k u k s = (cid:16) ´ R N ´ R N | u ( x ) − u ( y ) | p | x − y | N + ps d x d y + ´ R N V ( x ) | u | p d x (cid:17) , a, b ∈ R +0 with a + b > λ > s ∈ (0 , N > ps , θ ∈ [1 , NN − ps ), α ∈ (0 , N ), p ∗ α,s = p (2 N − α )2( N − sp ) is the critical exponentin the sense of Hardy–Littlewood–Sobolev inequality, and f : R N → R is a Caratheodory function, V : R N → R + is a potential function. By using variational methods, they established the existenceof nontrivial nonnegative solution to problem (1.4).2here is an open problem in [20]. We define the best constant: S p,s,α,µ := inf u ∈ W s,p ( R N ) \{ } ´ R N ´ R N | u ( x ) − u ( y ) | p | x − y | N + ps d x d y − µ ´ R N | u | p | x | ps d x (cid:18) ´ R N ´ R N | u ( x ) | p ∗ α,s | u ( y ) | p ∗ α,s | x − y | α d x d y (cid:19) p · p ∗ α,s , (1.5)where N > p ∈ (1 , N ), s ∈ (0 , α ∈ (0 , N ) and µ ∈ [0 , C N,s,p ), C N,s,p is defined in [6,Theorem 1.1]. And p ∗ α,s = p (2 N − α )2( N − sp ) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. Open problem:
Is the best constant S p,s,α,µ achieved? (Result 1) For p = 2, s = 1, µ = 0 and α ∈ (0 , N ), Gao and Yang [8] showed that S , ,α, isachieved in R N by the extremal function: w σ ( x ) = C σ − N − w ( x ) , w ( x ) = b ( b + | x − a | ) N − , where C > a ∈ R N and b ∈ (0 , ∞ ). (Result 2) For p = 2, s ∈ (0 , µ = 0 and α ∈ (0 , N ), Mukherjee and Sreenadh [8] proved that S ,s,α, is achieved in R N by the extremal function: w σ ( x ) = C σ − N − s w ( x ) , w ( x ) = b ( b + | x − a | ) N − s , where C > a ∈ R N and b ∈ (0 , ∞ ). (Result 3) For p = 2, s ∈ (0 , µ ∈ (cid:20) , s Γ ( N +2 s )Γ ( N − s ) (cid:19) and α ∈ (0 , N ), Yang and Wu [25] showedthat S ,s,α,µ is achieved in R N .For Open problem, we study the case of p ∈ (1 , N ), s = 1, µ ∈ h , (cid:16) N − pp (cid:17) p (cid:17) and α ∈ (0 , N ).By using the refinement of Sobolev inequality in [15, Theorem 2], we show that S p, ,α,µ is achievedin R N (see Theorem 1.1).For the case p = 2, one expects that the minimizers of S p,s,α,µ have a form similar to the function ω σ . However, it is not known the explicit formula of the extremal function. We give the estimationof extremal function (see Theorem 1.2 and Theorem 1.3).The first main result of this paper reads as follows. Theorem 1.1.
Let N > , p ∈ (1 , N ) , α ∈ (0 , N ) and µ ∈ h , (cid:16) N − pp (cid:17) p (cid:17) . Then S p, ,α,µ is achievedin R N by a radially symmetric, nonincreasing and nonnegative function. The second main result of this paper reads as follows. For p = 2 and s ∈ (0 , heorem 1.2. Let N > , p = 2 , α ∈ (0 , N ) , s ∈ (0 , and µ ∈ [0 , ¯ µ ) . Any nonnegative minimizer u of S ,s,α,µ is radially symmetric and nonincreasing, and it satisfies for x = 0 that C (cid:18)(cid:18) ¯ µ ¯ µ − µ (cid:19) S ,s,α,µ (cid:19) ( N − α )( N − s )2 N ( N +2 s − α ) (cid:18) Nω N − (cid:19) N − s N | x | N − s > u ( x ) , where ω N − is the area of the unit sphere in R N . The third main result of this paper reads as follows. For p = 2 and s = 1, we give a estimationof extremal function. Theorem 1.3.
