Multiple solutions for \(p\)-Kirchhoff equations in \(\mathbb R^N\). (English) Zbl 1283.35020
Summary: In this paper, we prove the existence of multiple ground-state solutions for the nonhomogeneous \(p\)-Kirchhoff elliptic equation
\[
\begin{aligned} M& \left(\int_{\mathbb R^N}(|\nabla u|^p+V(x)|u|^p)dx\right)(-\Delta_pu+V(x)|u|^{p-2}u)\\& = f(x,u)+g(x)\quad \text{in}\;\mathbb R^N, \end{aligned} \eqno{(0.1)}
\]
where \(V(x)\in C(\mathbb R^N)\) and \(V(x) \to+\infty\) as \(|x|\to+\infty\). The nonlinear function \(f(x,u)\) is continuous and satisfies some conditions. The solutions are obtained by the Mountain Pass Theorem, Ekeland’s variational principle and Krasnoselskii’s genus theory in [M. Struwe, Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. 3rd ed. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 34. Berlin: Springer (2000; Zbl 0939.49001)].
MSC:
| 35J20 | Variational methods for second-order elliptic equations |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
| 35J92 | Quasilinear elliptic equations with \(p\)-Laplacian |
| 58E30 | Variational principles in infinite-dimensional spaces |
| 49J40 | Variational inequalities |
Keywords:
\(p\)-Kirchhoff equation; Mountain Pass theorem; Ekeland’s variational principle; Krasnoselskii’s genus theory; multiple solutionsCitations:
Zbl 0939.49001References:
| [1] | Struwe, M., Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (2000), Springer-Verlag: Springer-Verlag New York · Zbl 0939.49001 |
| [3] | Li, Y. H.; Li, F. Y.; Shi, J. P., Existence of a positive solution to Kirchhoff type problems without compactness conditions, J. Differential Equations, 253, 2285-2294 (2012) · Zbl 1259.35078 |
| [4] | Alves, C. O.; Figueiredo, G. M., Nonlinear perturbations of a periodic Kirchhoff equation in \(R^N\), Nonlinear Anal., 75, 2750-2759 (2012) · Zbl 1264.45008 |
| [5] | 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 |
| [6] | Chipot, M.; Lovat, B., Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal., 30, 4619-4627 (1997) · Zbl 0894.35119 |
| [7] | Corrêa, F. J.S. A., On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal., 59, 1147-1155 (2004) · Zbl 1133.35043 |
| [8] | Dreher, M., The Kirchhoff equation for the \(p\)-Laplacian, Rend. Semin. Mat. Univ. Politec. Torino, 64, 217-238 (2006) · Zbl 1178.35006 |
| [9] | Liu, D.; Zhao, P., Multiple nontrivial solutions to a \(p\)-Kirchhoff equation, Nonlinear Anal., 75, 5032-5038 (2012) · Zbl 1248.35018 |
| [10] | Corrêa, F. J.S. A.; Figueiredo, G. M., On a \(p\)-Kirchhoff equation via Krasnoselskii’s genus, Appl. Math. Lett., 22, 819-822 (2009) · Zbl 1171.35371 |
| [11] | Colasuonno, F.; Pucci, P., Multiplicity of solutions for \(p(x)\)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74, 5962-5974 (2011) · Zbl 1232.35052 |
| [12] | Toan, H. Q.; Chung, N. T., Existence of weak solutions for a class of nonuniformly nonlinear elliptic equations in unbounded domains, Nonlinear Anal., 70, 3987-3996 (2009) · Zbl 1163.35394 |
| [13] | Mihăilescu, M.; Rădulescu, V., Ground state solutions of nonlinear singular Schrödinger equations with lack of compactness, Math. Methods Appl. Sci., 26, 897-906 (2003) · Zbl 1107.35102 |
| [14] | Gonçalves, J. V.; Miyagaki, O. H., Multiple positive solutions for semilinear elliptic equations in \(R^N\) involving subcritical exponents, Nonlinear Anal., 32, 1, 41-51 (1998) · Zbl 0891.35031 |
| [15] | Cazenave, T., Semilinear Schrödinger Equations, Courant lecture notes, vol. 10 (2003), American Math. Soc. Providence: American Math. Soc. Providence Rhole Island · Zbl 1055.35003 |
| [16] | Caffarelli, L.; Kohn, R.; Nirenberg, L., First order interpolation inequalities with weights, Compos. Math., 53, 3, 259-275 (1984) · Zbl 0563.46024 |
| [17] | Wu, T. F., Multiple positive solutions for a class of concave-convex elliptic problems in \(R^N\) involving sign-changing weight, J. Funct. Anal., 258, 99-131 (2010) · Zbl 1182.35119 |
| [18] | Wu, T. F., On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function, Commun. Pure Appl. Anal., 7, 383-405 (2008) · Zbl 1156.35046 |
| [19] | Brown, K. J.; Zhang, Y., The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193, 481-499 (2003) · Zbl 1074.35032 |
| [20] | Ambrosetti, A.; Rabinowitz, P. H., Dual variational methods in critical point theorey and applications, J. Funct. Anal., 14, 349-381 (1973) · Zbl 0273.49063 |
| [21] | Kuzin, I.; Pohozaev, S., (Entire Solutions of Semilinear Elliptic Equations. Entire Solutions of Semilinear Elliptic Equations, Progress in Nonlinear Differential Equations and Their Applications, vol. 33 (1997), Birkhäuser) · Zbl 0882.35003 |
| [22] | Chen, C. S.; Liu, S. L.; Yao, H. P., Existence of solutions for quasilinear elliptic exterior problem with the concave-convex nonlinearities and the nonlinear boundary conditions, J. Math. Anal. Appl., 383, 111-119 (2012) · Zbl 1222.35090 |
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