×

Multiplicity results for solutions of \(p\)-biharmonic problems. (English) Zbl 1427.35086

Summary: In this paper, we deal with the multiplicity existence of solutions for the following \(p\)-biharmonic equation: \[ \begin{cases} \varDelta_p^2 u = \lambda | u |^{p - 2} u + | u |^{q - 2} u,\qquad \text{in} \varOmega, u = \Delta u = 0,\qquad \qquad\qquad\text{on} \partial \varOmega, \end{cases}\] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N, \varDelta_p^2 u = \varDelta(| \varDelta u |^{p - 2} \varDelta u), p < q \leq p^\ast = \frac{N p}{N - 2 p}, \lambda \in \mathbb{R}\) is a parameter. When \(p < q < p^\ast \), we prove that the above problem possesses infinitely many solutions. While when \(q = p^\ast \), a multiplicity existence result is obtained.

MSC:

35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J30 Higher-order elliptic equations
35J40 Boundary value problems for higher-order elliptic equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adams, R. A.; Fournier, J. J.F., Sobolev Spaces (2003), Academic Press: Academic Press New York · Zbl 1098.46001
[2] Benedikt, J. R.; Drábek, P., Estimates of the principal eigenvalue of the \(p\)-biharmonic operator, Nonlinear Anal., 75, 13, 5374-5379 (2012) · Zbl 1244.35096
[3] Chen, Y.; Mckenna, P. J., Traveling waves in a nonlinearly suspended beam: theoretical results and numerical observations, J. Differential Equations, 136, 2, 325-355 (1997) · Zbl 0879.35113
[4] Colasuonno, F.; Pucci, P., Multiplicity of solutions for \(p(x)\)-polyharmonic elliptic Kirchhoff equations, Nonlinear Anal., 74, 17, 5962-5974 (2011) · Zbl 1232.35052
[5] Fadell, E. R.; Rabinowitz, P. H., Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math., 45, 139-174 (1978) · Zbl 0403.57001
[6] Guo, Z. C.; Yin, J. X.; Ke, Y. Y., Multiplicity of positive radially symmetric solutions for a quasilinear biharmonic equation in the plane, Nonlinear Anal., 74, 4, 1320-1330 (2011) · Zbl 1215.34030
[7] Huang, Y. S.; Liu, X. Q., Sign-changing solutions for \(p\)-biharmonic equations with Hardy potential, J. Math. Anal. Appl., 412, 1, 142-154 (2014) · Zbl 1317.35070
[8] Ji, C.; Wang, W. H., On the \(p\)-biharmonic equation involving concave-convex nonlinearities and sign-changing weight function, Electron. J. Qual. Theory Differ. Equ., 2 (2012), 17 pp · Zbl 1340.35077
[9] Lazer, A. C.; Mckenna, P. J., Large-amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev., 32, 4, 537-578 (1990) · Zbl 0725.73057
[10] Li, C.; Tang, C. L., Three solutions for a Navier boundary value problem involving the \(p\)-biharmonic, Nonlinear Anal., 72, 3-4, 1339-1347 (2010) · Zbl 1180.35210
[11] Li, G. B.; Ye, H. Y., The existence of infinitely many solutions for \(p\)-Laplacian type equations on \(R^N\) with linking geometry, Ann. Acad. Sci. Fenn. Math., 38, 2, 515-534 (2013) · Zbl 1291.35105
[12] Liu, S. B.; Medeiros, E.; Perera, K., Multiplicity results for p-biharmonic problems via Morse theory, Commun. Appl. Anal., 13, 3, 447-455 (2009) · Zbl 1192.35042
[13] Perera, K., Nontrivial critical groups in p-Laplacian problems via the Yang index, Topol. Methods Nonlinear Anal., 21, 301-309 (2003) · Zbl 1039.47041
[14] Perera, K.; Agarwal, R. P.; O’Regan, D., (Morse Theoretic Aspects of P-Laplacian Type Operators. Morse Theoretic Aspects of P-Laplacian Type Operators, Mathematical Surveys and Monographs, vol. 161 (2010), American Mathematical Society: American Mathematical Society Providence, RI) · Zbl 1192.58007
[15] Perera, K.; Squassina, M.; Yang, Y., Bifurcation and multiplicity results for critical p-Laplacian problems, Topol. Methods Nonlinear Anal., 47, 1, 187-194 (2016) · Zbl 1379.35153
[16] Pucci, P.; Radulescu, V. D., Remarks on a polyharmonic eigenvalue problem, C. R. Math. Acad. Sci. Paris, 348, 3-4, 161-164 (2010) · Zbl 1186.35127
[17] Pucci, P.; Serrin, J., Critical exponents and critical dimensions for polyharmonic operators, J. Math. Pures Appl. (9), 69, 1, 55-83 (1990) · Zbl 0717.35032
[18] 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, 1, 99-131 (2010) · Zbl 1182.35119
[19] Yan, S. S.; Yang, J. F., Fountain theorem over cones and applications, Acta Math. Sci. Ser. B, 30, 6, 1881-1888 (2010) · Zbl 1240.35185
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.