Nontrivial solutions for some fourth order semilinear elliptic problems. (English) Zbl 0929.35053

From the introduction: Let us consider the following problem: \[ \begin{cases} \Delta^2u+ a^2\Delta u= b[(u+ 1)^+- 1]\quad & \text{in }\Omega,\\ \Delta u= 0,\quad u=0\quad & \text{on }\partial\Omega,\end{cases} \] where \(\Delta^2\) is the biharmonic operator, \(u^+= \max\{u,0\}\), \(\Omega\subset \mathbb{R}^N\) is a smooth open bounded set and \(a\), \(b\) are constants.
We study the problem, when the nonlinearity \((u+1)^+-1\) is replaced by a more general function \(g\), by using a variational approach. If \(\lambda_1< a^2\) and \(b< \lambda_1\) \((\lambda_1- a^2)\) the existence of two solutions is proved by the classical mountain pass theorem. In two different cases we get the existence of two solutions using a “variation of linking” theorem, by studying the geometry of the functional. The first case is when \(\lambda_{j+1}\leq a^2\) and \(b\) is close to \(\lambda_{j+1}\) \((\lambda_{j+1}- a^2)\) with \(b<\lambda_{j+1}\) \((\lambda_{j+1}- a^2)\) for some \(j\geq 1\). The second case is when \(\lambda_1\leq a^2\leq \lambda_i\) and \(b\) is close to \(\lambda_i\) \((\lambda_i- a^2)\) with \(b> \lambda_i\) \((\lambda_i- a^2)\) for some \(i\geq 2\). Finally, we give some uniqueness results.


35J65 Nonlinear boundary value problems for linear elliptic equations
35A15 Variational methods applied to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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[1] A.C. Lazer, P.J. McKenna, Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Review 32 (1990) 537-578. · Zbl 0725.73057
[2] G. Tarantello, A note on a semilinear elliptic problem, Diff. Integ. Equat. 5(3) (1992) 561-565. · Zbl 0786.35060
[3] A.C. Lazer, P.J. McKenna, Global bifurcation and a theorem of Tarantello, J. Math. Anal. Appl. 181 (1994) 648-655. · Zbl 0797.34021
[4] Micheletti, A.M.; Pistoia, A., Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear analysis, 31, 895-903, (1998) · Zbl 0898.35032
[5] Ekeland, J.; Temam, R., Convex analysis and variational problems, (1976), North-Holland Amsterdam
[6] A. Marino A.M. Micheletti, A. Pistoia, Some variational results on semilinear problems with asympotically nonsimmetric behaviour. Nonlinear Analysis “A tribute in honour of G. Prodi”, S.N.S. Pisa, 1991, pp. 243-256. · Zbl 0849.35035
[7] Marino, A.; Micheletti, A.M.; Pistoia, A., A non symmetric asymptotically linear elliptic problem, Topol. meth. nonlin. anal., 4, 289-339, (1994) · Zbl 0844.35035
[8] P. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, C.B.M.S. Reg. Conf. Ser. in Math. 65. Amer. Math. Soc. Providence, RI, 1986. · Zbl 0609.58002
[9] M. Schechter, K. Tintarev, Pairs of critical points produced by linking subsets with application to semilinear elliptic problems, Bull. Soc. Math. Belg. 44 (3) ser B, (1992) 249-261. · Zbl 0785.58017
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