Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator


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1 Wan Boundary Value Problems (2015) 2015:239 DOI /s R E S E A R C H Open Access Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator Liquan Wan * * Correspondence: School of Accounting, Henan University of Economics and Law, Zhengzhou, , China Abstract In this paper, we not only give the asymptotic behavior at the origin for the maximum modulus of weak solutions of the stationary Schrödinger equation in a cone but also obtain the property of the negative parts of them, which generalize the PhragménLindelöf type theorems for subfunctions. Keywords: stationary Schrödinger equation; weak solution; cone 1 Introduction and main results Let R n be the ndimensional Euclidean space, where n 2. Let E be an open set in R n,the boundary and the closure of it are denoted by E and E,respectively.ApointP is denoted by (X, x n ), where X =(x 1, x 2,...,x n 1 ). For P R n and r >0,letB(P, r) denote the open ball with center at P and radius r in R n. We introduce a system of spherical coordinates (r, ), =(θ 1, θ 2,...,θ n 1 ), in R n which are related to the cartesian coordinates (X, x n )=(x 1, x 2,...,x n 1, x n )byx n = r cos θ 1. Let S n 1 and S n 1 + denote the unit sphere and the upper half unit sphere, respectively. For S n 1,apoint(1, ) ons n 1,andtheset{ ;(1, ) } are simply denoted by and respectively. The set {(r, ) R n ; r,(1, ) } in R n is simply denoted by, where R + and S n 1. Especially, the set R + by C n ( ), where R + is the set of positive real number and S n 1. Let C n ( ; I) ands n ( ; I) denotethesetsi and I,respectively,whereI is an interval on R and R is the set of real numbers. Especially, the set S n ( ) denotes S n ( ;(0,+ )), which is C n ( ) {O}. Let be the spherical part of the Laplace operator (see [1]), = n 1 r r + 2 r + 2 r, 2 and be a domain on S n 1 with smooth boundary. We consider the Dirichlet problem (see [2], p.41) ( + λ ) ϕ( )=0 on, ϕ( )=0 on Wan. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License ( which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
2 Wan Boundary Value Problems (2015) 2015:239 Page 2 of 10 The least positive eigenvalue of the above boundary value problem is denoted by λ and the normalized positive eigenfunction corresponding to λ by ϕ( ), ϕ2 ( ) d =1,where d denotes the (n 1)dimensional volume element. We put a rather strong assumption on : ifn 3, then is a C 2,α domain (0 < α <1) on S n 1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [3], pp.8889, for the definition of the C 2,α domain). Let A a denote the class of nonnegative radial potentials a(p), i.e. 0 a(p) =a(r), P = (r, ) C n ( ), such that a L b loc (C n( )) with some b > n/2 if n 4andwithb =2ifn =2 or n =3. Let I be the identical operator. If a A a, then the stationary Schrödinger operator SSE a = + a(p)i can be extended in the usual way from the space C 0 (C n( )) to an essentially selfadjoint operator on L 2 (C n ( )) (see [4],Chapter13).WewilldenoteitSSE a as well. This last one has a GreenSch function G a (P, Q)whichispositiveonC n( ) and its inner normal derivative G a (P, Q)/ n Q 0, where / n Q denotes the differentiation at Q along the inward normal into C n ( ). In this paper, we are concerned with the weak solutions of the inequality SSE a u(p) 0, (1.1) where P =(r, ) C n ( ). We will also consider the class B a, consisting of the potentials a A a such that there exists the finite limit lim r r 2 a(r)=k [0, ), moreover, r 1 r 2 a(r) k L(1, ). We denote by SbH a ( ) the class of all weak solutions of the inequality (1.1) for any P =(r, ) C n ( ), which are continuous when a B a (see [5]). We denote by SpH a ( )the class of u(p)satisfying u(p) SbH a ( ). If u(p) SbH a ( )andu(p) SpH a ( ), then u(p) is the solution of SSE a u(p) = 0 for any P =(r, ) C n ( ). In our terminology we follow Nirenberg [6]. Other authors have under similar circumstances used various terms such as subfunctions, subsolutions, submetaharmonic function, subelliptic functions, panharmonic functions, etc.; see, for example, Duffin, Littman, Qiao et al., Topolyansky, Vekua (see [7 11]). Solutions of the ordinary differential equation (r) n 1 ( ) λ (r)+ r r + a(r) (r)=0, 0<r <, (1.2) 2 play an essential role in this paper. It is well known (see, for example, [12]) that if the potential a A a,thenequation(1.2) has a fundamental system of positive solutions {V, W} such that V is nondecreasing with 0 V(0+) V(r) as r +, and W is monotonically decreasing with + = W(0+) > W(r) 0 asr +.
