## 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
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### 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
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