POSITIVE RADIAL SOLUTIONS OF A FOURTH-ORDER ELLIPTIC BOUNDARY VALUE PROBLEM ON UNIT BALL WITH NONLINEAR GRADIENT TERM

Yongxiang Li; Yang Liu

doi:10.11948/20260011

2026 Volume 16 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(6): 3307-3319

Next Article Previous Article

Yongxiang Li, Yang Liu. POSITIVE RADIAL SOLUTIONS OF A FOURTH-ORDER ELLIPTIC BOUNDARY VALUE PROBLEM ON UNIT BALL WITH NONLINEAR GRADIENT TERM[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3307-3319. doi: 10.11948/20260011

Citation:

Yongxiang Li, Yang Liu. POSITIVE RADIAL SOLUTIONS OF A FOURTH-ORDER ELLIPTIC BOUNDARY VALUE PROBLEM ON UNIT BALL WITH NONLINEAR GRADIENT TERM[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3307-3319. doi: 10.11948/20260011

POSITIVE RADIAL SOLUTIONS OF A FOURTH-ORDER ELLIPTIC BOUNDARY VALUE PROBLEM ON UNIT BALL WITH NONLINEAR GRADIENT TERM

Yongxiang Li^1, ,,
Yang Liu^1,

1.
Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Author Bio: Email: liuynwnu@163.com(Y. Liu)
Corresponding author: Email: liyxnwnu@163.com(Y. Li)
Fund Project: The authors were supported by NNSFs of China (12061062, 12161080)

Abstract

Abstract

In this paper, we discuss the existence of positive radial solution of the fourth-order elliptic equation $ {\Delta}^2 u = f(|x|,\,u,\,|\nabla u|,\,\Delta u) $ on the unit ball $ \Omega $ of $ {\mathbb{R}}^N $ with Navier boundary condition $ u|_{\partial \Omega}=0 $ and $ \Delta u|_{\partial \Omega}=0 $, where $ N\ge 2 $, $ f: [0,\,1]\times{\mathbb{R}}^+\times{\mathbb{R}}^+\times{\mathbb{R}}^-\to {\mathbb{R}} $ is a continuous function. Under certain local inequality conditions of $ f $, an existence result of positive radial solution is obtained. The inequality conditions relate to the principal eigenvalue of Laplacian $ -\Delta $ on $ u|_{\partial\Omega}=0 $. The discussion is based on the method of lower and upper solutions and truncating function technique.
- Fourth-order elliptic boundary value problem /
- positive radial solution /
- lower and upper solutions /
- truncating function technique
MSC: 35J40, 47H10, 47N20

FullText(HTML)

