Nonexistence of Positive Supersolution to a Class of Semilinear Elliptic Equations and Systems in an Exterior Domain
aa r X i v : . [ m a t h . A P ] D ec SCIENCE CHINAMathematics
CrossMark doi: 10.1007/s11425-000-0000-0 c (cid:13) Science China Press and Springer-Verlag Berlin Heidelberg 2017 math.scichina.com link.springer.com . ARTICLES .Nonexistence of positive supersolution to a class ofsemilinear elliptic equations and systems in anexterior domain
Huyuan Chen , Rui Peng ∗ & Feng Zhou School of Mathematics and Statistics, Jiangsu Normal University,Xuzhou, Jiangsu , China; Center for PDEs and Department of Mathematics, East China Normal University,Shanghai , ChinaEmail: [email protected], pengrui [email protected], [email protected]
Received March 27, 2018; accepted December 10, 2018
Abstract
In this paper, we primarily consider the following semilinear elliptic equation − ∆ u = h ( x, u ) in Ω ,u > ∂ Ω , where Ω is an exterior domain in R N with N > h : Ω × R + → R is a measurable function, and derive optimalnonexistence results of positive supersolution. Our argument is based on a nonexistence result of positivesupersolution of a linear elliptic problem with Hardy potential. We also establish sharp nonexistence results ofpositive supersolution to an elliptic system. Keywords
Semilinear elliptic problem; Supersolution; Nonexistence
MSC(2010)
Citation:
Huyuan Chen, Rui Peng, Feng Zhou. Nonexistence of positive supersolution to a class of semi-linear elliptic equations and systems in an exterior domain. Sci China Math, 2018, 60, doi:10.1007/s11425-000-0000-0
In this paper, we are mainly concerned with the nonexistence of positive supersolution to the followingsemilinear elliptic equation − ∆ u = h ( x, u ) in Ω ,u > ∂ Ω , (1.1)where Ω is a punctured or an exterior domain in R N with N >
3; that is, Ω = R N \ O , here O is abounded, closed smooth subset of R N and h : Ω × R + → R is a measurable function, here R + = [0 , + ∞ ).We assume, unless otherwise specified, that N > * Corresponding author Huyuan Chen et al. Sci China Math we also assume that
O ⊂ B R (0), where B R (0) represents the ball with the radius R , centered at theorigin. A typical punctured domain is Ω = R N \ { } , and a typical exterior domain is Ω = R N \ B R (0)with O = B R (0). A function u ∈ C (Ω) is said to be a positive supersolution of (1.1) if u ( x ) > and − ∆ u ( x ) > h ( x, u ) for all x ∈ Ω . The existence and nonexistence of solution or supersolution to problem (1.1) have attracted greatattentions for many years; see [2–6, 12, 13, 17, 19, 23, 25–27] and the references therein. In the special casethat h only depends on u , given R >
0, Alarcon, Melian and Quaas [2] proved that problem (1.1) withΩ = R N \ B R (0) admits a positive solution if and only if h satisfies Z σ h ( t ) t − N − N − dt < + ∞ for some σ >
0, by appealing to ODE techniques. Nevertheless, such an approach fails to apply to (1.1)if the nonlinear term h is not radially symmetric.In particular, when h ( x, u ) = V ( x ) u p , (1.1) becomes the following − ∆ u = V ( x ) u p in Ω ,u > ∂ Ω . (1.2)The traditional method to establish nonexistence results for solution or supersolution to (1.2) is to makeuse of the fundamental solution and Hadamard property ([5, 7, 8, 13]). It is worth mentioning that if p = −
2, problem (1.2) in a bounded domain is used to describe the MEMS model ([21, 24]), and if p = −
1, problem (1.2) is related to the study of singular minimal hypersurfaces with symmetry ([29,34]).When V ( x ) = (1+ | x | ) β and Ω is a punctured or an exterior domain, Bidaut-V´eron [7] and Bidaut-V´eronand Pohozaev [8] showed that problem (1.2) has no solution if p N + βN − p ∗ β with β ∈ ( − , . On the other hand, when Ω = R N \ { } and V ( x ) = | x | a (1 + | x | ) β − a with a ∈ ( − N, + ∞ ) and β ∈ ( −∞ , a ), Chen, Felmer and Yang [15] derived infinitely many positive solutions of (1.2) if p ∈ (cid:18) p ∗ β , N + a N − (cid:19) ∩ (0 , + ∞ ) . Armstrong and Sirakov [5] and Chen and Felmer [13] dealt with the more general potential V ( x ) > | x | β (ln | x | ) τ for | x | > e, where β > − τ ∈ R . Especially, [5, Theorem 3.1] and [13, Theorem 1.1] imply the following result: • Let β ∈ ( − , . Problem (1.2) with Ω = R N \ B e (0) has no positive supersolution provided thateither p < p ∗ β , τ ∈ R or p = p ∗ β , τ > . In the current paper, we will provide a sharp improvement of the above result; indeed, we can concludethe following: • Let β > − . Problem (1.2) with Ω = R N \ B e (0) has no positive supersolution provided that either p < p ∗ β , τ ∈ R or p = p ∗ β , τ > − (see Proposition 4.1); • Let β = − and p = p ∗ β = 1 . Problem (1.2) with Ω = R N \ B e (0) has no positive supersolutionprovided that lim inf | x |→ + ∞ V ( x ) | x | > ( N − (see Theorem 2.1); • Let β ∈ R . Problem (1.2) with Ω = R N \ B ℓ (0) has positive supersolution for properly large ℓ providedthat either p > p ∗ β , τ ∈ R or p = p ∗ β , τ < − (see Proposition 4.2). As a consequence, the above results show that both p = p ∗ β and τ = − h ( x, u ) > ˜ h ( x, u ) > x ∈ R N \ B e (0) , u > , and ˜ h : Ω × R + → R + is a function satisfying the following uyuan Chen et al. Sci China Math (H) (a) for any x ∈ R N \ B e (0), ˜ h ( x, s ) s > ˜ h ( x, s ) s if s > s > t >
0, lim inf | x |→ + ∞ ˜ h ( x, t | x | − N ) | x | N > ( N − σ ∈ (0 ,
1) such that for any t > | x |→ + ∞ ˜ h ( x, t | x | − N ) | x | N (ln | x | ) σ > σ > t > | x |→ + ∞ ˜ h ( x, t | x | − N (ln | x | ) σ ) t | x | − N (ln | x | ) σ > ( N − . Then we have
Theorem 1.1.
