Bound state solutions for a class of nonlinear Schrödinger equations. (English) Zbl 1156.35084

Bonheure, Denis; van Schaftingen, Jean

Rev. Mat. Iberoam. 24, No. 1, 297-351 (2008).

Summary: We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schrödinger equations of the form
\[ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, \]
where \(V, K\) are positive continuous functions and \(p > 1\) is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential \(V\) is allowed to vanish at infinity and the competing function \(K\) does not have to be bounded. In the semi-classical limit, i.e. for \(\varepsilon\sim 0\), we prove the existence of bound state solutions localized around local minimum points of the auxiliary function \(\mathcal{A} = V^\theta K^{-\frac{2}{p-1}}\), where \(\theta=(p+1)/(p-1)-N/2\). A special attention is devoted to the qualitative properties of these solutions as \(\varepsilon\) goes to zero.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35J60	Nonlinear elliptic equations
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

nonlinear Schrödinger equation; semi-classical states; concentration; vanishing potentials; unbounded competition functions

References:

[1]	Ambrosetti, A., Badiale, M. and Cingolani, S.: Semiclassical states of nonlinear Schrödinger equations. Arch. Rational Mech. Anal. 140 (1997), 285-300. · Zbl 0896.35042 · doi:10.1007/s002050050067
[2]	Ambrosetti, A., Felli, V. and Malchiodi, A.: Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Eur. Math. Soc. (JEMS) 7 (2005), 117-144. · Zbl 1064.35175 · doi:10.4171/JEMS/24
[3]	Ambrosetti, A. and Malchiodi, A.: Concentration phenomena for nonlinear Schrödinger equation: recent results and new perspectives. In Perspectives in Nonlinear PDEs, in honor of Haïm Brezis , 19-30. Contemp. Math. 446 . Amer. Math. Soc., Providence, RI, 2007. · Zbl 1200.35106
[4]	Ambrosetti, A., Malchiodi, A. and Ni, W.-M.: Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. I. Comm. Math. Phys. 235 (2003), 427-466. · Zbl 1072.35019 · doi:10.1007/s00220-003-0811-y
[5]	Ambrosetti, A., Malchiodi, A. and Ni, W.-M.: Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres. II. Indiana Univ. Math. J. 53 (2004), no. 2, 297-329. · Zbl 1081.35008 · doi:10.1512/iumj.2004.53.2400
[6]	Ambrosetti, A., Malchiodi, A. and Ruiz, D.: Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity. J. Anal. Math. 98 (2006), 317-348. · Zbl 1142.35082 · doi:10.1007/BF02790279
[7]	Ambrosetti, A. Malchiodi, A. and Secchi, S.: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal. 159 (2001), 253-271. · Zbl 1040.35107 · doi:10.1007/s002050100152
[8]	Ambrosetti, A. and Ruiz, D.: Radial solutions concentrating on spheres of nonlinear Schrödinger equations with vanishing potentials. Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 5, 889-907. · Zbl 1126.35059 · doi:10.1017/S0308210500004789
[9]	Bahri, A., Lions, P. L.: On the existence of a positive solution of semilinear elliptic equations in unbounded domains. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 365-413. · Zbl 0883.35045 · doi:10.1016/S0294-1449(97)80142-4
[10]	Bonheure, B. and Van Schaftingen, J. : Nonlinear Schrödinger equations with potentials vanishing at infinity. C. R. Math. Acad. Sci. Paris 342 (2006), no. 12, 903-908. · Zbl 1099.35127 · doi:10.1016/j.crma.2006.04.011
[11]	Byeon, J. and Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch. Ration. Mech. Anal. 165 (2002), no. 4, 295-316. · Zbl 1022.35064 · doi:10.1007/s00205-002-0225-6
[12]	Byeon, J. and Wang, Z.-Q.: Standing waves with a critical frequency for nonlinear Schrödinger equations. II. Calc. Var. Partial Differential Equations 18 (2003), no. 2, 207-219. · Zbl 1073.35199 · doi:10.1007/s00526-002-0191-8
[13]	Byeon, J. and Wang, Z.-Q.: Spherical semiclassical states of a critical frequency for Schrödinger equations with decaying potentials. J. Eur. Math. Soc. (JEMS) 8 (2006), no. 2, 217-228. · Zbl 1245.35036
[14]	Del Pino, M. and Felmer, P.: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var. Partial Differential Equations 4 (1996), 121-137. · Zbl 0844.35032 · doi:10.1007/s005260050031
[15]	Del Pino, M. and Felmer, P.: Semi classical states for Nonlinear Schrödinger equations. J. Funct. Anal. 149 (1997), no. 1, 245-265. · Zbl 0887.35058 · doi:10.1006/jfan.1996.3085
[16]	Del Pino, M. and Felmer, P.: Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no 2, 127-149. · Zbl 0901.35023 · doi:10.1016/S0294-1449(97)89296-7