References

[1]	I. Abid and S. Baraket, Construction of singular solutions for elliptic problem of fourth order derivative with a subcritical nonlinearity, Differ. Integr. Equ., 2008, 21(7–8), 653–664. Google Scholar
[2]	C. O. Alves and A. B. Nobrega, Nodal ground state solution to a biharmonic equation via dual method, J. Differential Equations, 2016, 260(6), 5174–5201. doi: 10.1016/j.jde.2015.12.014 CrossRef Google Scholar
[3]	Y. An and R. Liu, Existence of nontrivial solutions of an asymptotically linear fourth-order elliptic equation, Nonlinear Anal., 2008, 68(11), 3325–3331. doi: 10.1016/j.na.2007.03.028 CrossRef Google Scholar
[4]	A. Benhassine, S. Farhani and T. Talbi, On nodal solutions for a class of fourth-order elliptic equations, The Journal of Analysis, 2025, 33(5), 2081–2096. doi: 10.1007/s41478-025-00907-8 CrossRef Google Scholar
[5]	Y. Chen and P. J. McKenna, Traveling waves in a nonlinear suspension beam: Theoretical results and numerical observations, J. Differential Equations, 1997, 135(2), 325–355. Google Scholar
[6]	R. Dalmasso, Uniqueness theorems for some fourth order elliptic equations, Proc. Amer. Math. Soc., 1995, 123(4), 1177–1183. doi: 10.1090/S0002-9939-1995-1242078-X CrossRef Google Scholar
[7]	M. Feng, Positive solutions for biharmonic equations: Existence, uniqueness and multiplicity, Mediterr. J. Math., 2023, 20(6), Paper No. 309. doi: 10.1007/s00009-023-02513-z CrossRef Google Scholar
[8]	M. Feng and H. Chen, Positive solutions for a class of biharmonic equations: Existence and uniqueness, Applied Mathematics Letters, 2023, 143(1), Paper No. 108687. Google Scholar
[9]	M. Feng and Y. Lu, Positive solutions for a fourth order elliptic problem: Existence, uniqueness and nonexistence, Proc. Indian Acad. Sci. Math. Sci., 2024, 134(2), Paper No. 38. doi: 10.1007/s12044-024-00801-6 CrossRef Google Scholar
[10]	M. Feng and Y. Lu, Existence, uniqueness and multiplicity of nontrivial solutions for biharmonic equations, Electron. J. Differential Equations, 2025, Paper No. 52. Google Scholar
[11]	F. Gazzola, H. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 2003, 18(2), 117–143. doi: 10.1007/s00526-002-0182-9 CrossRef Google Scholar
[12]	F. Gazzola, H. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Lectures Notes in Mathematics, 1991, Springer-Verlag, Berlin, 2010. Google Scholar
[13]	R. Ghoudi and M. Lahrach, Sign-changing solutions for a perturbed biharmonic equation with critical exponent, Partial Differ. Equ. Appl., 2025, 6(6), Paper No. 44. doi: 10.1007/s42985-025-00355-w CrossRef Google Scholar
[14]	B. Gidas, W. -M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Commun. Math. Phys., 1979, 68(3), 209–243. doi: 10.1007/BF01221125 CrossRef Google Scholar
[15]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, New York, 1983. Google Scholar
[16]	Z. Guo, B. Lai and D. Ye, Revisiting the biharmonic equation modelling electrostatic actuation in low dimensions, Proc. Amer. Math. Soc., 2014, 142(6), 2027–2034. doi: 10.1090/S0002-9939-2014-11895-8 CrossRef Google Scholar
[17]	C. P. Gupta and Y. C. Kwong, Biharmonic eigenvalue problems and $L. p$ estimates, Int. J. Math. Sci., 1990, 13(3), 469–480. doi: 10.1155/S0161171290000692 CrossRef Google Scholar
[18]	S. Hu and L. Wang, Existence of nontrivial solutions for fourth-order asymptotically linear elliptic equations, Nonlinear Anal., 2014, 94(1), 120–132. Google Scholar
[19]	O. A. Ladyzhenskaya and N. N. Uraltseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. Google Scholar
[20]	Y. Li and Y. Wang, Existence results for nonlinear fourth-order elliptic boundary value problems, Journal of Inequalities and Applications, 2025, Paper No. 160. Google Scholar
[21]	Y. Li and Y. Wang, The existence and uniqueness of radial solutions for biharmonic elliptic equations in an annulus, Axioms, 2024, 13(6), Paper No. 383. doi: 10.3390/axioms13060383 CrossRef Google Scholar
[22]	Y. Li and S. Yang, Existence of positive solutions for the fourth-order elliptic boundary value problems, Boundary Value Problems, 2025, Paper No. 53. Google Scholar
[23]	Y. Li and S. Yang, Positive radial symmetric solutions of nonlinear biharmonic equations in an annulus, Symmetry, 2024, 16(7), Paper No. 793. doi: 10.3390/sym16070793 CrossRef Google Scholar
[24]	X. Liu and Y. Huang, On sign-changing solution for a fourth-order asymptotically linear elliptic problem, Nonlinear Anal., 2010, 72(5), 2271–2276. doi: 10.1016/j.na.2009.11.001 CrossRef Google Scholar
[25]	Y. Liu and Z. Wang, Biharmonic equations with asymptotically linear nonlinearities, Acta Math. Sci. Ser. B, 2007, 27(3), 549–560. doi: 10.1016/S0252-9602(07)60055-1 CrossRef Google Scholar
[26]	Z. Liu, Concentrating solutions for a biharmonic problem with supercritical growth, Topol. Methods Nonlinear Anal., 2023, 62(2), 455–484. Google Scholar
[27]	P. J. McKenna and W. Walter, Traveling waves in a suspension bridge, SIAM J. Appl. Math., 1990, 50(3), 703–715. doi: 10.1137/0150041 CrossRef Google Scholar
[28]	H. B. Omrane, M. Ghedamsi and S. Khenissy, Biharmonic equations under dirichlet boundary conditions with supercritical growth, Adv. Nonlinear Stud., 2016, 16(2), 175–184. doi: 10.1515/ans-2015-5028 CrossRef Google Scholar
[29]	C. V. Pao, On fourth-order elliptic boundary value problems, Proc. Amer. Math. Soc., 2000, 128(4), 1023–1030. Google Scholar
[30]	R. Pei and H. Xia, Multiplicity results for some fourth-order elliptic equations with combined nonlinearities, AIMS Mathematics, 2023, 8(6), 14704–14725. doi: 10.3934/math.2023752 CrossRef Google Scholar
[31]	Y. M. Wang, On fourth-order elliptic boundary value problems with nonmonotone nonlinear function, J. Math. Anal. Appl., 2005, 307(1), 1–11. Google Scholar
[32]	J. Zhang and Z. Wei, Multiple solutions for a class of biharmonic equations with a nonlinearity concave at the origin, J. Math. Anal. Appl., 2011, 383(2), 291–306. doi: 10.1016/j.jmaa.2011.05.030 CrossRef Google Scholar