Let N > , p ∈ (1 , N ) , α ∈ (0 , N ) and µ ∈ [0 , ˜ µ ) . Any nonnegative minimizer u of S p, ,α,µ is radially symmetric and nonincreasing, and it satisfies for x = 0 that α N ω N − ! · p ∗ α | x | N − pp > u ( x ) , where ω N − is the area of the unit sphere in R N . The Sobolev space W ,p ( R N ) is the completion of C ∞ ( R N ) with respect to the norm k u k pW = ˆ R N |∇ u | p d x. For s ∈ (0 ,
1) and p ∈ (1 , N ), the fractional Sobolev space W s,p ( R N ) is defined by W s,p ( R N ) := (cid:26) u ∈ L NpN − sp ( R N ) | ˆ R N ˆ R N | u ( x ) − u ( y ) | p | x − y | N + ps d x d y < ∞ (cid:27) . For s ∈ (0 ,
1) and p ∈ (1 , N ), we introduce the Hardy inequalities:¯ µ ˆ R N | u | | x | s d x ˆ R N ˆ R N | u ( x ) − u ( y ) | | x − y | N +2 s d x d y, for any u ∈ W s, ( R N ) and ¯ µ = 4 s Γ ( N +2 s )Γ ( N − s ) , and ˜ µ ˆ R N | u | p | x | p d x ˆ R N |∇ u | p d x, for any u ∈ W ,p ( R N ) and ˜ µ = (cid:18) N − pp (cid:19) p . The Coulomb–Sobolev space [5] is defined by E s,α, ∗ α,s ( R N ) = ( ˆ R N ˆ R N | u ( x ) − u ( y ) | | x − y | N +2 s d x d y < ∞ and ˆ R N ˆ R N | u ( x ) | ∗ α,s | u ( y ) | ∗ α,s | x − y | α d x d y < ∞ ) . (2.6)We endow the space E s,α, ∗ α,s ( R N ) with the norm k u k E ,α = ˆ R N ˆ R N | u ( x ) − u ( y ) | | x − y | N +2 s d x d y + ˆ R N ˆ R N | u ( x ) | ∗ α,s | u ( y ) | ∗ α,s | x − y | α d x d y ! ∗ α,s . (2.7)4e could define the best constant: S p, , ,µ := inf u ∈ W ,p ( R N ) \{ } k u k pW − µ ´ R N | u | p | x | p d x ( ´ R N | u | p ∗ d x ) pp ∗ , (2.8)where S p, , ,µ is attained in R N . Lemma 2.1. ( Hardy − Littlewood − Sobolev inequality, [ ]) Let t, r > and < µ < N with t + r + µN = 2 , f ∈ L t ( R N ) and h ∈ L r ( R N ) . There exists a sharp constant C > , independentof f, g such that ˆ R N ˆ R N | f ( x ) || h ( y ) || x − y | µ d x d y C k f k t k h k r . A measurable function u : R N → R belongs to the Morrey space k u k L r,̟ ( R N ) with r ∈ [1 , ∞ )and ̟ ∈ (0 , N ] if and only if k u k r L r,̟ ( R N ) = sup R> ,x ∈ R N R ̟ − N ˆ B ( x,R ) | u ( y ) | r d y < ∞ . Lemma 2.2. [15] For any < p < N , let p ∗ = NpN − p . There exists C > such that for θ and ϑ satisfying pp ∗ θ < , ϑ < p ∗ = NpN − p , we have (cid:18) ˆ R N | u | p ∗ d x (cid:19) p ∗ C k u k θW k u k − θ L ϑ, ϑ ( N − p ) p ( R N ) , for any u ∈ W ,p ( R N ) . Lemma 2.3. ( Endpoint ref ined Sobolev inequality, [ , T heorem . Let α ∈ (0 , N ) and s ∈ (0 , . Then there exists a constant C > such that the inequality k u k L NN − s ( R N ) C (cid:18) ˆ R N ˆ R N | u ( x ) − u ( y ) | | x − y | N +2 s d x d y (cid:19) ( N − α )( N − s )2 N ( N +2 s − α ) · ˆ R N ˆ R N | u ( x ) | N − αN − s | u ( y ) | N − αN − s | x − y | α d x d y ! s ( N − s ) N ( N +2 s − α ) holds for all u ∈ E s,α, ∗ α,s ( R N ) . We show the refinement of Hardy-Littlewood-Sobolev inequality. This inequality plays a key rolein the proof of Theorem 1.1.
Lemma 3.1.
