We investigate the bi-harmonic problem $$\left\{\begin{array}{ll} \Delta^{2}u - \alpha\nabla \cdot (f(\nabla u)) - \beta\Delta_{p}u = g(x,u) &\hbox{in}\ \ \Omega,\\[2mm] \frac{\partial u}{\partial n}=0, \frac{\partial(\Delta u)}{\partial n}=0 &\hbox{on}\ \ \partial\Omega,\end{array} \right. $$ where $\Delta^{2}u = \Delta(\Delta u), \Delta_{p}u =\div\left(|\nabla u|^{p-2}\nabla u\right)$ with $p > 2.$ $\Omega$ is a bounded smooth domain in $\mathbb{R}^{N},$ $N \geq 1.$ By using a special function space with the constraint $\int_{\Omega}u {\rm d}x = 0$, under suitable assumptions on $f$ and $g(x,u)$, we show the existence and multiplicity of sign-changing solutions to the above problem via the Mountain pass theorem and the Fountain theorem. Recent results from the literature are extended.
Wenqing WANG
,
Anmin MAO
. THE EXISTENCE AND NON-EXISTENCE OF SIGN-CHANGING SOLUTIONS TO BI-HARMONIC EQUATIONS WITH A p-LAPLACIAN[J]. Acta mathematica scientia, Series B, 2022
, 42(2)
: 551
-560
.
DOI: 10.1007/s10473-022-0209-6
[1] Lazer A, Mckenna P. Large-Amplitude periodic oscillations in suspension bridges:some new connections with nonlinear analysis. SIAM Review, 1990, 32:537-578
[2] King B, Stein O, Winkler M. A fourth-order parabolic equation modeling epitaxial thin film growth. J Math Anal Appl, 2003, 286:459-490
[3] Liu G. On asymptotic properties of Biharmonic Steklov eigenvalues. J Differential Equations, 2016, 261:4729-4757
[4] Zhang J, Li S. Multiple nontrivial solutions for some fourth-order semilinear elliptic problems. Nonlinear Anal, 2005, 60(2):221-230
[5] Yang R, Zhang W, Liu X. Sign-changing solutions for p-Biharmonic equations with hardy potential in RN. Acta Mathematica Scientia, 2017, 37(3):593-606
[6] Deng Z, Huang Y. Multiple symmetric results for a class of Biharmonic elliptic systems with critical homogeneous nonlinearity in RN. Acta Mathematica Scientia, 2017, 37(6):1665-1684
[7] Liu Y, Wang Z. Biharmonic equations with asymptotically linear nonlinearities. Acta Mathematica Scientia, 2007, 27(3):549-560
[8] Guo Z, Wei J, Zhou F. Singular radial entire solutions and weak solutions with prescribed singular set for a Biharmonic equation. J Differential Equations, 2017, 263:1188-1224
[9] Liu X, Huang Y. On sign-changing solution for a fourth-order asymptotically linear elliptic problem. Nonlinear Anal, 2010, 72:2271-2276
[10] Alves C, Nóbrega A. Nodal ground state solution to a Biharmonic equation via dual method. J Differential Equations, 2016, 260:5174-5201
[11] Sun J, Chu J, Wu T. Existence and multiplicity of nontrivial solutions for some Biharmonic equations with p-Laplacian. J Differential Equations, 2017, 262:945-977
[12] Mao A, Zhang Z. Sign-changing and multiple solutions of Kirchhoff type problems without the PS condition. Nonlinear Anal, 2009, 70:1275-1287
[13] Wang Z, Zhou H. Sign-changing solutions for the nonlinear Schrödinger-Poisson system in R3. Calc Var Partial Differential Equations, 2015, 52:927-943
[14] Shuai W, Wang Q. Existence and asymptotic behavior of sign-changing solutions for the nonlinear Schrödinger-Poisson system in R3. Z Angew Math Phys, 2015, 66:3267-3282
[15] 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 Letters, 2017, 73:128-135
[16] Ekeland I. Convexity Methods in Hamiltonian Mechanics. Springer, 1990
[17] Bartsch T. Infinitely many solutions of a symmetric Dirichlet problem. Nonlinear Anal, 1993, 20:1205-1216