Under the assumption (H) , problem (1.1) has no positive supersolution.
We would like to mention that (H)-(a) means that h is linear or superlinear, (H)-(b) is related to thesubcritical case while (H)-(b1)(b2) deal with the critical case. As one will see below, Theorem 1.1 allowsus to obtain some optimal nonexistence results of positive supersolution to problem (1.2).To prove Theorem 1.1, it turns out that a nonexistence result of positive supersolution of the linearHardy elliptic problem (2.1) (see Section 2) is vital in our analysis. Such a nonexistence result can beestablished by Agmon-Allegretto-Piepenbrink theory [1]; in this paper, we shall provide a different proofwhich seems simpler.Another focus of our paper is on the following semilinear elliptic system: − ∆ u = h ( x, u, v ) in Ω , − ∆ v = h ( x, u, v ) in Ω ,u, v > ∂ Ω . (1.3)It is known that Liouville-type theorems for system (1.3) have been established (see [16, 22, 32, 33, 35])primarily on the whole space Ω = R N . In particular, when h ( x, u, v ) = v p , h ( x, u, v ) = u q , thenonexistence of positive solution to (1.3) in Ω = R N has been investigated by [16,33,35] in the subcriticalcase that 1 p + 1 + 1 q + 1 > − N .
It seems that there is little research work devoted to the nonexistence/existence of solution/supersolutionto (1.3) when Ω is an exterior domain.
We say a function pair ( u, v ) ∈ C (Ω) × C (Ω) is a positive su-persolution of (1.3) if u ( x ) , v ( x ) > , − ∆ u ( x ) > h ( x, u, v ) and − ∆ v > h ( x, u, v ) for all x ∈ Ω . For system (1.3), two functions ˜ h , ˜ h are involved to control functions h , h respectively.(SH) Suppose that the nonnegative function ˜ h i ( i = 1 ,
2) defined on ( R N \ B e (0)) × [0 , + ∞ ) × [0 , + ∞ )satisfies that(a) the maps t → ˜ h ( x, s, t ), s → ˜ h ( x, s, t ) are nondecreasing and for any x ∈ R N \ B e (0),˜ h ( x, s , t ) s > ˜ h ( x, s , t ) s and ˜ h ( x, s , t ) t > ˜ h ( x, s , t ) t if s > s > , t > t > Huyuan Chen et al. Sci China Math (b) for any t >
0, lim inf | x |→ + ∞ ˜ h ( x, t | x | − N , t | x | − N ) | x | N > ( N − , or lim inf | x |→ + ∞ ˜ h ( x, t | x | − N , t | x | − N ) | x | N > ( N − σ , σ with either σ < σ <
1, such that for any t > | x |→ + ∞ ˜ h i ( x, t | x | − N , t | x | − N ) | x | N (ln | x | ) σ i +2 > , i = 1 , σ < σ > σ > t > | x |→ + ∞ ˜ h ( x, t | x | − N (ln | x | ) σ , t | x | − N ) t | x | − N (ln | x | ) σ > ( N − σ > σ <
1, there exists σ > t > | x |→ + ∞ ˜ h ( x, t | x | − N , t | x | − N (ln | x | ) σ )) t | x | − N (ln | x | ) σ > ( N − σ , σ <
1, there exist σ , σ > t > | x |→ + ∞ ˜ h ( x, t | x | − N (ln | x | ) σ , t | x | − N (ln | x | ) σ ) t | x | − N (ln | x | ) σ > ( N − | x |→ + ∞ ˜ h ( x, t | x | − N (ln | x | ) σ , t | x | − N (ln | x | ) σ )) t | x | − N (ln | x | ) σ > ( N − . Then we can state
Theorem 1.2.
Assume that for all ( x, u, v ) ∈ ( R N \ B e (0)) × [0 , + ∞ ) × [0 , + ∞ ) , the functions h , h satisfy h ( x, u, v ) > ˜ h ( x, u, v ) , h ( x, u, v ) > ˜ h ( x, u, v ) , with ˜ h , ˜ h fulfilling (SH) . Then system (1.3) has no positive supersolution. Assumption (SH)-(b) is related to the subcritical case, (SH)-(b2) says that one of the nonlinearities iscritical, and (SH)-(b3) represents that both nonlinear terms are critical. In particular, when the nonlin-earities h , h take the form h ( x, u, v ) = | x | β (ln | x | ) τ u p v q and h ( x, u, v ) = | x | β (ln | x | ) τ u p v q , weare able to clarify the nonexistence and existence of positive supersolution in terms of the parameters p , p , q , q , τ and τ ; see Proposition 4.3 and Proposition 4.4 for the precise details.The rest of the paper is organized as follows. In Section 2, we show the nonexistence of supersolutionof a linear Hardy problem. In Section 3, we prove our main results Theorems 1.1 and 1.2. In Section 4,we apply Theorems 1.1 and 1.2 to two concrete examples to obtain sharp nonexistence results. In this section, we shall investigate the linear elliptic problem with Hardy potential: − ∆ u = V ( x ) u in Ω ,u > ∂ Ω . (2.1)The nonexistence result of positive supersolution to (2.1) reads as follows. uyuan Chen et al. Sci China Math Theorem 2.1.
Assume that Ω is a punctured or an exterior domain, and V is a nonnegative functionsatisfying lim inf | x |→ + ∞ V ( x ) | x | > ( N − . (2.2) Then problem (2.1) has no positive supersolution.
