Abstract
We consider the following class of elliptic Kirchhoff-Boussinesq type problems given by
where \(\Omega \subset \mathbb {R}^{N}\) is a bounded and smooth domain, \(2< p\le \frac{2N}{N-2}\) for \(N\ge 3\), \(2_{**}=\frac{2N}{N-4}\) if \(N\ge 5\), \(2_{**}=\infty \) if \(3\le N <5\) and f is a continuous function. We show existence and multiplicity of nontrivial solutions using minimization technique on the Nehari manifold, Mountain Pass Theorem and Genus theory. In this paper we consider the subcritical case \(\beta =0\) and the critical case \(\beta =1\).
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References
Chueshov, I., Lasiecka, I.: On global attractor for \(2D\) Kirchhoff-Boussinesq model with supercritical nonlinearity. Commun. Partial Differ. Equ. 36, 67–99 (2011)
Chueshov, I., Lasiecka, I.: Existence, uniqueness of weak solutions and global attractors for a class of nonlinear 2D Kirchhoff-Boussinesq models. Discr. Cont. Dyn. Sys. 15, 777–809 (2006)
Lagnese, J.: Boundary Stabilization of Thin Plates. SIAM, Philadelphia (1989)
Lagnese, J., Lions, J.L.: Modeling, Analysis and Control of Thin Plates. Collection RMA. Masson, Paris (1988)
Sun, F., Liu, L., Wu, Y.: Infinitely many sign-changing solutions for a class of biharmonic equation with \(p-\)Laplacian and Neumann boundary condition. Appl. Math. Lett. 73, 128–135 (2017)
Sun, J., Wu, T.: Existence of nontrivial solutions for a biharmonic equation with \(p-\)Laplacian and singular sign-changing potential. Appl. Math. Lett. 66, 61–67 (2017)
Sun, J., Chu, J., Wu, T.: Existence and multiplicity of nontrivial solutions for some biharmonic equations with \(p-\) Laplacian. J. Differ. Equ. 262, 945–977 (2017)
Yang, T.: On a critical biharmonic system involving \(p-\)Laplacian and Hardy potential. Appl. Math. Lett. 121, 107433 (2021)
Ramos, M.: Uniform estimates for the biharmonic operator in \(\mathbb{R} ^{N}\) and applications. Commun. Appl. Anal. 8, 435–457 (2009)
Gagliardo, E.: Ulteriori proprietà di alcune classi di funzioni in più variabili. Ric. Mat. 8, 24–51 (1959)
Nirenberg, L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 3(13), 115–162 (1959)
Liu, Jia-quan, Wang, Ya.-qi, Wang, Zhi-Qiang.: Solutions for quasilinear Schrodinger equations via the Nehari method. Comm. PDE 29(5–6), 879–901 (2004)
Willem, M.: Minimax methods, Handbook of nonconvex analysis and applications, pp. 597–632. Int. Press, Somerville (2010)
Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36, 437–477 (1983)
Kavian, O.: Introduction à la théorie des points critiques. Springer-Verlag, Berlin (1991)
Alves, C.O., Carrião, P.C., Medeiros, E.S.: Multiplicity of solutions for a class of quasilinear problem in exterior domain with Neumann conditions. Abstr. Appl. Anal. Article ID 415780 (2004)
Ambrosetti, A., Rabinowitz, P.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)
Castro, A.: Metodos variacionales y analisis functional no linear, X Colóquio Colombiano de Matemáticas (1980)
Costa, D.G.: Tópicos em Ánalise não linear. Escola Latino-Americano de Matemática (1986)
Krasnolselskii, M.A.: Topological methods in the theory of nonlinear integral equations. Mac Millan, New York (1964)
Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence, RI (1986)
Clark, D.C.: A variant of the Lusternik-Schnirelman theory. Indiana Univ. Math. J. 22, 65–74 (1972)
Benci, V.: On critical points theory for indefinite functionals in the presence of symmetric. Trans. Amer. Math. Soc. 274, 533–572 (1982)
Alves, C.O., Souto, M.A.S.: Existence of least energy nodal solution for a Schrödinger-Poisson system in bounded domains. Z. Angew. Math, Phys (2013)
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Communicated by P. H. Rabinowitz.
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Romulo D. Carlos and Giovany M. Figueiredo were partially supported by CNPq, Capes and FapDF - Brazil.
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Carlos, R.D., Figueiredo, G.M. Existence and multiplicity of nontrivial solutions to a class of elliptic Kirchhoff-Boussinesq type problems. Calc. Var. 63, 120 (2024). https://doi.org/10.1007/s00526-024-02734-4
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DOI: https://doi.org/10.1007/s00526-024-02734-4