×

Multi-bump solutions and multi-tower solutions for equations on \(\mathbb R^N\). (English) Zbl 1171.35114

Summary: Let \(\epsilon>0\) be a small parameter. In this paper, we study existence of multiple multi-bump positive solutions for the semilinear Schrödinger equation
\[ -\Delta u+u=(1-\varepsilon a(x))|u|^{p-2}u,\quad u\in H^1(\mathbb R^N), \]
where \(N\geq 1\), \(2<p<2N/(N-2)\) if \(N\geq 3\), \(p>2\) if \(N=1\) or \(N=2\), \(a\in C(\mathbb R^N)\), \(a(x)>0\) for \(x\in\mathbb R^N\), and \(\lim_{|x|\to\infty}(x)=0.\) We also study existence of multiple multi-tower positive solutions for the prescribed scalar curvature equation
\[ -\Delta u=(1-\varepsilon K(|x|))u^{\frac{N+2}{N-2}},\quad u\in{\mathcal D}^{1,2}(\mathbb R^N), \]
where \(N\geq 3\), \(K\in C([0,\infty))\), \(K(r)>0\) for \(r>0\), \(\lim_{r\to 0}K(r)=0\), and \(\lim_{r\to\infty} K(r)=0.\)

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35J60 Nonlinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ambrosetti, A.; Badiale, M., Homoclinics: poincaré-melnikov type results via a variational approach, Ann. inst. H. Poincaré anal. non linéaire, 15, 233-252, (1998) · Zbl 1004.37043
[2] Ambrosetti, A.; Badiale, M.; Cingolani, S., Semiclassical states of nonlinear Schrödinger equations, Arch. ration. mech. anal., 140, 285-300, (1997) · Zbl 0896.35042
[3] Ambrosetti, A.; Garcia Azorero, J.; Peral, I., Perturbation of \(\operatorname{\Delta} u + u^{\frac{N + 2}{N - 2}} = 0\), the scalar curvature problem in \(\mathbb{R}^N\) and related topics, J. funct. anal., 165, 117-149, (1999) · Zbl 0938.35056
[4] Ambrosetti, A.; Garcia Azorero, J.; Peral, I., Remarks on a class of semilinear elliptic equations on \(R^n\) via perturbation methods, Adv. nonlinear stud., 1, 1-13, (2001) · Zbl 1001.35038
[5] Ambrosetti, A.; Malchiodi, A., On the symmetric scalar curvature problem on \(S^n\), J. differential equations, 170, 228-245, (2001) · Zbl 1006.35044
[6] Ambrosetti, A.; Malchiodi, A., Perturbation methods and semilinear elliptic problem on \(\mathbb{R}^N\), Progr. math., vol. 240, (2005), Birkhäuser Boston
[7] 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
[8] Ambrosetti, A.; Malchiodi, A.; Secchi, S., Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. ration. mech. anal., 159, 253-271, (2001) · Zbl 1040.35107
[9] Bahri, A.; Lions, P.L., On the existence of a positive solution of semilinear elliptic equations on unbounded domains, Ann. inst. H. Poincaré anal. non linéaire, 14, 365-413, (1997) · Zbl 0883.35045
[10] Berestycki, H.; Lions, P.L., Nonlinear scalar field equations: I, II, Arch. ration. mech. anal., 82, 313-375, (1983)
[11] Byeon, J.; Wang, Z.-Q., Standing waves with a critical frequency for nonlinear Schrödinger equations, Arch. ration. mech. anal., 165, 295-316, (2002) · Zbl 1022.35064
[12] 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
[13] Cao, D.M.; Noussair, E.S.; Yan, S.S., On the scalar curvature equation \(- \operatorname{\Delta} u = (1 + \operatorname{\&z.epsi;} K) u^{\frac{N + 2}{N - 2}}\) in \(\mathbb{R}^N\), Calc. var. partial differential equations, 15, 403-419, (2002)
[14] Catrina, F.; Wang, Z.-Q., On the caffarelli – nirenberg inequalities: sharp constants, existence (and nonexistence) and symmetry of extremal functions, Comm. pure appl. math., 54, 229-258, (2001) · Zbl 1072.35506
[15] Cerami, G.; Passaseo, D., Existence and multiplicity results for semilinear elliptic Dirichlet problems in exterior domains, Nonlinear anal., 24, 1533-1547, (1995) · Zbl 0845.35026