We remark that ( N − / Z R N |∇ u | dx > ( N − Z R N u | x | dx. Related Hardy problems have been studied extensively; one may refer to, for instance, [9–11, 14, 20]. Inparticular, the authors [14] considered the Hardy problem − ∆ u = µ | x | u + f ( x ) in D \ { } , subject tothe zero Dirchlet boundary condition, where f > D is a bounded domain containing the origin,and proved that this problem has no positive solution once µ > ( N − . To this end, they used newdistributional identities to classify the isolated singular solution of − ∆ u = µ | x | u + f in D \ { } and findfundamental solutions of − ∆ u = µ | x | u in R N .It is worth noting that [18, 30, 31] indicate that the nonexistence of positive supersolution to (2.1)can be obtained by using Agmon-Allegretto-Piepenbrink theory [1]. We will provide a different andelementary proof. Our strategy is to employ the Kelvin transform to transfer the unbounded domain Ωinto a bounded one containing the origin.For the linear elliptic equation involving the general homogeneous potential in the punctured domain R N \ { } : − ∆ u = µ | x | − α u in R N \ { } , (2.3)we obtain that Theorem 2.2.
Problem (2.3) has no positive supersolution provided that one of the following conditionsholds: (i) α = 2 , µ > α = 2 , µ > ( N − . Theorem 2.2 is optimal in a certain sense, and it also reveals essential differences between problem(1.1) with a punctured domain and problem (1.1) with an exterior domain; see the following remark.
Remark 2.3.
Concerning Theorem 2.2, we would like to make some comments as follows.(i) When α < , µ >
0, one can easily see from the proof of Theorem 2.2 that the linear problem − ∆ u = µ | x | − α u in R N \ B ℓ (0) ,u > ∂B ℓ (0) (2.4)has no positive supersolution for any ℓ > α > , µ >
0, problem (2.4) has positive supersolution for properly large ℓ ; this can be seenfrom Proposition 4.2(i) below (by taking p = 1 , β < − , τ = 0 there). Such a result is in sharp contrastwith the above (i) and Theorem 2.2(i);(iii) When α = 2 , µ ( N − /
4, problem (2.3) (and so problem (2.4)) has positive supersolution byconsidering the fundamental solutions of Hardy operators; one can refer to, for example, [14]. Hence, for α = 2, µ = ( N − / In the case that β = − p ∗ β = 1, problem (2.1) is related to the Hardy-Leraypotentials. The following result plays an essential role in the proof of Theorem 2.1. Huyuan Chen et al. Sci China Math
Lemma 2.4. [14, Proposition 5.2] Assume that µ > ( N − / , D is a bounded smooth domaincontaining the origin and f ∈ L ∞ loc ( D \ { } ) is a nonnegative function. Then the Hardy problem − ∆ u = µ | x | u + f in D \ { } ,u = 0 on ∂ D (2.5) has no positive solution. Proof of Theorem 2.1.
We argue indirectly and suppose that u is a positive supersolution of (2.1).The main idea below is to reflect Ω to a bounded punctured domain through the Kelvin transform andthen to obtain a contradiction by Lemma 2.4.Without loss of generality, we may assume that Ω is a connected exterior domain satisfying 0 Ω.Denote by Ω ♯ = (cid:26) x ∈ R N : x | x | ∈ Ω (cid:27) and v ( x ) = u (cid:18) x | x | (cid:19) for x ∈ Ω ♯ . Clearly, Ω ♯ is a bounded punctured domain. By direct computation, for x ∈ Ω ♯ , we have ∇ v ( x ) = ∇ u (cid:18) x | x | (cid:19) | x | − (cid:18) ∇ u (cid:18) x | x | (cid:19) · x (cid:19) x | x | and ∆ v ( x ) = 1 | x | ∆ u (cid:18) x | x | (cid:19) + 2(2 − N ) | x | (cid:18) ∇ u (cid:18) x | x | (cid:19) · x (cid:19) . Let u ♯ ( x ) = | x | − N v ( x ) , V ♯ ( x ) = | x | − V (cid:18) x | x | (cid:19) . Then for x ∈ Ω ♯ , using the fact that ∆( | x | − N ) = 0, we observe that − ∆ u ♯ ( x ) = − ∆ v ( x ) | x | − N − ∇ v ( x ) · ( ∇| x | − N )= − (cid:20) | x | ∆ u (cid:18) x | x | (cid:19) + 2(2 − N ) | x | (cid:18) ∇ u (cid:18) x | x | (cid:19) · x (cid:19)(cid:21) | x | − N − − N ) x | x | (cid:20) ∇ u (cid:18) x | x | (cid:19) | x | − (cid:18) ∇ u (cid:18) x | x | (cid:19) · x (cid:19) x | x | (cid:21) | x | − N = | x | − − N − ∆ u (cid:18) x | x | (cid:19) > V ♯ ( x ) u ♯ ( x ) . Set V ∗ ( x ) = − ∆ u ♯ ( x ) u ♯ ( x ) . Thus, we notice that V ∗ is continuous in Ω ♯ and V ∗ > V ♯ in Ω ♯ . Therefore, V ∗ ( x ) > | x | − V ( x | x | ) , ∀ x ∈ Ω ♯ . Because of (2.2), it follows that lim