3 Wan Boundary Value Problems (2015) 2015:239 Page 3 of 10 Denote ι ± k = 2 n ± (n 2) 2 +4(k + λ), 2 then the solutions to equation (1.2) have the following asymptotic (see [3]): V(r) r ι+ k, W(r) r ι k, asr. Let u(p) (P =(r, ) C n ( )) be a function. We introduce the following notations: u + = max{u,0}, u = min{u,0}, =sup u(p), l = max ϕ( ), u(p) S u (r)=sup ϕ( ), L u = lim sup S u (r) W(r), J u = sup P C n ( ) u(p) W(r)ϕ( ). For any two positive numbers δ an r,we put E u 0 (r; δ)={ : u(p) δw(r) } and ξ u (δ)=lim sup ϕ( ) d. E0 u(r;δ) The integral u(r, )ϕ( ) d, is denoted by N u (r), when it exists. The finite or infinite limits N u (r) lim r V(r) and N u (r) lim W(r) are denoted by μ u and η u, respectively, when they exist. We shall say that u(p) (P =(r, ) C n ( )) satisfies the PhragménLindelöf boundary condition on S n ( ), if lim sup u(p) 0 P Q,Q S n ( ) for every Q S n ( ). Throughout this paper, unless otherwise specified, we will always assume that u(p) SbH a ( ) and satisfy the PhragménLindelöf boundary condition on S n ( ). Recently, about the PhragménLindelöf theorems for subfunctions in a cone, Qiao and Deng (see [9], Theorem 3) proved the following result. Theorem A If μ u + = η u + =0,
4 Wan Boundary Value Problems (2015) 2015:239 Page 4 of 10 then u(p) 0 for any P =(r, ) C n ( ). A stronger version of a PhragménLindelöf type theorem is also due to Qiao and Deng (see [9], Theorem 3). Theorem B If and then lim inf r lim inf V(r) W(r) <+ (1.3) <+, (1.4) u(p) ( μ u V(r)+η u W(r) ) ϕ( ) (1.5) for any P =(r, ) C n ( ). However, they do not tell us in [9] whether or not the limit B u = lim W(r) exists. In this paper, we first of all answer this question positively and prove the following result. Theorem 1 If (1.3) is satisfied, then the limit B u (0 B u + ) exists and B u =(L u ) + l, (1.6) where (L u ) + = η u +. (1.7) Remark It is obvious that η u L u. On the other hand, we have η u L u from (1.5). Thus, if (1.3)and(1.4) are satisfied, then we have η u = L u. As an application of Theorem 1 we immediately have the following result by using Lemma 3 in Section 2. Corollary If lim inf r V(r) 0, (1.8)
5 Wan Boundary Value Problems (2015) 2015:239 Page 5 of 10 then B u =(J u ) + l. (1.9) In [9], the authors gave the properties of the positive part of weak solutions satisfying the PhragménLindelöf boundary condition on S n ( ). Finally, we shall show one of the properties of its negative part. From the remark, we have η u + = L u + =(L u ) + =(η u ) +. Since N u (r)=n u +(r) N u (r), Theorem 2 follows immediately. Theorem 2 Under the conditions of Theorem B, if η u 0, then N u (r) lim W(r) =0. 2 Some lemmas Lemma 1 (see [9], Lemma 8) (1) Both of the limits μ u and η u ( < μ u, η u + ) exist. (2) If η u 0, then V 1 (r)n u (r) is nondecreasing on (0, + ). (3) If μ u 0, then W 1 (r)n u (r) is nonincreasing on (0, + ). Lemma 2 If (1.3) is satisfied and there exists a positive number R such that u(p) 0 for any P =(r, ) C n ( ;(0,R)), then for any positive number δ, we have u(p) ( μ u V(r) δξ u (δ)w(r) ) ϕ( ) (2.1) for any P =(r, ) C n ( ). Proof Let δ be any given positive number and {r k } be a sequence such that lim r k =0 and lim ϕ( ) d = ξ u (δ). k k E0 u(r k;δ) Then we have N u (r k ) E u 0 (r k;δ) for any 0 < r k < R and hence u(r k, )ϕ( ) d δw(r k ) ϕ( ) d E0 u(r k;δ) η u δξ u (δ). Thus we obtain (2.1)fromTheoremB.
