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Existence of a positive solution to Kirchhoff-type problems without compactness conditions. (English) Zbl 1259.35078

Summary: The existence of a positive solution to a Kirchhoff-type problem on \(\mathbb R^{N}\) is proved by using variational methods, and the new result does not require usual compactness conditions. A cut-off functional is utilized to obtain the bounded Palais-Smale sequences.

MSC:

35J20 Variational methods for second-order elliptic equations
35B09 Positive solutions to PDEs
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[1] Berestycki, H.; Lions, P.-L., Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal., 82, 313-345 (1983) · Zbl 0533.35029
[2] Chen, C.; Kuo, Y.; Wu, T., The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations, 250, 1876-1908 (2011) · Zbl 1214.35077
[3] Ding, W.-Y.; Ni, W.-M., On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. Anal., 91, 283-308 (1986) · Zbl 0616.35029
[4] He, X.; Zou, W., Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear Anal., 70, 1407-1414 (2009) · Zbl 1157.35382
[5] He, X.; Zou, W., Multiplicity of solutions for a class of Kirchhoff type problems, Acta Math. Appl. Sin. Engl. Ser., 26, 387-394 (2010) · Zbl 1196.35077
[6] Jeanjean, L., Local condition insuring bifurcation from the continuous spectrum, Math. Z., 232, 651-664 (1999) · Zbl 0934.35047
[7] Jeanjean, L.; Le Coz, S., An existence and stability result for standing waves of nonlinear Schrödinger equations, Adv. Differential Equations, 11, 813-840 (2006) · Zbl 1155.35095
[8] Jin, J.; Wu, X., Infinitely many radial solutions for Kirchhoff-type problems in \(R^N\), J. Math. Anal. Appl., 369, 564-574 (2010) · Zbl 1196.35221
[9] Kikuchi, H., Existence and stability of standing waves for Schrödinger-Poisson-Slater equation, Adv. Nonlinear Stud., 7, 403-437 (2007) · Zbl 1133.35013
[10] Li, Y.; Wang, Z.-Q.; Zeng, J., Ground states of nonlinear Schrödinger equations with potentials, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23, 829-837 (2006) · Zbl 1111.35079
[11] Liu, Z.; Wang, Z.-Q., On the Ambrosetti-Rabinowitz superlinear condition, Adv. Nonlinear Stud., 4, 563-574 (2004) · Zbl 1113.35048
[12] Mao, A.; Zhang, Z., Sign-changing and multiple solutions of Kirchhoff type problems without the (PS) condition, Nonlinear Anal., 70, 1275-1287 (2009) · Zbl 1160.35421
[13] Perera, K.; Zhang, Z., Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations, 221, 246-255 (2006) · Zbl 1357.35131
[14] Strauss, W. A., Existence of solitary waves in higher dimensions, Comm. Math. Phys., 55, 149-162 (1977) · Zbl 0356.35028
[15] Struwe, M., Variational Methods and Their Applications to Nonlinear Differential Equations and Hamiltonian Systems (1990), Springer-Verlag: Springer-Verlag Berlin
[16] Sun, J.; Tang, C., Existence and multiplicity of solutions for Kirchhoff type equations, Nonlinear Anal., 74, 1212-1222 (2011) · Zbl 1209.35033
[17] Willem, M., Minimax Theorems (1996), Birkhäuser · Zbl 0856.49001
[18] Wu, X., Existence of nontrivial solutions and high energy solutions for Schrödinger-Kirchhoff-type equations in \(R^N\), Nonlinear Anal. Real World Appl., 12, 1278-1287 (2011) · Zbl 1208.35034
[19] Yang, Y.; Zhang, J., Nontrivial solutions of a class of nonlocal problems via local linking theory, Appl. Math. Lett., 23, 377-380 (2010) · Zbl 1188.35084
[20] Zhang, Z.; Perera, K., Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl., 317, 456-463 (2006) · Zbl 1100.35008
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