inf | x |→ + V ∗ ( x ) | x | > ( N − , which in turn implies that there exist µ > ( N − / r > V ∗ ( x ) > µ | x | − , ∀ x ∈ B r (0) \ { } . Denote u = u ♯ − ϕ , where ϕ is the unique positive solution of − ∆ ϕ = 0 in B r (0) ,ϕ = u ♯ on ∂B r (0) . uyuan Chen et al. Sci China Math Then u satisfies u = 0 on ∂B r (0) and − ∆ u ( x ) − µ | x | u = V ∗ ( x ) u ♯ ( x ) − µ | x | u ♯ ( x ) + µ | x | ϕ ( x ) := f ∗ ( x ) > , ∀ x ∈ B r (0) \ { } . As a consequence, u is a positive solution of (2.5) with D = B r (0), µ = µ and f = f ∗ . This contradictsLemma 2.4. Thus, (2.1) admits no positive supersolution, and the proof is complete. ✷ On the contrary, suppose that problem − ∆ u = µ | x | − α u in R N \ { } has a positive supersolution u , that is, − ∆ u > µ | x | − α u in R N \ { } pointwise . (2.6)When α = 2 and µ > ( N − , a contradiction can be seen directly from Theorem 2.1.When α <
2, (2.6) can be written as − ∆ u > µ | x | − α | x | − u > ( N − | x | − u in R N \ B r µ (0) , where r µ = (cid:20) ( N − µ (cid:21) − α > . This contradicts Theorem 2.1.When α >
2, it follows from (2.6) that there exists ˜ f > − ∆ u = µ | x | − α | x | − u + ˜ f > ( N − | x | − u in B r µ (0) \ { } . Denote by ϕ the unique positive solution of − ∆ ϕ = 0 in B r µ (0) ,ϕ = u on ∂B r µ (0) . Then ϕ ∈ C ( B r µ (0)) ∩ C ( B r µ (0)), and v := u − ϕ is bounded from below and is a solution of − ∆ v = f in B r µ (0) \ { } ,v = 0 on ∂B r µ (0) , where f = µ | x | − α u + ˜ f > v > B r µ (0) \ { } . Indeed, consider − ∆ z = µ | x | − α u χ Brµ (0) \ Brµ/ in B r µ (0) ,z = 0 on ∂B r µ (0) , which admits a unique positive bounded solution, denoted by z . For any small ǫ >
0, set ψ ǫ ( x ) = ǫ ( r − Nµ − | x | − N ) + z ( x ) . Then there exists r ǫ ∈ (0 , r µ ) such that lim ǫ → + r ǫ = 0 and ψ ǫ v on B r ǫ (0). By the classical comparisonprinciple, we have v > ψ ǫ in B r µ (0) \ B r ǫ (0) . Huyuan Chen et al. Sci China Math
Sending ǫ → v > z > B r µ (0) \ { } .Furthermore, we see that v = 0 on ∂B r µ (0) and − ∆ v = µ | x | − α u + ˜ f > ( N − | x | − v + F in B r µ (0) \ { } , where F ( x ) = µ | x | − α ϕ + ˜ f + ( µ | x | − α − ( N − | x | − ) u > < | x | < r µ . Thus, we obtain a contradiction with Theorem 2.4 and the proof ends. ✷ Remark 2.5.
With a slight modification of the proof for the case α > µ > ( N − / N >
2, and f ∈ L ∞ loc ( D \ { } ) anonnegative function with D a bounded smooth domain containing the origin, then the Hardy problem(2.5) has no nontrivial nonnegative supersolution. Our proof is based on Theorem 2.1 and the following comparison principle.
Lemma 3.1.
Assume that Ω is a smooth exterior domain, f , f are continuous in Ω , g , g arecontinuous on ∂ Ω , and f > f in Ω and g > g on ∂ Ω . Let u and u satisfy − ∆ u > f in Ω , u > g on ∂ Ω , and − ∆ u f in Ω , u g on ∂ Ω . If lim inf | x |→ + ∞ u ( x ) > lim sup | x |→ + ∞ u ( x ) , then wehave u > u on Ω . Proof.
Such a comparison principle may be folklore; we here provide a simple proof for sake ofcompleteness. Letting w = u − u , then w satisfies − ∆ w , w ∂ Ω and lim sup | x |→ + ∞ w ( x ) . Thus, given ǫ >
0, there exists r ǫ > ǫ → w ǫ ( r − Nǫ + 1) on ∂B r ǫ (0) . As a result, we have w ( x ) ǫ < ǫ ( | x | − N + 1) on ∂ (Ω ∩ B r ǫ (0)) . Note that ∆( | x | − N ) = 0. It then follows from the classical comparison principle in any bounded smoothdomain that w ( x ) ǫ ( | x | − N + 1) in Ω ∩ B r ǫ (0) . According to the arbitrariness of ǫ >
0, we can conclude that w ✷ In order to prove Theorem 1.1, we introduce the auxiliary function˜ w ( r ) = r − N (ln r ) σ with σ > . Elementary computation yields˜ w ′ ( r ) = (2 − N ) r − N (ln r ) σ + σr − N (ln r ) σ − and ˜ w ′′ ( r ) = (2 − N )(1 − N ) r − N (ln r ) σ + σ (3 − N ) r − N (ln r ) σ − + σ ( σ − r − N (ln r ) σ − . uyuan Chen et al. Sci China Math Let w ( x ) = ˜ w ( | x | ). Then, if x satisfies | x | > max n , e − σ )( N − o , we have ∇ w ( x ) = ˜ w ′ ( | x | ) ∇| x | = ˜ w ′ ( | x | ) x | x | , and so − ∆ w ( x ) = − h w ′′ ( | x | ) + ˜ w ′ ( | x | ) · N − | x | i = σ ( N − | x | − N (ln | x | ) σ − − σ ( σ − | x | − N (ln | x | ) σ − σ ( N − | x | − N (ln | x | ) σ − . (3.1)Now, with the aid of the function w , we are ready to prove Theorem 1.1. Proof of Theorem 1.1.
