Semiclassical states for nonlinear Schrödinger equations with sign-changing potentials. (English) Zbl 1131.35075

Ding, Yanheng; Wei, Juncheng

J. Funct. Anal. 251, No. 2, 546-572 (2007).

Summary: We establish the existence and multiplicity of semiclassical bound states of the following nonlinear Schrödinger equation:
\[ \begin{cases} -\varepsilon^2\Delta u+V(x)u=g(x,u) \quad & \text{for }x\in\mathbb R^N,\\ u(x)\to\infty & \text{as }|x|\to\infty\end{cases} \]
where \(V\) changes sign and \(g\) is superlinear with critical or supercritical growth as \(|u|\to\infty\).

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
81Q05	Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q20	Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory

Keywords:

superlinear; supercritical growth; existence and multiplicity of semiclassical bound states

References:

[1]	Ambrosetti, A.; Badiale, M.; Cingolani, S., Semiclassical states of nonlinear Schrödinger equations, Arch. Ration. Mech. Anal., 140, 285-300 (1997) · Zbl 0896.35042
[2]	Ambrosetti, A.; Malchiodi, A.; Ni, W.-M., Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, part I, Comm. Math. Phys., 235, 427-466 (2003) · Zbl 1072.35019
[3]	Bartsch, T.; Pankov, A.; Wang, Z. Q., Nonlinear Schrödinger equations with steep potential well, Commun. Contemp. Math., 3, 549-569 (2001) · Zbl 1076.35037
[4]	Benci, V., On critical point theory of indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc., 274, 533-572 (1982) · Zbl 0504.58014
[5]	Byeon, J.; Wang, Z. Q., Standing waves with a critical frequency for nonlinear Schrödinger equations, II, Calc. Var. Partial Differential Equations, 18, 207-219 (2003) · Zbl 1073.35199
[6]	Coti-Zelati, V.; Rabinowitz, P., Homoclinic type solutions for a semilinear elliptic PDE on \(R^n\), Comm. Pure Appl. Math., 46, 1217-1269 (1992) · Zbl 0785.35029
[7]	Ding, W. Y.; Ni, W. M., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91, 283-308 (1986) · Zbl 0616.35029
[8]	Ding, Y. H.; Lin, F. H., Solutions of perturbed Schrödinger equations with critical nonlinearity, Calc. Var. Partial Differential Equations, 30, 231-249 (2007) · Zbl 1206.35102
[9]	Ding, Y. H.; Szulkin, A., Bound states for semilinear Schrödinger equations with sign-changing potential, Calc. Var. Partial Differential Equations, 29, 397-419 (2007) · Zbl 1119.35082
[10]	Ding, Y. H.; Szulkin, A., Existence and number of solutions for a class of semilinear Schrödinger equations, Progr. Nonlinear Differential Equations Appl., 66, 221-231 (2006) · Zbl 1131.35332
[11]	del Pino, M.; Felmer, P., Multipeak bound states of nonlinear Schrödinger equations, Ann. Inst. Poincaré Anal. Non Linéaire, 15, 127-149 (1998) · Zbl 0901.35023
[12]	del Pino, M.; Felmer, P., Semi-classical states of nonlinear Schrödinger equations: a variational reduction method, Math. Ann., 324, 1-32 (2002) · Zbl 1030.35031
[13]	del Pino, M.; Kowalczyk, K.; Wei, J., Nonlinear Schrödinger equations: concentration on weighted geodesics in the semi-classical limit, Comm. Pure Appl. Math., 60, 1, 113-146 (2007) · Zbl 1123.35003
[14]	Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal., 69, 397-408 (1986) · Zbl 0613.35076
[15]	Grossi, M., Some results on a class of nonlinear Schrödinger equations, Math. Z., 235, 687-705 (2000) · Zbl 0970.35039
[16]	Gui, C., Existence of multi-bump solutions for nonlinear Schrödinger equations via variational method, Comm. Partial Differential Equations, 21, 787-820 (1996) · Zbl 0857.35116
[17]	Jeanjean, L.; Tanaka, K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. Var. Partial Differential Equations, 21, 287-318 (2004) · Zbl 1060.35012
[18]	Kang, X.; Wei, J., On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. Differential Equations, 5, 899-928 (2000) · Zbl 1217.35065
[19]	Li, Y. Y., On singularly perturbed elliptic equation, Adv. Differential Equations, 2, 955-980 (1997) · Zbl 1023.35500
[20]	Oh, Y. G., Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class \((V)_\alpha \), Comm. Partial Differential Equations, 13, 1499-1519 (1988) · Zbl 0702.35228
[21]	Oh, Y. G., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys., 131, 223-253 (1990) · Zbl 0753.35097
[22]	Pistoia, A., Multi-peak solutions for a class of nonlinear Schrödinger equations, NoDEA Nonlinear Differential Equations Appl., 9, 69-91 (2002) · Zbl 1001.35030
[23]	Reed, M.; Simon, B., Methods of Modern Mathematical Physics, IV. Analysis of Operators (1978), Academic Press: Academic Press New York · Zbl 0401.47001
[24]	Sirakov, B., Standing wave solutions of the nonlinear Schrödinger equations in \(R^N\), Ann. Mat. Pura Appl. (4), 183, 73-83 (2002) · Zbl 1223.35291
[25]	Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Comm. Math. Phys., 153, 229-244 (1993) · Zbl 0795.35118
[26]	Willem, M., Minimax Theorems (1996), Birkhäuser Boston: Birkhäuser Boston Boston, MA · Zbl 0856.49001