For any < p < N and α ∈ (0 , N ) , there exists C > such that for θ and ϑ satisfying pp ∗ θ < , ϑ < p ∗ = NpN − p , we have (cid:18) ˆ R N ˆ R N | u ( x ) | p ∗ α | u ( y ) | p ∗ α | x − y | α d x d y (cid:19) p ∗ α C k u k θW k u k − θ ) L ϑ, ϑ ( N − p ) p ( R N ) , for any u ∈ W ,p ( R N ) . roof. By using Lemma 2.2, we have (cid:18) ˆ R N | u | p ∗ d x (cid:19) p ∗ C k u k θW k u k − θ L ϑ, ϑ ( N − p ) p ( R N ) . (3.9)By Hardy-Littlewood-Sobolev inequality and (3.9), we obtain (cid:18) ˆ R N ˆ R N | u ( x ) | p ∗ α | u ( y ) | p ∗ α | x − y | α d x d y (cid:19) p ∗ α C p ∗ α k u k L p ∗ ( R N ) C p ∗ α C k u k θW k u k − θ ) L ϑ, ϑ ( N − p ) p ( R N ) . (cid:3) In [15], there is a misprint, the authors point out it by themselves. The right one is L p ∗ ( R N ) ֒ → L r,r N − pp ( R N ) , (3.10)for any p ∈ (1 , N ) and r ∈ [1 , p ∗ ). This embedding plays a key role in the proof of Theorem 1.1. Proof of Theorem 1.1: Step 1.
Suppose now 0 µ < ˜ µ = (cid:16) N − pp (cid:17) p . Applying Lemma 3.1 with ϑ = p , we have (cid:18) ˆ R N ˆ R N | u ( x ) | p ∗ α | u ( y ) | p ∗ α | x − y | α d x d y (cid:19) p ∗ α C (cid:18) k u k pW − µ ˆ R N | u | p | x | p d x (cid:19) θp k u k − θ ) L p,N − p ( R N ) , (3.11)for u ∈ W ,p ( R N ). Let { u n } be a minimizing sequence of S p, ,α,µ , that is k u n k pW − µ ˆ R N | u n | p | x | p d x → S p, ,α,µ , as n → ∞ , and ˆ R N ˆ R N | u n ( x ) | p ∗ α | u n ( y ) | p ∗ α | x − y | α d x d y = 1 . Inequality (3.11) enables us to find
C > n such that k u n k L p,N − p ( R N ) > C > . (3.12)We have the chain of inclusions W ,p ( R N ) ֒ → L p ∗ ( R N ) ֒ → L p,N − p ( R N ) , (3.13)which implies that k u n k L p,N − p ( R N ) C. (3.14)Applying (3.12) and (3.14), there exists C > < C k u n k L p,N − p ( R N ) C − . n ∈ N , we get the existenceof λ n > x n ∈ R N such that1 λ pn ˆ B ( x n ,λ n ) | u n ( y ) | p d y > k u n k p L p,N − p ( R N ) − C n > ˜ C > C that does not depend on n .Let v n ( x ) = λ N − pp n u n ( λ n x ). Notice that, by using the scaling invariance, we have k v n k pW − µ ˆ R N | v n | p | x | p d x → S p, ,α,µ , as n → ∞ , and ˆ R N ˆ R N | v n ( x ) | p ∗ α | v n ( y ) | p ∗ α | x − y | α d x d y = 1 . Then ˆ B ( xnλn , | v n ( y ) | p d y > ˜ C > . We can also show that v n is bounded in W ,p ( R N ). Hence, we may assume v n ⇀ v in W ,p ( R N ) , v n → v a . e . in R N , v n → v in L qloc ( R N ) for all q ∈ [ p, p ∗ ) . We claim that { x n λ n } is uniformly bounded in n . Indeed, for any 0 < β < p , by H¨older’s inequality,we observe that0 < ˜ C ˆ B ( xnλn , | v n | p d y = ˆ B ( xnλn , | y | pβp ( N − β ) N − p | v n | p | y | pβp ( N − β ) N − p d y ˆ B ( xnλn , | y | β ( N − p ) p − β d y ! − N − pN − β ˆ B ( xnλn , | v n | p ( N − β ) N − p | y | β d y N − pN − β . By the rearrangement inequality, see [10, Theorem 3.4], we have ˆ B ( xnλn , | y | β ( N − p ) p − β d y ˆ B (0 , | y | β ( N − p ) p − β d y C. Therefore, 0 < C ˆ B ( xnλn , | v n | p ( N − β ) N − p | y | β d y. (3.15)7ow, suppose on the contrary, that x n λ n → ∞ as n → ∞ . Then, for any y ∈ B ( x n λ n , | y | > | x n λ n | − n large. Thus, ˆ B ( xnλn , | v n | p ( N − β ) N − p | y | β d y | x n λ n | − β ˆ B ( xnλn , | v n | p ( N − β ) N − p d y (cid:12)(cid:12)(cid:12) B ( x n λ n , (cid:12)(cid:12)(cid:12) βN ( | x n λ n | − β ˆ B ( xnλn , | v n | NpN − p d y ! N − βN (cid:12)(cid:12)(cid:12) B ( x n λ n , (cid:12)(cid:12)(cid:12) βN ( | x n λ n | − β · k v n k N − βN W S N − βN − p p, , , C ( | x n λ n | − β → n → ∞ , which contradicts (3.15). Hence, { x n λ n } is bounded, and there exists R > ˆ B (0 ,R ) | v n ( y ) | p d y > ˆ B ( xnλn , | v n ( y ) | p d y > ˜ C > . Since the embedding W ,p ( R N ) ֒ → L qloc ( R N ) q ∈ [ p, p ∗ ) is compact, we deduce that ˆ B (0 ,R ) | v ( y ) | p d y > ˜ C > , which means v Step 2.
