## Multiplicity and concentration of positive solutions for a Kirchhoff type problem with critical growth.(English)Zbl 1402.35119

Summary: We are concerned with the multiplicity and concentration of positive solutions for the semilinear Kirchhoff type equation $\begin{cases} -\left(\epsilon^2 a+b\epsilon \int_{\mathbb{R}^3}|\nabla u|^2\right) \Delta u+M(x)u=\lambda f(u)+|u|^4u,&x\in{\mathbb{R}^3}\\ u \in H^1 (\mathbb{R}^3),~u>0&x\in{\mathbb{R}^3}, \end{cases}$ where $$\epsilon$$ is a small parameter, $$a,b>0$$ are positive constants and $$\lambda>0$$ is a parameter, and $$f$$ is a continuous superlinear and subcritical nonlinearity. Suppose that $$M(x)$$ has at least one minimum. We first prove that the system has a positive ground state solution $$u_\epsilon$$ for sufficiently large $$\lambda>0$$ and $$\epsilon>0$$ sufficiently small. Then we show that $$u_\epsilon$$ converges to the positive ground state solution of the associated limit problem and concentrates to a minimum point of $$M(x)$$ in certain sense as $$\epsilon\to 0$$. Moreover, some further properties of the ground state solutions are also studied. Finally, we investigate the relation between the number of positive solutions and the topology of the set of the global minima of the potentials by minimax theorems and the Ljusternik-Schnirelmann theory.

### MSC:

 35J60 Nonlinear elliptic equations 35B09 Positive solutions to PDEs 35B25 Singular perturbations in context of PDEs 35J20 Variational methods for second-order elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces

Zbl 1235.35093
Full Text:

### References:

 [1] Ding, Y.H., Variational methods for strongly indefinite problems, (2008), World Scientific Press [2] Szulkin, A.; Weth, T., The method of Nehari manifold, (), 597-632 · Zbl 1218.58010 [3] He, X.M.; Zou, W.M., Existence and concentration behavior of positive solutions for a Kirchhoff equation in $$\mathbb{R}^3$$, J. differential equations, 2, 1813-1834, (2012) · Zbl 1235.35093 [4] Mawhin, J.; Willen, M., Critical point theory and Hamiltonian systems, (1989), Springer-Verlag [5] V. Coti-Zelati, A short Introduction to critical point theory, Second school on nonlinear functional analysis and applications to differential equations, ICTP-Trieste, SMR 990-15, 1997. [6] Alves, C.O.; Souto, M.A., On existence and concentration behavior of ground state solutions for a class of problems with critical growth, Commun. pure appl. anal., 1, 417-431, (2002) · Zbl 1183.35121 [7] Rabinowitz, P.H., On a class of nonlinear Schrödinger equations, Z. angew. math. phys., 43, 270-291, (1992) · Zbl 0763.35087 [8] Jeanjean, L.; Tanaka, K., Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities, Calc. var. partial differential equations, 21, 287-318, (2004) · Zbl 1060.35012 [9] Lions, P.L., The concentration compactness principle in the calculus of variations: the locally compact case. parts 1, 2, Ann. inst. H. Poincaré anal. non linéaire, Ann. inst. H. Poincaré anal. non linéaire, 2, 223-283, (1984) · Zbl 0704.49004 [10] Willem, M., Minimax theorems, Progr. nonlinear differential equations appl., vol. 24, (1996), Birkhäuser Basel · Zbl 0856.49001 [11] J. Wang, L.X. Tian, J.X. Xu, F.B. Zhang, Multiplicity and concentration of positive ground state solutions for Schrödinger-Poisson systems with critical growth, preprint. [12] Tolksdorf, P., Regularity for some general class of quasilinear elliptic equations, J. differential equations, 51, 126-150, (1984) · Zbl 0488.35017 [13] 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 [14] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, Grundlehren math. wiss., vol. 224, (1983), Springer Berlin · Zbl 0691.35001 [15] Pankov, A., On decay of solution to nonlinear Schrödinger equations, Proc. amer. math. soc., 136, 2565-2570, (2008) · Zbl 1143.35093 [16] Strauss, W., Existence of solitary waves in higher dimensions, Comm. math. phys., 55, 149-162, (1977) · Zbl 0356.35028 [17] de Figueiredo, D.G.; Yang, J., Decay, symmetry and existence of solutions of semilinear elliptic systems, Nonlinear anal., 33, 211-234, (1998) · Zbl 0938.35054 [18] Kryszewski, W.; Szulkin, A., Generalized linking theorem with an application to semilinear Schrödinger equations, Adv. differential equations, 3, 441-472, (1998) · Zbl 0947.35061 [19] Ding, Y.H., Multiple homoclinics in a Hamiltonian system with asymptotically or superlinear terms, Commun. contemp. math., 8, 453-480, (2006) · Zbl 1104.70013 [20] Cingolani, S.; Lazzo, M., Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions, J. differential equations, 160, 118-138, (2000) · Zbl 0952.35043 [21] Benci, V.; Cerami, G., The effect of the domain topology on the number of positive solutions of nonlinear elliptic problems, Arch. ration. mech. anal., 114, 79-93, (1991) · Zbl 0727.35055 [22] 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 [23] Kirchhoff, G., Mechanik, (1883), Teubner Leipzig · JFM 08.0542.01 [24] Chipot, M.; Lovat, B., Some remarks on non local elliptic and parabolic problems, Nonlinear anal., 30, 4619-4627, (1997) · Zbl 0894.35119 [25] Alves, C.O.; Corrêa, F.J.S.A.; Ma, T.F., Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. math. appl., 49, 85-93, (2005) · Zbl 1130.35045 [26] Alves, C.O.; Corrêa, F.J.S.A.; Figueiredo, G.M., On a class of nonlocal elliptic problems with critical growth, Differ. equ. appl., 2, 409-417, (2010) · Zbl 1198.35281 [27] Chen, Ching-yu; Kuo, Yueh-cheng; Wu, Tsung-fang, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. differential equations, 250, 1876-1908, (2011) · Zbl 1214.35077 [28] Lions, J.-L., On some questions in boundary value problems of mathematical physics, (), 284-346 [29] DʼAncona, P.; Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. math., 108, 247-262, (1992) · Zbl 0785.35067 [30] Arosio, A.; Panizzi, S., On the well-posedness of the Kirchhoff string, Trans. amer. math. soc., 348, 305-330, (1996) · Zbl 0858.35083 [31] 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 [32] Perera, K.; Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. differential equations, 221, 246-255, (2006) · Zbl 1357.35131 [33] He, X.; Zou, W., Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear anal., 70, 1407-1414, (2009) · Zbl 1157.35382 [34] del Pino, M.; Felmer, P.L., Local mountain passes for semilinear elliptic problems in unbounded domains, Calc. var. partial differential equations, 4, 121-137, (1996) · Zbl 0844.35032 [35] del Pino, M.; Felmer, P.L., Multi-peak bound states for nonlinear Schrödinger equations, Ann. inst. H. Poincaré anal. non linéaire, 15, 127-149, (1998) · Zbl 0901.35023 [36] 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 [37] Wang, X.F., On concentration of positive bound states of nonlinear Schrödinger equations, Comm. math. phys., 153, 229-244, (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.