[17]	Del Pino, M. and Felmer, P.: Semi-classical states of nonlinear Schrödinger equations: a variational reduction method. Math. Ann. 324 (2002), no. 1, 1-32. · Zbl 1030.35031 · doi:10.1007/s002080200327
[18]	Floer, A. and Weinstein, A.: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 69 (1986), 397-408. · Zbl 0613.35076 · doi:10.1016/0022-1236(86)90096-0
[19]	Fraenkel, L. E.: An introduction to maximum principles and symmetry in elliptic problems . Cambridge Tracts in Mathematics 128 . Cambridge University Press, Cambridge, 2000 · Zbl 0947.35002
[20]	Gidas, B., Ni, W. M. and Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in \(\R^N\). In Mathematical analysis and applications, Part A , 369-402. Advances in Math. Suppl. Studies 7a . Academic Press, New York-London, 1981. · Zbl 0469.35052
[21]	Grisvard, P.: Elliptic problems in nonsmooth domains . Monographs and Studies in Mathematics 24 . Pitman (Advanced Publishing Program), Boston, MA, 1985. · Zbl 0695.35060
[22]	Kufner, A.: Weighted Sobolev spaces . Translated from the Czech. A Wiley-Interscience Publication. John Wiley & Sons, New York, 1985. · Zbl 0567.46009
[23]	Lions, P. L.: The concentration-compactness principle in the calculus of variations. The locally compact case. II. Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), 223-283. · Zbl 0704.49004
[24]	Li, Y.: Remarks on a semilinear elliptic equation on \(\R^N\). J. Differential Equations 74 (1988), 34-49. · Zbl 0662.35038 · doi:10.1016/0022-0396(88)90017-4
[25]	Li, Y. Y.: On a singularly perturbed elliptic equation. Adv. Differential Equations 2 (1997), 955-980. · Zbl 1023.35500
[26]	Oh, Y.-G.: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class \((V)_a\). Comm. Partial Differential Equations 13 (1988), 1499-1519. · Zbl 0702.35228 · doi:10.1080/03605308808820585
[27]	Oh, Y.-G.: Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials. Comm. Math. Phys. 121 (1989), 11-33. · Zbl 0693.35132 · doi:10.1007/BF01218621
[28]	Oh, Y.-G.: On positive multi-bump bound states of nonlinear Schödinger equations under multiple well potential. Comm. Math. Phys. 131 (1990), 223-253. · Zbl 0753.35097 · doi:10.1007/BF02161413
[29]	Opic, B. and Kufner, A.: Hardy-type Inequalities . Pitman Research Notes in Mathematics Series 219 . Longman Scientific & Technical, Harlow, 1990. · Zbl 0698.26007
[30]	Pistoia, A.: Multi-peak solutions for a class of nonlinear Schrödinger equations. NoDEA Nonlinear Differential Equations Appl. 9 (2002), no. 1, 69-91. · Zbl 1001.35030 · doi:10.1007/s00030-002-8119-8
[31]	Pomponio, A. and Secchi, S.: On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results. J. Differential Equations 207 (2004), 229-266. · Zbl 1129.35403 · doi:10.1016/j.jde.2004.06.015
[32]	Rabinowitz, P.: Minimax methods in critical point theory with applications to differential equations . CBMS Regional Conference Series in Mathematics 65 . American Mathematical Society, Providence, RI, 1986. · Zbl 0609.58002
[33]	Rabinowitz, P.: On a class of nonlinear Schrödinger equations. Z. Angew Math Phys 43 (1992), 270-291. · Zbl 0763.35087 · doi:10.1007/BF00946631
[34]	Souplet, P. and Zhang, Q. S.: Stability for semilinear parabolic equations with decaying potentials in \(\R^n\) and dynamical approach to the existence of ground states. Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 683-703. · Zbl 1017.35033 · doi:10.1016/S0294-1449(02)00098-7
[35]	Su, J., Wang, Z.-Q. and Willem, M.: Nonlinear Schrödinger equations with unbounded and decaying potentials. Commun. Contemp. Math. 9 (2007), no. 4, 571-583. · Zbl 1141.35056 · doi:10.1142/S021919970700254X
[36]	Van Schaftingen, J. and Willem, M.: Symmetry of solutions of semilinear elliptic problems. J. Eur. Math. Soc. (JEMS) 10 (2008), no. 2, 439-456. · Zbl 1148.35025
[37]	Wang, X.: On concentration of positive bound states of nonlinear Schrödinger equations. Comm. Math. Phys. 153 (1993), 229-244. · Zbl 0795.35118 · doi:10.1007/BF02096642
[38]	Wang, X. and Zeng, B.: On concentration of positive bound states of nonlinear Schrödinger equations with competing potential functions. Siam J. Math. Anal. 28 (1997), no. 3, 633-655. · Zbl 0879.35053 · doi:10.1137/S0036141095290240
[39]	Willem, M.: Minimax theorems . Progress in Nonlinear Differential Equations and their Applications 24 . Birkhäuser Boston, Boston, MA, 1996.