×

Standing waves for supercritical nonlinear Schrödinger equations. (English) Zbl 1124.35082

The authors treat standing waves of the nonlinear Schrödinger equation in \(\mathbb{R}^N\) with a supercritical nonlinearity, namely solutions of the problem \[ \Delta u-V(x)u+u^p=0,\qquad u>0,\;\lim_{| x| \to\infty}u(x)=0.\tag{1} \] Here it is assumed that \(V\) is bounded and nonnegative, \(N\geq3\), and \(p>(N+2)/(N-2)\). To prove existence, in the case \(N\geq4\) and \(p>(N+1)/(N-3)\), the only additional assumption on \(V\) is that of superquadratic decay at \(\infty\). In the general supercritical case the authors have to assume a somewhat faster decay of \(V\) at \(\infty\). Under these hypotheses it is proved that there exists a continuum of small solutions of (1).
This result stands in sharp contrast to the subcritical case, where one only expects solutions if \(V\) decays slower that quadratic at \(\infty\). Moreover, it is remarkable that a continuum of solutions is presented without employing a singular perturbation parameter. The method rests on the existence of a scaled family \(w_\lambda(x)=\lambda^{\frac{2}{p-1}}w(\lambda x)\) of radially symmetric positive solutions of the problem \[ \Delta w +w^p=0\qquad\text{in }\mathbb{R}^N. \] By a fixed point argument it is shown that a solution of (1) exists near some \(w_\lambda(\cdot-\xi)\) if \(\lambda\) is sufficiently small. For \(p> (N+1)/(N-3)\) the center of symmetry \(\xi\) can be chosen arbitrarily within expanding domains of \(\mathbb{R}^N\), essentially giving an \(N+1\)-dimensional set of solutions. In the general case \(\xi\) needs to be chosen from a fixed point set depending on \(\lambda\).

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems
35Q51 Soliton equations
PDF BibTeX XML Cite
Full Text: DOI

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.; Felli, V.; Malchiodi, A., Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity, J. eur. math. soc. (JEMS), 7, 1, 117-144, (2005) · Zbl 1064.35175
[3] 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
[4] 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
[5] Badiale, M.; D’Aprile, T., Concentration around a sphere for a singularly perturbed Schrödinger equation, Nonlinear anal., 49, 947-985, (2002) · Zbl 1018.35021
[6] Benci, V.; D’Aprile, T., The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. differential equations, 184, 109-138, (2002) · Zbl 1060.35129
[7] Benci, V.; Cerami, G., Existence of positive solutions of the equation \(- \operatorname{\Delta} u + a(x) u = u^{(N + 2) /(N - 2)}\) in \(\mathbb{R}^N\), J. funct. anal., 88, 1, 90-117, (1990) · Zbl 0705.35042
[8] Benci, V.; Grisanti, C.R.; Micheletti, A.M., Existence of solutions for the nonlinear Schrödinger equation with \(V(\infty) = 0\), (), 53-65 · Zbl 1231.35225
[9] Benci, V.; Grisanti, C.R.; Micheletti, A.M., Existence and non-existence of the ground state solution for the nonlinear Schrödinger equations with \(V(\infty) = 0\), Topol. methods nonlinear anal., 26, 2, 203-219, (2005) · Zbl 1105.35112
[10] Byeon, J.; Wang, Z.-Q., Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. ration. mech. anal., 65, 295-316, (2002) · Zbl 1022.35064
[11] Cingolani, S.; Pistoia, A., Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Schrödinger exponent, Z. angew. math. phys., 55, 201-215, (2004) · Zbl 1120.35308
[12] Castro, R.; Felmer, P., Semi-classical limit for radial non-linear Schrödinger equation, Comm. math. phys., 256, 2, 411-435, (2005) · Zbl 1075.35010
[13] J. Dávila, M. del Pino, M. Musso, The supercritical Lane-Emden-Fowler equation in exterior domains, Comm. Partial Differential Equations, in press · Zbl 1137.35023
[14] J. Dávila, M. del Pino, M. Musso, J. Wei, Fast and slow decay solutions for supercritical elliptic problems in exterior domains, preprint · Zbl 1147.35030
[15] 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
[16] del Pino, M.; Felmer, P., Semi-classical states for nonlinear Schrödinger equations, J. funct. anal., 149, 245-265, (1997) · Zbl 0887.35058
[17] 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)
[18] 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
[19] Gui, C.; Ni, W.-M.; Wang, X., On the stability and instability of positive steady states of a semilinear heat equation in \(\mathbb{R}^n\), Comm. pure appl. math., 45, 1153-1181, (1992) · Zbl 0811.35048
[20] Jeanjean, L.; Tanaka, K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. var. partial differential equations, 21, 3, 287-318, (2004) · Zbl 1060.35012
[21] Kang, X.; Wei, J., On interacting bumps of semi-classical states of nonlinear Schrödinger equations, Adv. differential equations, 5, 7-9, 899-928, (2000) · Zbl 1217.35065
[22] Li, Y.Y., On a singularly perturbed elliptic equation, Adv. differential equations, 2, 6, 955-980, (1997) · Zbl 1023.35500
[23] Micheletti, A.M.; Musso, M.; Pistoia, A., Super-position of spikes for a slightly supercritical elliptic equation in \(\mathbb{R}^N\), Discrete contin. dyn. syst., 12, 4, 747-760, (2005) · Zbl 1135.35036
[24] Norimichi, H.; Micheletti, A.M.; Pistoia, A., Existence of sign changing solutions for some critical problems on \(\mathbb{R}^N\), Comm. pure appl. anal., 4, 1, 143-164, (2005) · Zbl 1123.35019
[25] Oh, Y.-G., Stability of semiclassical bound states of nonlinear Schrödinger equations with potentials, Comm. math. phys., 121, 11-33, (1989) · Zbl 0693.35132
[26] Rabinowitz, P.H., On a class of nonlinear Schrödinger equations, Z. angew. math. phys., 43, 270-291, (1992) · Zbl 0763.35087
[27] Souplet, P.; Zhang, Q., Stability for semilinear parabolic equations with decaying potentials in \(\mathbb{R}^n\) and dynamical approach to the existence of ground states, Ann. inst. H. Poincaré anal. non linéaire, 19, 5, 683-703, (2002) · Zbl 1017.35033
[28] Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Comm. math. phys., 153, 229-243, (1993) · Zbl 0795.35118
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.