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
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[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
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