zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Asymptotically linear Schrödinger equation with potential vanishing at infinity. (English) Zbl 1188.35181

Summary: We are concerned with the existence of bound states and ground states of the following nonlinear Schrödinger equation

-Δu(x)+V(x)u(x)=K(x)f(u),x N ,uH 1 ( N ),u(x)>0,N3,(1)

where the potential V(x) may vanish at infinity, f(s) is asymptotically linear at infinity, that is, f(s)O(s) as s+. For this kind of potential, it seems difficult to find solutions in H 1 ( N ), i.e. bound states of (1). If f(s)=s p and p(σ,(N+2)/(N-2)) with σ1, A. Ambrosetti, V. Felli and A. Malchiodi [J. Eur. Math. Soc. (JEMS) 7, No. 1, 117–144 (2005; Zbl 1064.35175)] showed that (1) has a solution H 1 ( N ) and (1) has no ground states if p is out of the above range. We are interested in what happens if f(s) is asymptotically linear. Under appropriate assumptions on K, we prove that (1) has a bound state and a ground state.

MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
35J20Second order elliptic equations, variational methods
35J60Nonlinear elliptic equations
81Q05Closed and approximate solutions to quantum-mechanical equations
References:
[1]Ambrosetti, A.; Felli, V.; Malchiodi, A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. eur. Math. soc. 7, 117-144 (2005) · Zbl 1064.35175 · doi:10.4171/JEMS/24 · doi:http://www.ems-ph.org/journals/show_abstract.php?issn=1435-9855&vol=7&iss=1&rank=6
[2]Ambrosetti, A.; Malchiodi, A.: Perturbation methods and semilinear elliptic problems on RN, Progr. math. Ser. 240 (2006)
[3]Ambrosetti, A.; Malchiodi, A.; Ruiz, D.: Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. anal. Math. 98, 317-348 (2006) · Zbl 1142.35082 · doi:10.1007/BF02790279
[4]Ambrosetti, A.; Ruiz, D.: Radial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials, Proc. roy. Soc. Edinburgh sect. A 136, 889-907 (2006) · Zbl 1126.35059 · doi:10.1017/S0308210500004789
[5]Ambrosetti, A.; Wang, Z. Q.: Nonlinear Schrödinger equations with vanishing and decaying potentials, Differential integral equations 18, 1321-1332 (2005) · Zbl 1210.35087
[6]Bonheure, D.; Schaftingen, J. V.: Nonlinear Schrödinger equations with potentials vanishing at infinity, C. R. Acad. sci. Paris 342, 903-908 (2006) · Zbl 1099.35127 · doi:10.1016/j.crma.2006.04.011
[7]Byeon, J.; Wang, Z. Q.: Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials, J. eur. Math. soc. 8, 217-228 (2006)
[8]Costa, D. G.; Tehrani, H.: On a class of asymptotically linear elliptic problems in RN, J. differential equations 173, 470-494 (2001)
[9]Ding, Y. H.; Luan, S. X.: Multiple solutions for a class of nonlinear Schrödinger equations, J. differential equations 207, 423-457 (2004) · Zbl 1072.35166 · doi:10.1016/j.jde.2004.07.030
[10]Ekeland, I.: Convexity methods in Hamiltonian mechanics, (1990)
[11]Van Heerden, F. A.; Wang, Z. Q.: Schrödinger type equations with asymptotically linear nonlinearities, Differential integral equations 16, 257-280 (2003) · Zbl 1030.35067
[12]Jeanjean, L.: On the existence of bounded palais – Smale sequences and applications to a landesman – lazer-type set on RN, Proc. roy. Soc. Edinburgh sect. A 129, 787-809 (1999) · Zbl 0935.35044 · doi:10.1017/S0308210500013147
[13]Jeanjean, L.; Tanaka, K.: A positive solution for an asymptotically linear elliptic problem on RN autonomous at infinity, ESAIM control optim. Calc. var. 7, 597-614 (2002) · Zbl 1225.35088 · doi:10.1051/cocv:2002068 · doi:numdam:COCV_2002__7__597_0
[14]Jeanjean, L.; Tanaka, K.: A positive solution for a nonlinear Schrödinger equation on RN, Indiana univ. Math. J. 54, 443-464 (2005) · Zbl 1143.35321 · doi:10.1512/iumj.2005.54.2502
[15]Lieb, Elliott H.; Loss, M.: Analysis, Graduate stud. Math. 14 (1996)
[16]Li, G. B.; Zhou, H. S.: The existence of a positive solution to asymptotically linear scalar field equations, Proc. roy. Soc. Edinburgh sect. A 130, 81-105 (2000) · Zbl 0942.35075 · doi:10.1017/S0308210500000068
[17]Lions, P. L.: The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. inst. H. Poincaré anal. Non linéaire 1, 109-145 (1984) · Zbl 0541.49009 · doi:numdam:AIHPC_1984__1_2_109_0
[18]Li, S. J.; Wu, S. P.; Zhou, H. S.: Solutions to semilinear elliptic problems with combined nonlinearities, J. differential equations 185, 200-224 (2002) · Zbl 1032.35072 · doi:10.1006/jdeq.2001.4167
[19]Li, Y. Q.; Wang, Z. Q.; Zeng, J.: Ground states of nonlinear Schrödinger equations with potentials, Ann. inst. H. Poincaré anal. Non linéaire 23, 829-837 (2006) · Zbl 1111.35079 · doi:10.1016/j.anihpc.2006.01.003 · doi:numdam:AIHPC_2006__23_6_829_0
[20]Liu, Z. L.; Wang, Z. Q.: Existence of a positive solution of an elliptic equation on RN, Proc. roy. Soc. Edinburgh sect. A 134, 191-200 (2004) · Zbl 1067.35029 · doi:10.1017/S0308210500003152
[21]Del Pino, M.; Felmer, P.: Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. var. Partial differential equations 4, 121-137 (1996) · Zbl 0844.35032 · doi:10.1007/s005260050031
[22]Rabinowitz, P. H.: On a class of nonlinear Schrödinger equations, Z. angew. Math. phys. 43, 270-290 (1992) · Zbl 0763.35087 · doi:10.1007/BF00946631
[23]Stuart, C. A.; Zhou, H. S.: Applying the mountain pass theorem to an asymptotically linear elliptic equation on RN, Comm. partial differential equations 24, 1731-1758 (1999) · Zbl 0935.35043 · doi:10.1080/03605309908821481
[24]Stuart, C. A.; Zhou, H. S.: Axisymmetric TE-modes in a self-focusing dielectric, SIAM J. Math. anal. 37, 218-237 (2005) · Zbl 1099.78013 · doi:10.1137/S0036141004441751
[25]Stuart, C. A.; Zhou, H. S.: Global branch of solutions for non-linear Schrödinger equations with deepening potential well, Proc. London math. Soc. 92, 655-681 (2006) · Zbl 1225.35091 · doi:10.1017/S0024611505015637
[26]Su, J. B.; Wang, Z. Q.; Willem, M.: Weighted Sobolev embedding with unbounded and decaying radial potentials, J. differential equations 238, 201-219 (2007) · Zbl 1220.35026 · doi:10.1016/j.jde.2007.03.018
[27]Z.P. Wang, H.S. Zhou, Positive solution for nonlinear Schrödinger equation with deepening potential well, J. Eur. Math. Soc. (2008), in press
[28]Zou, W. M.: Sign-changing saddle point, J. funct. Anal. 219, 433-468 (2005) · Zbl 1174.35343 · doi:10.1016/j.jfa.2004.10.002