We use a contradiction argument and suppose that (1.1) has a positivesupersolution u . Since h ( x, u ) > ˜ h ( x, u ) for ( x, u ) ∈ ( R N \ B e (0)) × [0 , + ∞ ), then u fulfills − ∆ u > ˜ h ( x, u ) in R N \ B e (0) . (3.2)By the positivity of u , one can find a constant c > u ( x ) > c e − N , ∀ x ∈ ∂B e (0) . Hence, Lemma 3.1 gives u ( x ) > c | x | − N , ∀ | x | > e. (3.3)By our assumption (H)-(a), one can see that˜ h ( x, u ( x )) u ( x ) > ˜ h ( x, c | x | − N ) c | x | − N , ∀ | x | > e, and if (H)-(b) holds, then lim inf | x |→ + ∞ ˜ h ( x, c | x | − N ) c | x | − N | x | > ( N − . (3.4)Take V ( x ) = ˜ h ( x,u ( x )) u ( x ) , which satisfies lim inf | x |→ + ∞ V ( x ) | x | > ( N − /
4. It is easily observed that u is a positive supersolution of − ∆ u = V ( x ) u in R N \ B e (0) . This contradicts Theorem 2.1. Thus, (3.2) has no positive supersolution provided (H) holds.If (H)-(b) fails, in the sequel we shall establish the nonexistence result in Theorem 1.1 using theassumption (H)-(b1)(b2). Under (H)-(b1), by (3.3), there exists ̺ > e such that˜ h ( x, u ( x )) > ˜ h ( x, c | x | − N ) u ( x ) c | x | − N > ˜ h ( x, c | x | − N ) > m | x | − N (ln | x | ) − σ , ∀ | x | > ̺ , (3.5)where m = min n , lim inf | x |→ + ∞ ˜ h ( x, c | x | − N ) | x | N (ln | x | ) σ o > . In light of (3.1) (by taking σ = 1 − σ > t ∈ (0 , m N − σ ) and ̺ > ̺ such that u ( x ) > t w ( x ) , | x | = ̺ and − ∆ u ( x ) > m | x | − N (ln | x | ) − σ > − ∆( t w ( x )) , ∀ | x | > ̺ . et al. Sci China Math Then by Lemma 3.1, it follows that u ( x ) > t | x | − N (ln | x | ) θ , ∀ | x | > ̺ (3.6)with θ = 1 − σ .As a next step, we are going to improve the decay of u at infinity by an induction argument. To thisend, let { θ j } j be the sequence generated by θ j +1 = θ j + θ = ( j + 1) θ , j = 1 , , · · · . (3.7)In view of (H)-(a), we may assume that σ >
1, where σ > j → + ∞ θ j = + ∞ , one can assert that there exists j ∈ N such that θ j > σ and θ j − < σ . (3.8)We now claim that for any integer j , there exist ̺ j > ̺ and t j > u ( x ) > t j | x | − N (ln | x | ) θ j , ∀ | x | > ̺ j . (3.9)Indeed, when j = 1, (3.9) has been proved above. Assume that (3.9) holds for some j , we will show that(3.9) holds true for j + 1. Notice that there exists r j > ̺ j such that t j | x | − N (ln | x | ) θ j > c | x | − N , ∀ | x | > r j . By (H)-(a)(b1), (3.5) and (3.9), we obtain˜ h ( x, u ( x )) > ˜ h ( x, t j | x | − N (ln | x | ) θ j ) u ( x ) t j | x | − N (ln | x | ) θ j > ˜ h ( x, t j | x | − N (ln | x | ) θ j ) > ˜ h ( x, c | x | − N ) t j | x | − N (ln | x | ) θ j c | x | − N > m t j c | x | − N (ln | x | ) θ j − σ , ∀ | x | > r j . Taking σ = 1+ θ j − σ > u with | x | − N (ln | x | ) θ j + θ ),we can conclude that u ( x ) > t j +1 | x | − N (ln | x | ) θ j + θ , ∀ | x | > ̺ j +1 , for some t j +1 > ̺ j +1 > r j . This verifies the previous claim (3.9).Therefore, (3.8) and (3.9) imply that u ( x ) > t | x | − N (ln | x | ) σ , ∀ | x | > ̺ j , and in turn by (H)-(a),˜ h ( x, u ( x )) u ( x ) > ˜ h ( x, t | x | − N (ln | x | ) σ ) t | x | − N (ln | x | ) σ , ∀ x ∈ R N \ B ̺ j (0) . (3.10)By (H)-(b2), there exist ̺ > ̺ j and ǫ > h ( x, u ( x )) u ( x ) > (cid:16) ( N − ǫ (cid:17) | x | , ∀ x ∈ R N \ B ̺ (0) . Let V ( x ) = h ( x,u ( x )) u ( x ) . Then V ( x ) | x | > ( N − ǫ , ∀ x ∈ R N \ B ̺ (0) . Clearly, u is a positive supersolution of − ∆ u = V ( x ) u in R N \ B ̺ (0) , which is a contradiction with Theorem 2.1. Thus, (1.1) admits no positive supersolution. The proof iscomplete. ✷ uyuan Chen et al. Sci China Math Suppose that (1.3) has a positive supersolution ( u, v ). Clearly, ( u, v ) fulfills − ∆ u > ˜ h ( x, u, v ) > − ∆ v > ˜ h ( x, u, v ) > R N \ B e (0) . (3.11)Since one can find a small c > u ( x ) , v ( x ) > c e − N , ∀ x ∈ ∂B e (0) , an analysis similar to that of obtaining (3.3) yields that u ( x ) , v ( x ) > c | x | − N , ∀ | x | > e. (3.12)Thus, it follows from (SH)-(a) that˜ h ( x, u ( x ) , v ( x )) u ( x ) > ˜ h ( x, u ( x ) , c | x | − N ) u ( x ) > ˜ h ( x, c | x | − N , c | x | − N ) c | x | − N , ∀ | x | > e and ˜ h ( x, u ( x ) , v ( x )) v ( x ) > ˜ h ( x, c | x | − N , v ( x )) v ( x ) > ˜ h ( x, c | x | − N , c | x | − N ) c | x | − N , ∀ | x | > e. If (SH)-(b) holds, we then havelim inf | x |→ + ∞ ˜ h ( x, c | x | − N , c | x | − N ) c | x | − N | x | > ( N − | x |→ + ∞ ˜ h ( x, c | x | − N , c | x | − N ) c | x | − N | x | > ( N − . (3.14)By taking V ( x ) = ˜ h ( x, u ( x ) , v ( x )) u ( x ) if (3.13) holds (or V ( x ) = ˜ h ( x,u ( x ) , v ( x )) v ( x ) if (3.14) holds), we see that u (or v ) is a positive supersolution of − ∆ u = V ( x ) u in R N \ B e (0)with lim inf | x |→ + ∞ V ( x ) | x | > ( N − . This is impossible due to Theorem 2.1. Therefore, (1.3) has nopositive supersolution.If (SH)-(b) fails, we continue to prove the nonexistence result in Theorem 1.2 under the assumption(SH)-(b1)(b2)(b3). There are three cases to distinguish as follows. Case 1: < σ < , σ > . By (3.12) and the assumption (SH)-(a), we have˜ h ( x, u ( x ) , v ( x )) > ˜ h ( x, u ( x ) , c | x | − N ) := ˜ h ( x, u ) , ∀ | x | > e. Then u verifies that − ∆ u > ˜ h ( x, u ) in R N \ B e (0) . (3.15)By virtue of the assumption (SH)-(a)(b1), there exists ρ > e such that˜ h ( x, u ) > ˜ h ( x, c | x | − N , c | x | − N ) (3.16) > m | x | − N (ln | x | ) − σ , ∀ | x | > ρ , where m = min n , lim inf | x |→ + ∞ ˜ h ( x, c | x | − N , c | x | − N ) | x | N (ln | x | ) σ o . et al. Sci China Math This implies that (H)-(b1) holds. Moreover, (SH)-(b2) indicates that ˜ h satisfies (H)-(b2). Thus, anapplication of Theorem 1.1 to problem (3.15) leads to a contradiction. Hence, (1.3) has no positivesupersolution in Case 1. Case 2: σ > , < σ < . The proof is similar to
Case 1.Case 3: < σ , σ < . Due to (SH)-(b1), we can deduce˜ h i ( x, u ( x ) , v ( x )) > m i | x | − N (ln | x | ) − σ i , ∀ | x | > ρ , for some ρ > e and m i = min n , lim inf | x |→ + ∞ ˜ h i ( x, c | x | − N , c | x | − N ) | x | N (ln | x | ) σ i o . Proceeding similarly as in (3.6), one can assert that for some t > ρ > ρ , u ( x ) > t | x | − N (ln | x | ) − σ and v ( x ) > t | x | − N (ln | x | ) − σ in R N \ B ρ (0) . (3.17)Then, reasoning as in the part of the claim (3.9), we have u ( x ) > t j | x | − N (ln | x | ) j (1 − σ ) and v ( x ) > t j | x | − N (ln | x | ) j (1 − σ ) , ∀ | x | > ρ j for a sequence { ( t j , ρ j ) } ∞ j =1 . Therefore, there exists a large integer j ∗ such that j ∗ (1 − σ ) > σ and j ∗ (1 − σ ) > σ . Thus, we obtain u ( x ) > t j ∗ | x | − N (ln | x | ) σ , v ( x ) > t j ∗ | x | − N (ln | x | ) σ , ∀ | x | > ρ j ∗ . This then yields˜ h ( x, u ( x ) , v ( x )) u ( x ) > ˜ h ( x, t j ∗ | x | − N (ln | x | ) σ , t j ∗ | x | − N (ln | x | ) σ ) t j ∗ | x | − N (ln | x | ) σ , ∀ | x | > ρ j ∗ and ˜ h ( x, u ( x ) , v ( x )) u ( x ) > ˜ h ( x, t j ∗ | x | − N (ln | x | ) σ , t j ∗ | x | − N (ln | x | ) σ ) t j ∗ | x | − N (ln | x | ) σ , ∀ | x | > ρ j ∗ . By the assumption (SH)-(b3), there exist ̺ > ρ j ∗ and ǫ > h ( x, u ( x ) , v ( x )) u ( x ) or ˜ h ( x, u ( x ) , v ( x )) v ( x ) ! > ( ( N − ǫ ) 1 | x | , ∀ x ∈ R N \ B ̺ (0) . Taking V ( x ) = h ( x,u ( x ) ,v ( x )) u ( x ) (or V ( x ) = h ( x,u ( x ) ,v ( x )) v ( x ) ), u is a positive supersolution of − ∆ u = V ( x ) u in R N \ B ̺ (0) , contradicting Theorem 2.1. As a consequence, (1.3) has no positive supersolution in Case 3. The proofis now complete. ✷ In this section, we shall use two typical examples to illustrate the optimality of the nonexistence resultsobtained by this paper.
Example 1: h ( x, u ) = | x | β (ln | x | ) τ u p .When β > −
2, then p ∗ β = N + βN − >
1. We have uyuan Chen et al. Sci China Math Proposition 4.1.
Assume that h ( x, u ) = | x | β (ln | x | ) τ u p , x ∈ R N \ B e (0) with β > − . Then thefollowing assertions hold. (i) Problem (1.1) has no positive supersolution provided that either p < p ∗ β , τ ∈ R or p = p ∗ β , τ > − Problem (1.1) has no positive bounded supersolution provided that p ∈ ( −∞ , , τ ∈ R . Proof.