6 Wan Boundary Value Problems (2015) 2015:239 Page 6 of 10 Lemma 3 Under the conditions of the corollary, L u > and J u = L u. Proof It is evident that J u L u. (2.2) Hence, we shall prove that J u = L u under the assumption that L u <+. Since(1.3) and (1.4) are satisfied and (1.8)givesμ u 0, we have u(p) η u W(r)ϕ( ) for any P =(r, ) C n ( )fromtheoremb,whichgives η u J u. (2.3) Since Lemma 1 and the remark give η u > and η u = L u,respectively,wehavethe conclusion from (2.2)and(2.3). Given a continuous function ψ defined on the truncated cone C n ( ;(R 1, R 2 )), where R 1 and R 2 are two positive real numbers satisfying R 1 < R 2, then the solution of the Dirichlet Sch problem on C n ( ;(R 1, R 2 )) with ψ is denoted by H ψ (P; C n ( ;(R 1, R 2 ))). Lemma 4 If μ u + <+ and η u + <+, (2.4) are satisfied, then B u η u +. Proof Take any P =(r, ) C n ( ) and any pair of numbers R 1, R 2 satisfying 0 < 2R 1 < r < 1 2 R 2 <.Ifψ(P) is a boundary function on C n ( ;(R 1, R 2 )) satisfying ψ(p)= then we have { u(r i, ) on{r i } (i =1,2), 0 ons n ( ;(R 1, R 2 )), ( ( u(p) H ψ P; Cn ;(R1, R 2 ) )) = u + (R 1, ) Ga C n ( ;(R 1,R 2 )) (P,(R 1, )) y R n 1 1 d u + (R 2, ) Ga C n ( ;(R 1,R 2 )) (P,(R 2, )) R n 1 2 d, y where G a C n ( ;(R 1,R 2 )) (, ) is the GreenSch function on C n( ;(R 1, R 2 )) with the pole at P. Here we use the following inequalities (see [1], p.124): (G a C n ( ;(R 1,R 2 )) (P,(R 1, ))) R c 1 W(r) W(R 1 ) ϕ( )ϕ( ) R n 1 1
7 Wan Boundary Value Problems (2015) 2015:239 Page 7 of 10 and (G a C n ( ;(R 1,R 2 )) (P,(R 2, ))) R c 2 V(r) V(R 2 ) where c 1 and c 2 are two positive constants. Then we have ϕ( )ϕ( ) R n 1, 2 u(p) c 3 W 1 (R 1 )N u +(R 1 )W(r)ϕ( )+c 4 V 1 (R 2 )N u +(R 2 )V(r)ϕ( ), (2.5) where c 3 and c 4 are two positive constants. As R 1 0andR 2 in (2.5), we obtain ( c 3 η u +W(r)+c 4 μ u +V(r) ) max ϕ( ) from Lemma 1, which gives the conclusion of Lemma 4 from (2.4). 3 Proof of Theorem 1 Put τ = lim inf W(r). Since N u (r) ϕ( ) d, and Lemma 1 gives η u >, (3.1) we immediately see that τ >. Now we distinguish two cases. Case 1 τ =+. In this case B u exists and is equal to +.Itisobviousthatforanypositivenumberr M u +(r) W(r) l S u +(r) W(r), (3.2) which gives L u =+. These results show that (1.7) holds in this case. Case 2 τ <+. From Theorem B,weseethatL u <+. On the other hand, we have L u > from (3.1). Subcase L u <+. There exists a positive number R ɛ such that u(p) (L u + ɛ)w(r)ϕ( ) for any ɛ >0,whereP =(r, ) C n ( ;(0,R ɛ )).