[16] Coti Zelati, V.; Rabinowitz, P.H., Homoclinic type solutions for a semilinear elliptic PDE on \(\mathbb{R}^N\), Comm. math. pure appl., 45, 1217-1269, (1992) · Zbl 0785.35029
[17] D’Aprile, T.; Wei, J.C., Standing waves in the maxwell – schrödinger equation and an optimal configuration problem, Calc. var. partial differential equations, 25, 105-137, (2005) · Zbl 1207.35129
[18] 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
[19] del Pino, M.; Felmer, P., Semi-classical states for nonlinear Schrödinger equations, J. funct. anal., 149, 245-265, (1997) · Zbl 0887.35058
[20] del Pino, M.; Felmer, P., Multi-peak bound states of nonlinear Schrödinger equations, Ann. inst. H. Poincaré anal. non linéaire, 15, 127-149, (1998) · Zbl 0901.35023
[21] 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
[22] del Pino, M.; Kowalczyk, M.; Wei, J.C., Concentration on curves for nonlinear Schrödinger equations, Comm. pure appl. math., 60, 113-146, (2007) · Zbl 1123.35003
[23] Floer, A.; Weinstein, A., Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential, J. funct. anal., 69, 397-408, (1986) · Zbl 0613.35076
[24] Korevaar, N.; Mazzeo, R.; Pacard, F.; Schoen, R., Refined asymptotics for constant scalar curvature metric with isolated singularities, Invent. math., 135, 233-272, (1999) · Zbl 0958.53032
[25] Kwong, M.K., Uniqueness of positive solutions of \(\operatorname{\Delta} u - u + u^p = 0\) in \(\mathbb{R}^N\), Arch. ration. mech. anal., 105, 243-266, (1989) · Zbl 0676.35032
[26] Li, Y.Y., Prescribing scalar curvature on \(S^3\), \(S^4\) and related problems, J. funct. anal., 118, 43-118, (1993) · Zbl 0790.53040
[27] Li, Y.Y., On Nirenberg’s problem and related topics, Topol. methods nonlinear anal., 3, 221-233, (1994) · Zbl 0812.53039
[28] Li, Y.Y., Prescribing scalar curvature on \(S^n\) and related problems, part I, J. differential equations, 120, 319-410, (1995) · Zbl 0827.53039
[29] Li, Y.Y., Prescribing scalar curvature on \(S^n\) and related problems. II. existence and compactness, Comm. pure appl. math., 49, 541-597, (1996) · Zbl 0849.53031
[30] Li, Y.Y., On a singularly perturbed elliptic equation, Adv. differential equations, 2, 955-980, (1997) · Zbl 1023.35500
[31] L.S. Lin, Z.L. Liu, S.W. Chen, Multi-bump solutions for a semilinear Schrödinger equation, Indiana Univ. Math. J., in press
[32] Liu, Z.L.; Wang, Z.-Q., Multi-bump type nodal solutions having a prescribed number of nodal domains: I, II, Ann. inst. H. Poincaré anal. non linéaire, 22, 597-631, (2005) · Zbl 1330.35154
[33] Malchiodi, A., The scalar curvature problem on \(S^n\): an approach via Morse theory, Calc. var. partial differential equations, 14, 429-445, (2002) · Zbl 1012.53035
[34] Ndiaye, C.B., Multiple solutions for the scalar curvature problem on the sphere, Comm. partial differential equations, 31, 1667-1678, (2006) · Zbl 1215.35055
[35] Ni, W.-M.; Takagi, I., Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke math. J., 70, 247-281, (1993) · Zbl 0796.35056
[36] Oh, Y.G., Existence of semi-classical bound states of nonlinear Schrödinger equations with potential on the class \((V)_a\), Comm. partial differential equations, 13, 1499-1519, (1988) · Zbl 0702.35228
[37] 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
[38] Rabinowitz, P.H., On a class of nonlinear Schrödinger equations, Z. angew. math. phys., 43, 270-291, (1992) · Zbl 0763.35087
[39] Wang, Z.-Q., Existence and symmetry of multi-bump solutions for nonlinear Schrödinger equations, J. differential equations, 159, 102-137, (1999) · Zbl 1005.35083
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.