Set h ( t ) = t · p ∗ αp , t > < p < N ) . Since p ∈ (1 , N ) and α ∈ (0 , N ), we get2 · p ∗ α p = 2 N − αN − p > N + p − α > . We know that h ′′ ( t ) = (2 N − α )( N + p − α )( N − p ) t p − αN − p > , which implies that h ( t ) is a convex function. By using h (0) = 0 and l ∈ [0 , h ( lt ) = h ( lt + (1 − t ) · lh ( t ) + (1 − l ) h (0) = lh ( t ) . (3.16)For any t , t ∈ [0 , ∞ ), applying last inequality with l = t t + t and l = t t + t , we get h ( t ) + h ( t ) = h (cid:18) ( t + t ) t t + t (cid:19) + h (cid:18) ( t + t ) t t + t (cid:19) t t + t h ( t + t ) + t t + t h ( t + t ) (by (3 . h ( t + t ) . (3.17)8ow, we claim that v n → v strongly in W ,p ( R N ) . Set K ( u, v ) = ˆ R N |∇ u | p − ∇ u ∇ v d x − µ ˆ R N | u | p − uv | x | p d x. Since { v n } is a minimizing sequence, lim n →∞ K ( v n , v n ) = S p, ,α,µ . By using Br´ezis–Lieb type lemma [4] and [20, Theorem 2.3], we know K ( v, v ) + lim n →∞ K ( v n − v, v n − v ) = lim n →∞ K ( v n , v n ) + o (1) = S p, ,α,µ + o (1) , (3.18)and ˆ R N ˆ R N | v n ( x ) | p ∗ α | v n ( y ) | p ∗ α | x − y | α d x d y − ˆ R N ˆ R N | v n ( x ) − v ( x ) | p ∗ α | v n ( y ) − v ( y ) | p ∗ α | x − y | α d x d y = ˆ R N ˆ R N | v ( x ) | p ∗ α | v ( y ) | p ∗ α | x − y | α d x d y + o (1) . (3.19)where o (1) denotes a quantity that tends to zero as n → ∞ . Therefore,1 = lim n →∞ ˆ R N ˆ R N | v n ( x ) | p ∗ α | v n ( y ) | p ∗ α | x − y | α d x d y = lim n →∞ ˆ R N ˆ R N | v n ( x ) − v ( x ) | p ∗ α | v n ( y ) − v ( y ) | p ∗ α | x − y | α d x d y + ˆ R N ˆ R N | v ( x ) | p ∗ α | v ( y ) | p ∗ α | x − y | α d x d y S − · p ∗ αp p, ,α,µ (cid:16) lim n →∞ K ( v n − v, v n − v ) (cid:17) · p ∗ αp + S − · p ∗ αp p, ,α,µ ( K ( v, v )) · p ∗ αp S − · p ∗ αp p, ,α,µ (cid:16) lim n →∞ K ( v n − v, v n − v ) + K ( v, v ) (cid:17) · p ∗ αp (by (3 . . . Therefore, all the inequalities above have to be equalities. We know that (cid:16) lim n →∞ K ( v n − v, v n − v ) (cid:17) · p ∗ αp + ( K ( v, v )) · p ∗ αp = (cid:16) lim n →∞ K ( v n − v, v n − v ) + K ( v, v ) (cid:17) · p ∗ αp . (3.20)We show that lim n →∞ K ( v n − v, v n − v ) = 0. Combining (3.17) and (3.20), we know thateithor lim n →∞ K ( v n − v, v n − v ) = 0 or K ( v, v ) = 0 . Since v
0, so K ( v, v ) = 0. Therefore,lim n →∞ K ( v n − v, v n − v ) = 0 . (3.21)9his implies that v n → v strongly in W ,p ( R N ) . Moreover, we getlim n →∞ ˆ R N ˆ R N | v n ( x ) − v ( x ) | p ∗ α | v n ( y ) − v ( y ) | p ∗ α | x − y | α d x d y = 0 . (3.22) Step 3.