8 Wan Boundary Value Problems (2015) 2015:239 Page 8 of 10 This gives lim sup W(r) L ul. (3.3) Now, assume that τ < L u l.thereexistapositivenumberδ 1 and a set E u such that ϕ( ) d >0 E u and L u ϕ( ) τ 2δ 1 (3.4) for E u. We define v 1 (P)(P =(r, ) C n ( )) by v 1 (P)=u(P) (L u + ɛ)w(r)ϕ( ) (3.5) and apply Lemma 2 to v 1 (P). It gives u(p) [{ L u + ɛ δ 1 ξ v1 (δ 1 ) } W(r)+μ v1 V(r) ] ϕ( ) for any P =(r, ) C n ( ). So we have L u L u δ 1 ξ v1 (δ 1 ). If we can show that ξ v1 (δ 1 )>0, (3.6) then we have a contradiction. To prove (3.6), take a sequence {r k },withlim k r k =0,suchthat M u (r k ) W(r k ) τ + δ 1 (k =1,2,3,...). From (3.4)and(3.5)wehave v 1 (r k, ) W(r k ) for any E u,whichgives u(r k, ) W(r k ) L uϕ( ) δ 1 E u E v 1 0 (r k; δ 1 ) (k =1,2,3,...). Hence ξ v1 (δ 1 ) ϕ( ) d >0. E u
9 Wan Boundary Value Problems (2015) 2015:239 Page 9 of 10 Thus from (3.3) we can simultaneously prove the existence of B u and (1.6). Subcase 2.2 L u <0. Take any small number ɛ > 0 satisfying L u + ɛ < 0. There exists a positive number R ɛ such that u(p) (L u + ɛ)w(r)ϕ( ) for any P =(r, ) C n ( ;(0,R ɛ )). This gives lim sup W(r) 0. (3.7) Now suppose that τ <0.Thereareasequence{r k } tending to 0 and a positive number δ 2 such that M u (r k ) W(r k ) 2δ 2 (k =1,2,3,...). Define v 2 (P)(P =(r, ) C n ( )) by v 2 (P)=u(P) (L u + ɛ)w(r)ϕ( ) and apply Lemma 2 to v 2 (P). Then we obtain u(p) [{ L u + ɛ δ 2 ξ v2 (δ 2 ) } W(r)+μ v2 V(r) ] ϕ( ) for any P =(r, ) C n ( ), which gives L u L u δ 2 ξ v2 (δ 2 ). If we can show that ξ v2 (δ 2 )>0, (3.8) then we have a contradiction. To prove (3.8), write F u = { ; L u ϕ( ) δ 2 }. It is evident that ϕ( ) d >0. F u For every F u,wehave v 1 (r k, ) W(r k ) u(r k, ) W(r k ) L uϕ( ) δ 2,
10 Wan Boundary Value Problems (2015) 2015:239 Page 10 of 10 which shows that F u E v 2 0 (r k; δ 2 ) (k =1,2,3,...). Hence we have ξ v2 (δ 2 ) ϕ( ) d >0. F u Thus we can prove that τ 0. With (3.7), this also gives the existence of B u and B u =0=(L u ) + l. Lastly, we shall show that (1.7)holds. If η u + =+, thenitisevidentthatb u + =+. This together with (3.2) givesl u + =+. Since L u + =(L u ) +, (3.9) we know that (1.7)holds. Next suppose that η u + <+.WehaveB u <+ by Lemma 4 and hence L u + = η u + by the remark. With (3.9), this gives (1.7). Competing interests The author declares that there is no conflict of interests regarding the publication of this article. Acknowledgements The author would like to thank the referees for their comments. This work was supported by the National Social Science Foundation of China (No. 13BGL047), the Humanities and Social Science Fund of Ministry of Education (No ZD011) and the 2015 Universities Philosophy Social Sciences Innovation Team of Henan Province: the Institutional Arrangements for Mixed Ownership Reform of Stateowned Enterprises in Henan Province (No CXTD09). Received: 29 October 2015 Accepted: 8 December 2015 References 1. Azarin, V: Generalization of a theorem of Hayman on subharmonic functions in an mdimensional cone. Transl. Am. Math. Soc. 80, (1969) 2. Rosenblum, G, Solomyak, M, Shubin, M: Spectral Theory of Differential Operators. VINITI, Moscow (1989) 3. Gilbarg, D, Trudinger, N: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (1977) 4. Reed, M, Simon, B: Methods of Modern Mathematical Physics, vol. 3. Academic Press, New York (1970) 5. Simon, B: Schrödinger semigroups. Bull. Am. Math. Soc. 7, (1982) 6. Nirenberg, L: Existence Theorems in Partial Differential Equations. NYU Notes (1954) 7. Duffin, R: Yukawan potential theory. J. Math. Anal. Appl. 35, (1971) 8. Littman, W: A strong maximum principle for weakly Lsubharmonic functions. J. Math. Mech. 8, (1959) 9. Qiao, L, Deng, G: A theorem of PhragménLindelöf type for subfunctions in a cone. Glasg. Math. J. 53(3), (2011) 10. Topolyansky, D: A note on submetaharmonic functions. Dopov. Ukr. Akad. Nauk 4, (1960) (in Ukrainian) 11. Vekua, I: Metaharmonic functions. Proc. Tbilisi Math. Inst. 12, (1943) (inrussian) 12. Verzhbinskii, G, Maz ya, V: Asymptotic behavior of solutions of elliptic equations of the second order close to a boundary. I. Sib. Mat. Zh. 12, (1971)
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