## 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
Full Text:

### References:

 [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
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.