We shall apply Theorem 1.1 to obtain the desired results. It suffices to check the condition (H).We first verify (i). When p = 1, the nonexistence follows from Theorem 2.1 directly. Clearly, thecondition (H)-(a) is fulfilled once p >
1. We also note that h ( x, | x | − N ) | x | N = | x | β + N − ( N − p (ln | x | ) τ , ∀ | x | > e. If p < p ∗ β , then β + N − ( N − p >
0. So for any t > τ ∈ R , thenlim | x |→ + ∞ h ( x, t | x | − N ) | x | N = + ∞ . Thus, the assumption (H)-(b) is satisfied.If p = p ∗ β and τ ∈ ( − , σ = − τ >
0, we have h ( x, t | x | − N ) | x | N (ln | x | ) σ = t p ∗ β (ln | x | ) τ + σ = t p ∗ β , ∀ | x | > e. If p = p ∗ β and τ >
0, by taking σ = >
0, we have h ( x, t | x | − N ) | x | N (ln | x | ) σ = t p ∗ β (ln | x | ) τ + > t p ∗ β , ∀ | x | > e. Furthermore, let us choose σ > τ + σ ( p ∗ β − >
0. Then h ( x, t | x | − N (ln | x | ) σ ) t | x | − N (ln | x | ) σ = t p ∗ β − (ln | x | ) τ + σ ( p ∗ β − , ∀ | x | > e. Therefore, h satisfies the assumption (H)-(b1)(b2). Thus, the assertion (i) is proved.We next verify the assertion (ii). Arguing indirectly, we suppose that (1.1) has a positive boundedsupersolution u . Denote ˜ h ( x, u ) = M p − | x | β (ln | x | ) τ u, if p ∈ ( −∞ , , τ ∈ R , where M = 1 + sup x ∈ Ω u ( x ) > h ( x, u ) = | x | β (ln | x | ) τ u p > M p − | x | β (ln | x | ) τ u = ˜ h ( x, u ) , ∀ | x | > e, u ∈ (0 , M ] , and ˜ h fulfills the assumption (H)-(a)(b) for any τ ∈ R . Consequently, u is a positive supersolution of − ∆ u = ˜ h ( x, u ) in R N \ B e (0) . (4.1)However, (4.1) has no positive supersolution by Theorem 1.1. Such a contradiction implies that (1.1) hasno positive bounded supersolution. The assertion (ii) follows. ✷ Proposition 4.2.
Assume that h ( x, u ) = | x | β (ln | x | ) τ u p , ∀ x ∈ R N \ B ℓ (0) , where β ∈ R . Then forsome large ℓ > e , problem (1.1) with Ω = R N \ B ℓ (0) has a positive bounded supersolution if one of thefollowing conditions is satisfied: (i) p > p ∗ β and τ ∈ R ; (ii) p = p ∗ β and τ < − . et al. Sci China Math Proof.
Recall that w ( x ) = | x | − N (ln | x | ) σ with σ >
0. If | x | > max n , e σ − N − o , it immediately follows from (3.1) that − ∆ w ( x ) = σ ( N − | x | − N (ln | x | ) σ − − tσ ( σ − | x | − N (ln | x | ) σ − > σ ( N − | x | − N (ln | x | ) σ − and h ( x, w ) = | x | β (ln | x | ) τ ( | x | − N (ln | x | ) σ ) p = | x | β +(2 − N ) p (ln | x | ) τ + σp . When p > p ∗ β , then β + (2 − N ) p < − N . One can easily see that there exists a large constant ℓ > e such that − ∆ w ( x ) > h ( x, w ( x )) , ∀ | x | > ℓ. Hence, w is a desired supersolution.When p = p ∗ β , we have β + (2 − N ) p = − N . Similarly as above, for some large ℓ > e , problem (1.1)has a positive supersolution w if σ − > τ + σp ∗ β , that is, σ ( p ∗ β − < − τ − , (4.2)where − τ − > τ < −
1. So when β > −
2, (4.2) holds if we take σ > − τ − p ∗ β − . When β −
2, then p ∗ β − σ = 1.In each case, the supersolution w is bounded in R N \ B ℓ (0). ✷ Our second example is the following one:
Example 2: h ( x, u, v ) = | x | β (ln | x | ) τ u p v q , h ( x, u, v ) = | x | β (ln | x | ) τ u p v q . Proposition 4.3.
Assume that h ( x, u, v ) = | x | β (ln | x | ) τ u p v q , h ( x, u, v ) = | x | β (ln | x | ) τ u p v q , ∀ x ∈ R N \ B e (0) , where p , q > , p , q > , β , β > − and τ , τ ∈ R . Problem (1.3) has no positive supersolution ifone of the following conditions holds: (i) p + q < p ∗ β , τ , τ ∈ R ;(ii) p + q < p ∗ β , τ , τ ∈ R ;(iii) p + q = p ∗ β , p + q = p ∗ β , p > , τ > − , τ ∈ R ;(iv) p + q = p ∗ β , p + q = p ∗ β , q > , τ > − , τ ∈ R ;(v) p + q = p ∗ β , p + q = p ∗ β , τ > − , τ > − . Proof.
In order to apply Theorem 1.2, we only need to check that the nonlinearities h , h satisfy(SH).First of all, when p , q > h , h satisfy (SH)-(a). We further note that h ( x, s | x | − N , s | x | − N ) | x | N = s p + q | x | β + N − ( N − p + q ) (ln | x | ) τ , ∀ | x | > e,h ( x, s | x | − N , s | x | − N ) | x | N = s p + q | x | β + N − ( N − p + q ) (ln | x | ) τ , ∀ | x | > e. Condition (i) implies that β + N − ( N − p + q ) > β + N − ( N − p + q ) >
0. Hence, in each of these cases, (SH)-(b) holds, and so (1.3) has no positive supersolution.Thus, the Proposition holds in case (i) and case (ii).When p + q = p ∗ β and p + q = p ∗ β , we see that h ( x, t | x | − N , t | x | − N ) | x | N = t p + q (ln | x | ) τ , ∀ | x | > e, uyuan Chen et al. Sci China Math h ( x, t | x | − N , t | x | − N ) | x | N = t p + q (ln | x | ) τ , ∀ | x | > e. Let σ = − τ if − < τ < σ = − τ if − < τ <
0; otherwise, we let σ = σ = . Then for | x | > e , we find that h ( x, t | x | − N , t | x | − N ) | x | N (ln | x | ) σ > t p + q , h ( x, t | x | − N , t | x | − N ) | x | N (ln | x | ) σ > t p + q . Furthermore, if p >
1, for any t > h ( x, t | x | − N (ln | x | ) σ , t | x | − N ) t | x | − N (ln | x | ) σ = t p + q − (ln | x | ) τ + σ ( p − → + ∞ , as | x | → + ∞ by choosing σ > τ + σ ( p − > q >
1, for any t > h ( x, t | x | − N , t | x | − N (ln | x | ) σ ) t | x | − N (ln | x | ) σ = s p + q − (ln | x | ) τ + σ ( q − → + ∞ , as | x | → + ∞ by choosing σ > τ + σ ( q − > p = q = 1, then for any t >
0, it holds that h ( x, t | x | − N (ln | x | ) σ , t | x | − N (ln | x | ) σ ) t | x | − N (ln | x | ) σ = t q (ln | x | ) τ + σ q , ∀ | x | > e,h ( x, t | x | − N (ln | x | ) σ , t | x | − N (ln | x | ) σ ) t | x | − N (ln | x | ) σ = t q (ln | x | ) τ + σ p , ∀ | x | > e. Since p , q > p + q >
0, we may choose either σ > σ > τ + σ p > τ + σ q > . According to the above analysis, it is easily seen that (SH)-(b1)(b2)(b3) are fulfilled when one of thecases (iii), (iv) and (v) holds. Thus, Theorem 1.2 applies to assert that (1.3) has no positive supersolutionin each of such cases. The proof is complete. ✷ In what follows, we will establish the existence of positive supersolutions to system (1.3). For sake ofconvenience, denote σ = ( τ + 1) q − ( τ + 1)( q − p − q − − p q and σ = ( τ + 1) p − ( τ + 1)( p − p − q − − p q . (4.3) Proposition 4.4.
