zbMATH — the first resource for mathematics

Existence results for some fourth-order nonlinear elliptic problems. (English) Zbl 0981.35016
From the introduction: Let \(\Omega\) be a bounded open set in \(\mathbb{R}^n\). We are concerned with the fourth-order semilinear elliptic boundary value problem \[ \begin{aligned} &\Delta^2 u+ c\Delta u= f(x,u)\quad\text{in }\Omega,\\ & u|_{\partial\Omega}=\Delta u|_{\partial\Omega}= 0,\end{aligned}\tag{1} \] where \(\Delta^2\) denotes the biharmonic operator and \(c\in \mathbb{R}\).
We also consider the fourth-order quasilinear elliptic boundary value problem \[ \begin{aligned} &\Delta(g_1((\Delta u)^2)\Delta u)+ c\text{ div}(g_2(|\nabla u|^2)\nabla u)= f(x,u)\quad\text{in }\Omega,\\ & u|_{\partial\Omega}=\Delta u|_{\partial\Omega} =0.\end{aligned}\tag{2} \] It is the purpose of this paper to use variational methods for the fourth-order semilinear problem and the fourth-order quasilinear problem. We show the existence of solutions of problems (1) and (2) for a more general nonlinearity \(f\) under weak assumptions.

35J60 Nonlinear elliptic equations
35J35 Variational methods for higher-order elliptic equations
35J30 Higher-order elliptic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
Full Text: DOI
[1] Drabek, P.; Kufner, A.; Nicolosi, F., On the solvability of degenerated quasilinear elliptic equations of higher order, J. differential equations, 109, 325-347, (1994) · Zbl 0847.35054
[2] Grigis, A.; Rothchila, L.P., A criterion for analytic hypoellipticity of a class of differential operators with polynormal coefficients, Ann. math., 108, 443-460, (1983) · Zbl 0541.35017
[3] Lazer, A.C.; Mckenna, P.J., Large amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM rev., 32, 537-578, (1990) · Zbl 0725.73057
[4] Lazer, A.C.; Mckenna, P.J., Global bifurcation and a theorem of tarantello, J. math. anal. appl., 181, 648-655, (1994) · Zbl 0797.34021
[5] Mckenna, P.J.; Walter, W., Nonlinear oscillations in a suspension bridge, Arch. rational mech. anal., 98, 167-177, (1987) · Zbl 0676.35003
[6] Mckenna, P.J.; Walter, W., Travelling waves in a suspension bridge, SIAM J. appl. math., 50, 703-715, (1990) · Zbl 0699.73038
[7] Micheletti, A.M.; Pistoia, A., Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear anal., 31, 895-908, (1998) · Zbl 0898.35032
[8] Tarantello, G., A note on a semilinear elliptic problem, Differential integral equations, 5, 561-566, (1992) · Zbl 0786.35060
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.