Since v
0, putting (3.21) into (3.18), and inserting (3.22) into (3.19), we knowlim n →∞ (cid:18) k v n k pW − µ ˆ R N | v n | p | x | p d x (cid:19) → S p, ,α,µ = k v k pW − µ ˆ R N | v | p | x | p d x, and ˆ R N ˆ R N | v ( x ) | p ∗ α | v ( y ) | p ∗ α | x − y | α d x d y = 1 . Then v is an extremal.In addition, | v | ∈ W ,p ( R N ) and |∇| v || = |∇ v | a.e. in R N , therefore, | v | is also an extremal,and then there exist non–negative extremals.Let ¯ v > v ∗ the symmetric–decreasing rearrangement of ¯ v (See [10,Section 3]). From [19] it follows that ˆ R N |∇ ¯ v ∗ | p d x ˆ R N |∇ ¯ v | p d x. (3.23)According to the simplest rearrangement inequality in [10, Theorem 3.4], we get ˆ R N | ¯ v | p | x | p d x ˆ R N | ¯ v ∗ | p | x | p d x. (3.24)By using Riesz’s rearrangement inequality in [10, Theorem 3.7], we have ˆ R N ˆ R N | ¯ v ( x ) | p ∗ α | ¯ v ( y ) | p ∗ α | x − y | α d x d y ˆ R N ˆ R N | ¯ v ∗ ( x ) | p ∗ α | ¯ v ∗ ( y ) | p ∗ α | x − y | α d x d y. (3.25)Combining (3.23), (3.24) and (3.25), and the fact that µ >
0, we get that ¯ v ∗ is also an extremal,and then there exist radially symmetric and nonincreasing extremal. (cid:3) For p = 2 and s ∈ (0 , u ( x ). The proof of Theorem 1.2is based on the Coulomb–Sobolev space E s,α, ∗ α,s ( R N ) and the endpoint refined Sobolev inequalityin Lemma 2.3. Proof of Theorem 1.2:
In this step, we show some properties of radially symmetric, nonincreasingand nonnegative function u ( x ). Let ¯ µ = 4 s Γ ( N +2 s )Γ ( N − s ) . By the definition of extremal u (see the proofof Theorem 1.1), we know ˆ R N ˆ R N | u ( x ) − u ( y ) | | x − y | N +2 s d x d y − µ ˆ R N | u | | x | d x = S ,s,α,µ , (4.26)10nd ˆ R N ˆ R N | u ( x ) | ∗ α,s | u ( y ) | ∗ α,s | x − y | α d x d y = 1 . (4.27)Applying (4.26), (4.27) and the definition of Coulomb–Sobolev space E s,α, ∗ α,s ( R N ), we get u ∈E s,α, ∗ α,s ( R N ).By using (4.26), (4.27), u ∈ E s,α, ∗ α,s ( R N ) and Lemma 2.3, we have k u k L NN − s ( R N ) C (cid:18) ˆ R N ˆ R N | u ( x ) − u ( y ) | | x − y | N +2 s d x d y (cid:19) ( N − α )( N − s )2 N ( N +2 s − α ) · ˆ R N ˆ R N | u ( x ) | N − αN − s | u ( y ) | N − αN − s | x − y | α d x d y ! s ( N − s ) N ( N +2 s − α ) = C (cid:18) ˆ R N ˆ R N | u ( x ) − u ( y ) | | x − y | N +2 s d x d y (cid:19) ( N − α )( N − s )2 N ( N +2 s − α ) C (cid:18)(cid:18) ¯ µ ¯ µ − µ (cid:19) S ,s,α,µ (cid:19) ( N − α )( N − s )2 N ( N +2 s − α ) . (4.28)For any 0 < R < ∞ and B ( R ) = B (0 , R ) ⊂ R N , we obtain C (cid:18)(cid:18) ¯ µ ¯ µ − µ (cid:19) S ,s,α,µ (cid:19) ( N − α )( N − s )2 N ( N +2 s − α ) > (cid:18) ˆ R N | u ( x ) | NN − s d x (cid:19) N − s N > ˆ B ( R ) | u ( x ) | NN − s d x ! N − s N > | u ( R ) | ω N − s N N − (cid:18) ˆ R ρ N − d ρ (cid:19) N − s N = | u ( R ) | (cid:16) ω N − N (cid:17) N − s N R N − s , which implies C (cid:18)(cid:18) ¯ µ ¯ µ − µ (cid:19) S ,s,α,µ (cid:19) ( N − α )( N − s )2 N ( N +2 s − α ) (cid:18) Nω N − (cid:19) N − s N | x | N − s > | u ( x ) | . (cid:3) For p = 2 and s ∈ (0 , u ( x ). From Theorem 1.1, weknow that u ( x ) is a radially symmetric, nonincreasing and nonnegative function.The proof of Theorem 1.3 is different from Theorem 1.2. The endpoint refined Sobolev inequalityin Lemma 2.3 is true for p = 2. However, we don’t know that the endpoint refined Sobolev inequalityis true or not for p = 2. 11 roof of Theorem 1.3: Let ˜ µ = (cid:16) N − pp (cid:17) p . By the definition of extremal u , we know k u k pW − µ ˆ R N | u | p | x | p d x = S p, ,α,µ , (5.29)and ˆ R N ˆ R N | u ( x ) | p ∗ α | u ( y ) | p ∗ α | x − y | α d x d y = 1 . (5.30)For any R ∈ (0 , ∞ ) and B ( R ) = B (0 , R ) ⊂ R N , by H¨older’s inequality, we obtain ˆ B ( R ) | u | p ∗ α d x ! p ∗ α ˆ B ( R ) d x ! − p ∗ αp ∗ ˆ B ( R ) | u | p ∗ α · p ∗ p ∗ α d x ! p ∗ αp ∗ p ∗ α = | B ( R ) | p ∗ α − p ∗ ˆ B ( R ) | u | p ∗ d x ! p ∗ | B ( R ) | p ∗ α − p ∗ S − p p, , ,µ k u k W | B ( R ) | p ∗ α − p ∗ S − p p, , ,µ (cid:18)(cid:18) ˜ µ ˜ µ − µ (cid:19) S p, ,α,µ (cid:19) p < ∞ . By Fubini’s theorem, we get(2 R ) − α ˆ B ( R ) | u ( x ) | p ∗ α d x ! =(2 R ) − α ˆ B ( R ) | u ( x ) | p ∗ α d x ˆ B ( R ) | u ( y ) | p ∗ α d y =(2 R ) − α ˆ B ( R ) ˆ B ( R ) | u ( x ) | p ∗ α | u ( y ) | p ∗ α d x d y ˆ B ( R ) ˆ B ( R ) | u ( y ) | p ∗ α | u ( x ) | p ∗ α | x − y | α d x d y, which implies ˆ B ( R ) | u ( x ) | p ∗ α d x ! (2 R ) α ˆ R N ˆ R N | u ( y ) | p ∗ α | u ( x ) | p ∗ α | x − y | α d x d y. (5.31)According to (5.29), (5.30) and (5.31), we have1 = ˆ R N ˆ R N | u ( x ) | p ∗ α | u ( y ) | p ∗ α | x − y | α d x d y > (2 R ) − α ˆ B ( R ) | u | p ∗ α d x ! > (2 R ) − α | u ( R ) | · p ∗ α (cid:18) ω N − ˆ R ρ N − d ρ (cid:19) > ω N − α N | u ( R ) | · p ∗ α R N − α . α N ω N − ! · p ∗ α R N − pp > | u ( R ) | . Hence, for any 0 < | x | < ∞ , we obtain α N ω N − ! · p ∗ α | x | N − pp > u ( x ) . (cid:3) The results in this paper set the foundation for the study of a number of questions related tominimizing problem S p, ,α,µ := inf u ∈ W ,p ( R N ) \{ } ´ R N |∇ u | p d x − µ ´ R N | u | p | x | p d x (cid:16) ´ R N ´ R N | u ( x ) | p ∗ α | u ( y ) | p ∗ α | x − y | α d x d y (cid:17) p · p ∗ α , where N > p ∈ (1 , N ), α ∈ (0 , N ) and µ ∈ (cid:16) , (cid:16) N − pp (cid:17) p (cid:17) .During the preparation of the manuscript we faced several problems which are worth to betackled in forthcoming investigations. In the sequel, we shall formulate some of them: (a) The challenging problems are to prove the rest of Open problem: the case of N > p ∈ (1 , N ), s ∈ (0 , α ∈ (0 , N ) and µ ∈ [0 , C N,s,p ), and C N,s,p is defined in [6, Theorem 1.1]. (b)
