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Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb R^3\). (English) Zbl 1235.35093

Summary: We study the existence, multiplicity and concentration behavior of positive solutions for the nonlinear Kirchhoff type problem \[ -\left(\varepsilon^2a+\varepsilon b\int_{\mathbb{R}^3}|\nabla u|^2\right)\Delta u+V(x)u=f(u),\;u\in H^1(\mathbb{R}^3),\;u>0\quad\text{in }\mathbb{R}^3, \] where \(\varepsilon>0\) is a parameter and \(a,b>0\) are constants; \(V\) is a positive continuous potential satisfying some conditions and \(f\) is a subcritical nonlinear term. We relate the number of solutions with the topology of the set where \(V\) attains its minimum. The results are proved by using variational methods.

MSC:

35J20 Variational methods for second-order elliptic equations
35B09 Positive solutions to PDEs
35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35A15 Variational methods applied to PDEs
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[1] Alves, C.O.; Figueiredo, G.M., On multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in \(\mathbb{R}_N\), J. differential equations, 246, 1288-1311, (2009) · Zbl 1160.35024
[2] Alves, C.O.; Carrião, P.C.; Medeiros, E.S., Multiplicity of solutions for a class of quasilinear problem in exterior domains with Neumann conditions, Abstr. appl. anal., 3, 251-268, (2004) · Zbl 1133.35304
[3] Ambrosetti, A.; Badiale, M.; Cingolani, S., Semiclassical states of nonlinear Schrödinger equations with potentials, Arch. ration. mech. anal., 140, 285-300, (1997) · Zbl 0896.35042
[4] 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
[5] Ambrosetti, A.; Malchiodi, A., Perturbation methods and semilinear elliptic problems on \(\mathbb{R}^N\), (2006), Birkhäuser Verlag · Zbl 1115.35004
[6] DʼAncona, P.; Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. math., 108, 247-262, (1992) · Zbl 0785.35067
[7] Arosio, A.; Panizzi, S., On the well-posedness of the Kirchhoff string, Trans. amer. math. soc., 348, 305-330, (1996) · Zbl 0858.35083
[8] Bartsch, T.; Wang, Z.-Q., Existence and multiplicity results for some superlinear elliptic problems on \(\mathbb{R}^N\), Comm. partial differential equations, 20, 1725-1741, (1995) · Zbl 0837.35043
[9] Benci, V.; Cerami, G., Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology, Calc. var. partial differential equations, 2, 29-48, (1994) · Zbl 0822.35046
[10] Benedetto, E.D., \(C^{1 + \alpha}\) local regularity of weak solutions of degenerate results elliptic equations, Nonlinear anal., 7, 827-850, (1983) · Zbl 0539.35027
[11] Bernstein, S., Sur une classe dʼ équations fonctionnelles aux dérivées partielles, Bull. acad. sci. URSS. Sér. math., 4, 17-26, (1940) · JFM 66.0471.01
[12] Brezis, H.; Lieb, E.H., A relation between pointwise convergence of functions and convergence of functionals, Proc. amer. math. soc., 8, 486-490, (1983) · Zbl 0526.46037
[13] Byeon, J.; Jeanjean, L., Standing waves for nonlinear Schrödinger equations with a general nonlinearity, Arch. ration. mech. anal., 185, 185-200, (2007) · Zbl 1132.35078
[14] Cavalcanti, M.M.; Domingos Cavalcanti, V.N.; Soriano, J.A., Global existence and uniform decay rates for the Kirchhoff-carrier equation with nonlinear dissipation, Adv. differential equations, 6, 701-730, (2001) · Zbl 1007.35049
[15] Cingolani, S.; Lazzo, N., Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. differential equations, 160, 118-138, (2000) · Zbl 0952.35043
[16] Cingolani, S.; Lazzo, N., Multiple semiclassical standing waves for a class of nonlinear Schrödinger equations, Topol. methods nonlinear anal., 10, 1-13, (1997) · Zbl 0903.35018
[17] Del Pino, M.; Felmer, P.L., Semi-classical states for nonlinear Schrödinger equations: a variational reduction method, Math. ann., 324, 1-32, (2002) · Zbl 1030.35031
[18] Del Pino, M.; Felmer, P.L., Local mountain pass for semilinear elliptic problems in unbounded domains, Calc. var. partial differential equations, 4, 121-137, (1996) · Zbl 0844.35032
[19] Ekeland, I., On the variational principle, J. math. anal. appl., 47, 324-353, (1974) · Zbl 0286.49015
[20] 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
[21] He, X.; Zou, W., Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear anal., 70, 1407-1414, (2009) · Zbl 1157.35382
[22] He, X.; Zou, W., Multiplicity of solutions for a class of Kirchhoff type problems, Acta math. appl. sin., 26, 387-394, (2010) · Zbl 1196.35077
[23] Kirchhoff, G., Mechanik, (1883), Teubner Leipzig · JFM 08.0542.01
[24] Li, G., Some properties of weak solutions of nonlinear scalar fields equation, Ann. acad. sci. fenn. math., 14, 27-36, (1989)
[25] Lions, P.L., The concentration-compactness principle in the calculus of variations. the locally compact case. part II, Ann. inst. H. Poincaré non linéaire, 1, 223-283, (1984) · Zbl 0704.49004
[26] Lions, J.-L., On some questions in boundary value problems of mathematical physics, (), 284-346
[27] Ma, T.F.; Munoz Rivera, J.E., Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. math. lett., 16, 243-248, (2003) · Zbl 1135.35330
[28] Moser, J., A new proof of de giorgiʼs theorem concerning the regularity problem for elliptic differential equations, Comm. pure appl. math., 13, 457-468, (1960) · Zbl 0111.09301
[29] Oh, J.Y., Existence of semi-linear bound state of nonlinear Schrödinger equations with potentials on the class \((V)_\alpha\), Comm. partial differential equations, 13, 1499-1519, (1988) · Zbl 0702.35228
[30] Oh, J.Y., On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. math. phys., 131, 223-253, (1990) · Zbl 0753.35097
[31] Perera, K.; Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. differential equations, 221, 246-255, (2006) · Zbl 1357.35131
[32] Pohožaev, S.I., A certain class of quasilinear hyperbolic equations, Mat. sb. (N.S.), 96, 138, 152-166, (1975), 168
[33] Rabinowitz, P.H., On a class of nonlinear Schrödinger equations, Z. angew. math. phys., 43, 270-291, (1992) · Zbl 0763.35087
[34] Tolksdorf, P., Regularity for some general class of quasilinear elliptic equations, J. differential equations, 51, 126-150, (1984) · Zbl 0488.35017
[35] Trudinger, N.S., On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. pure appl. math., XX, 721-747, (1967) · Zbl 0153.42703
[36] Wang, X., On concentration of positive bound states of nonlinear Schrödinger equations, Comm. math. phys., 53, 224-229, (1993) · Zbl 0795.35118
[37] Willem, M., Minimax theorems, (1996), Birkhäuser Boston · Zbl 0856.49001
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