Assume that h ( x, u, v ) = | x | β (ln | x | ) τ u p v q , h ( x, u, v ) = | x | β (ln | x | ) τ u p v q . Problem (1.3) has a positive supersolution if β , β ∈ R , and p + q > p ∗ β , p + q > p ∗ β , σ > , σ > . Proof.
Set w i ( x ) = | x | − N (ln | x | ) σ i with σ i > , i = 1 , . If | x | > max (cid:8) , e { σ ,σ }− N − (cid:9) , it follows from (3.1) that − ∆( tw i ( x )) = tσ i ( N − | x | − N (ln | x | ) σ i − − tσ i ( σ i − | x | − N (ln | x | ) σ i − > tσ i ( N − | x | − N (ln | x | ) σ i − , i = 1 , p + q > p ∗ β , h ( x, tw , tw ) = t p + q | x | β (ln | x | ) τ | x | (2 − N )( p + q ) (ln | x | ) σ p + σ q et al. Sci China Math > t p + q | x | − N (ln | x | ) τ + σ p + σ q . Similarly, by p + q > p ∗ β , we have that h ( x, tw , tw ) > t p + q | x | − N (ln | x | ) τ + σ p + σ q . In view of the definitions of σ , σ , τ + σ p + σ q = σ − , τ + σ p + σ q = σ − . (4.4)By choosing t = min ((cid:20) σ ( N − (cid:21) p q , (cid:20) σ ( N − (cid:21) p q ) , we then see that ( tw , tw ) is a supersolution of (1.3). ✷ Remark 4.5.
Concerning the condition σ , σ > p = q = 1, then σ = − τ + 1 p > σ = − τ + 1 q > τ , τ < − , p , q > q = 1 and τ = −
1, we have σ = − τ + 1 p > σ = ( τ + 1)( p − p q > τ < − , p < , p , q > . Acknowledgements
H. Chen is supported by NSF of China (No. 11726614, 11661045), and Jiangxi ProvincialNatural Science Foundation (No. 20161ACB20007). R. Peng is supported by NSF of China (No. 11671175,11571200), the Priority Academic Program Development of Jiangsu Higher Education Institutions, Top-notchAcademic Programs Project of Jiangsu Higher Education Institutions (No. PPZY2015A013) and Qing LanProject of Jiangsu Province. F. Zhou is supported by NSF of China (No. 11726613, 11271133, 11431005) andSTCSM (No. 13dZ2260400). The authors would like to thank Professor Yehuda Pinchover for bringing to ourattention some existing work on the linear Hardy potential problem (2.1) such as [18, 30, 31] and the two refereesfor their careful reading and valuable suggestions, which helped to improve the exposition of the paper.
References uyuan Chen et al. Sci China Math R N . J Diff Eq, 2009,246: 670-68018 Devyver B, Fraas M, Pinchover Y. Optimal Hardy Weight for Second-Order Elliptic Operator: an answer to a problemof Agmon. J Funct Anal, 2014, 266: 4422-448919 Du Y, Guo Z. Positive solutions of an elliptic equation with negative exponent: Stability and critical power. J DiffEq, 2009, 246: 2387-241420 Dupaigne L. A nonlinear elliptic PDE with the inverse square potential. J D’Analyse Math, 2002, 86: 359-39821 Esposito P, Ghoussoub N, Guo Y. Compactness along the branch of semistable and unstable solutions for an ellipticproblem with a singular nonlinearity. Comm Pure Appl Math., 2007, 60: 1731-176822 Fazly M, Ghoussoub N. On the H´enon-Lane-Emden conjecture. Disc Cont Dyn Syst, 2013, 34: 2513-253323 Garc´ıa-Huidobro M, Yarur C. Classification of positive singular solutions for a class of semilinear elliptic systems. AdvDiff Eq, 1997, 2: 383-40224 Guo Y. Global solutions of singular parabolic equations arising from electrostatic MEMS. J Diff Eq, 2008, 245: 809-84425 Li Y. Asymptotic behavior of positive solutions of equation − ∆ u + K ( x ) u p = 0 in R N . J Diff Eq, 1992, 95: 304-33026 Li Y. On the positive solutions of the Matukuma equation. Duke Math J, 1993, 70: 575-59027 Liu Y, Li Y, Deng Y. Separation property of solutions for a semilinear elliptic equation. J Diff Eq, 2000, 163: 381-40628 Ma L, Wei J. Properties of positive solutions to an elliptic equation with negative exponent. J Funct Anal, 2008, 254:1058-108729 Meadows A. Stable and singular solutions of the equation − ∆ u = uu