In [21], the authors studied the following minimizing problem: I ,s,α,µ ( u, v ) := inf u,v ∈ W s, ( R N ) \{ } ´ R N ´ R N | u ( x ) − u ( y ) | + | v ( x ) − v ( y ) | | x − y | N +2 s d x d y − µ ´ R N (cid:16) | u | | x | s + | v | | x | s (cid:17) d x (cid:18) ´ R N ´ R N | u ( x ) | ∗ α,s | u ( y ) | ∗ α,s + | v ( x ) | ∗ α,s | v ( y ) | ∗ α,s | x − y | α d x d y (cid:19) ∗ α , where N > p = 2, s ∈ (0 , µ ∈ (cid:20) , s Γ ( N +2 s )Γ ( N − s ) (cid:19) and α ∈ (0 , N ).It is worth to extend the study of I ,s,α,µ ( u, v ) to the following minimizing problem: I p,s,α,µ ( u, v ) := inf u ∈ W s,p ( R N ) \{ } ´ R N ´ R N | u ( x ) − u ( y ) | p + | v ( x ) − v ( y ) | p | x − y | N + ps d x d y − µ ´ R N (cid:16) | u | p | x | ps + | v | p | x | ps (cid:17) d x (cid:18) ´ R N ´ R N | u ( x ) | p ∗ α,s | u ( y ) | p ∗ α,s + | v ( x ) | p ∗ α,s | v ( y ) | p ∗ α,s | x − y | α d x d y (cid:19) p · p ∗ α,s , where N > p ∈ (1 , N ), s ∈ (0 , µ ∈ [0 , C N,s,p ) and α ∈ (0 , N ). (c) By using Theorem 1.1 and Lemma 3.1, we could study the Choquard–equation involvingtwo critical nonlinearities: − ∆ p u − µ | u | p − u | x | p = (cid:18) ˆ R N | u | p ∗ α | x − y | α d y (cid:19) | u | p ∗ α − u + | u | p ∗ − u, in R N , N > p ∈ (1 , N ), µ ∈ [0 , (cid:16) N − pp (cid:17) p ) and α ∈ (0 , N ). (d) Ambrosetti, Brezis and Cerami [1] proved the existence of infinity many solutions to thefollowing problem − ∆ u = | u | ∗ − u + λ | u | q − u in Ω ,u = 0 in ∂ Ω , where Ω ⊂ R N is a smooth bounded domain, N > λ > q ∈ (1 , − ∆ p u = | u | p ∗ − u + λ | u | q − u in Ω ,u = 0 in ∂ Ω , where − ∆ p is the p –Laplacian operator, Ω ⊂ R N is a smooth bounded domain, N > λ > q ∈ (1 , p ) and p ∗ = NPN − p . Gao and Yang [9] proved the existence of infinity many solutions tofollowing problem − ∆ u = (cid:16) ´ Ω | u | ∗ α | x − y | α d y (cid:17) | u | ∗ α − u + λ | u | q − u in Ω ,u = 0 in R N \ Ω , where Ω ⊂ R N is a bounded domain with C , bounded boundary, N > λ > q ∈ (1 , < α < N and 2 ∗ α = N − αN − is the critical Hardy–Littlewood–Sobolev upper exponent.It is natural to ask: Does there exist a solution to following problem? − ∆ p u = (cid:16) ´ Ω | u | p ∗ α | x − y | α d y (cid:17) | u | p ∗ α − u + λ | u | q − u in Ω ,u = 0 in R N \ Ω , where Ω ⊂ R N is a bounded domain with C , bounded boundary, N > λ > p ∈ (1 , N ), q ∈ (1 , p ) and 0 < α < N . References [1] A. Ambrosetti, H. Brezis, G. Cerami,
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