Multiple solutions to boundary-value problems for fourth-order elliptic equations

Duong Trong  Luyen; Mai Thi Thu  Trang

doi:10.37863/umzh.v75i6.6958

Duong Trong Luyen Department of Mathematics, Hoa Lu University, Ninh Nhat, Ninh Binh City, Vietnam, International Center for Research and Postgraduate Training in Mathematics, Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam
Mai Thi Thu Trang Department of Basic, Academy of Finance, Duc Thang Wrd., Bac Tu Liem Dist., Hanoi, Vietnam

DOI: https://doi.org/10.37863/umzh.v75i6.6958

Keywords: Biharmonic, boundary value problems, critical points, perturbation methods, multiple solutions.

Abstract

UDC 517.9

We study the existence of multiple solutions for the biharmonic problem\begin{gather*}\Delta^2 u = f(x, u) + g(x, u)\quad \mbox{in}\quad \Omega,\\ u = \partial_\nu u = 0\quad \text{on}\quad \partial\Omega,\end{gather*} where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N,$ $ N >4,$ $f(x, \xi)$ is odd in $\xi,$ and $g(x, \xi)$ is a perturbation term. Under certain growth conditions on $f$ and $g,$ we show that there are infinitely many weak solutions to the problem.

References

C. T. Anh, B. K. My, Existence of solutions to $Delta_lambda$-Laplace equations without the Ambrosetti–Rabinowitz condition, Complex Var. and Elliptic Equat., 61, № 1, 137–150 (2016). DOI: https://doi.org/10.1080/17476933.2015.1068762

S. Agmon, On kernels, eigenvalues, and eigenfunctions of operators related to elliptic problems.} Commun. Pure and Appl. Math., 18, 627–663 (1965). DOI: https://doi.org/10.1002/cpa.3160180405

P. Bartolo, V. Benci, D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with ``strong'' resonance at infinity, Nonlinear Anal., 7, № 9, 981–1012 (1983). DOI: https://doi.org/10.1016/0362-546X(83)90115-3

M. Chipot, Variational inequalities and flow in porous media, Applied Mathematical Sciences, 52, Springer-Verlag, New York (1984). DOI: https://doi.org/10.1007/978-1-4612-1120-4

A. C. Lazer, P. J. Mckenna, Large amplitude periodic oscillation in suspension bridges: some new connections with nonlinear analysis, SIAM Rev., 32, 537–578 (1990). DOI: https://doi.org/10.1137/1032120

X. Q. Liu, Y. S. Huang, One sign-changing solution for a fourth-order asymptotically linear elliptic problem, Nonlinear Anal., 72, 2271–2276 (2010). DOI: https://doi.org/10.1016/j.na.2009.11.001

A. M. Micheletti, A. Pistoia, Multiplicity solutions for a fourth order semilinear elliptic problems, Nonlinear Anal., 31, 895–908 (1998). DOI: https://doi.org/10.1016/S0362-546X(97)00446-X

Å. Pleijel, On the eigenvalues and eigenfunctions of elastic plates, Commun. Pure and Appl. Math., 3, 1–10 (1950). DOI: https://doi.org/10.1002/cpa.3160030102

P. H. Rabinowitz, Multiple critical points of perturbed symmetric functionals, Trans. Amer. Math. Soc., 272, № 2, 753–769 (1982). DOI: https://doi.org/10.1090/S0002-9947-1982-0662065-5

J. H. Zhang, S. J. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal., 60, 221–230 (2005). DOI: https://doi.org/10.1016/S0362-546X(04)00313-X

J. Zhang, Z. L. Wei, Infinitely many nontrivial solutions for a class of biharmonic equations via variant fountain theorems, Nonlinear Anal., 74, 7474–7485 (2011). DOI: https://doi.org/10.1016/j.na.2011.07.067

J. W. Zhou, X. Wu, Sign-changing solutions for some fourth-order nonlinear elliptic problems, J. Math. Anal. and Appl., 342, 542–558 (2008). DOI: https://doi.org/10.1016/j.jmaa.2007.12.020

H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Math. Ann., 71, 441–479 (1912). DOI: https://doi.org/10.1007/BF01456804

A. E. Kogoj, E. Lanconelli, On semilinear $Delta_lambda$-Laplace equation, Nonlinear Anal., 75, № 12, 4637–4649 (2012). DOI: https://doi.org/10.1016/j.na.2011.10.007

D. T. Luyen, D. T. Huong, L. T. H. Hong, Existence of infinitely many solutions for $Delta_gamma$-Laplace problems, Math. Notes, 103, № 5, 724–736 (2018). DOI: https://doi.org/10.1134/S000143461805005X

D. T. Luyen, Two nontrivial solutions of boundary value problems for semilinear $Delta_{gamma}$ differential equations, Math. Notes, 101, № 5, 815–823 (2017). DOI: https://doi.org/10.1134/S0001434617050078

D. T. Luyen, Infinitely many solutions for a fourth-order semilinear elliptic equations perturbed from symmetry, Bull. Malays. Math. Sci. Soc., 44, № 3, 1701–1725 (2021). DOI: https://doi.org/10.1007/s40840-020-01031-5

D. T. Luyen, Picone's identity for $Delta_{gamma}$-Laplace operators and its applications, Ukr. Math. J., 73, № 4, 515–522 (2021). DOI: https://doi.org/10.37863/umzh.v73i4.639

D. T. Luyen, N. M. Tri, Existence of solutions to boundary value problems for semilinear $Delta_{gamma}$ differential equations, Math. Notes, 97, № 1, 73–84 (2015). DOI: https://doi.org/10.1134/S0001434615010101

D. T. Luyen, N. M. Tri, Large-time behavior of solutions to damped hyperbolic equation involving strongly degenerate elliptic differential operators, Sib. Math. J., 57, № 4, 632–649 (2016). DOI: https://doi.org/10.1134/S0037446616040078

D. T. Luyen, N. M. Tri, Global attractor of the Cauchy problem for a semilinear degenerate damped hyperbolic equation involving the Grushin operator, Ann. Polon. Math., 117, № 2, 141–162 (2016). DOI: https://doi.org/10.4064/ap3831-3-2016

D. T. Luyen, N. M. Tri, On the existence of multiple solutions to boundary value problems for semilinear elliptic degenerate operators, Complex Var. and Elliptic Equat., 64, № 6, 1050–1066 (2019). DOI: https://doi.org/10.1080/17476933.2018.1498086

P. T. Thuy, N. M. Tri, Nontrivial solutions to boundary value problems for semilinear strongly degenerate elliptic differential equations, NoDEA Nonlinear Different. Equat. and Appl., 19, № 3, 279–298 (2012). DOI: https://doi.org/10.1007/s00030-011-0128-z

P. T. Thuy, N. M. Tri, Long time behavior of solutions to semilinear parabolic equations involving strongly degenerate elliptic differential operators, NoDEA Nonlinear Different. Equat. and Appl., 20, № 3, 1213–1224 (2013). DOI: https://doi.org/10.1007/s00030-012-0205-y

N. M. Tri, Semilinear degenerate elliptic differential equations, local and global theories, Lambert Acad. Publ. (2010).

N. M. Tri, Recent progress in the theory of semilinear equations involving degenerate elliptic differential operators, Publ. House Sci. and Technology Vietnam Acad. Sci. and Technology (2014).