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Multiplicity of solutions of a two point boundary value problem for a fourth-order equation. (English) Zbl 1294.34016
The authors study the existence of multiple solutions for the semi-linear fourth order differential equation \[ u^{(4)}(x)+\lambda f(x,u(x))=0, 0<x<1, u(0)=u'(0)=u''(1)=0, u'''(1)=\lambda g(u(1)), \] describing elastic deflections, where \(\lambda\geq 0\) is a real parameter. Under some conditions on \(f(x,t)\) and \(g(t)\), they prove that there is a set \(A\subset (0,+\infty)\) such that for each \(\lambda\in A\), the above problem has at least two nontrivial solutions by a three critical point theorem. They also point out that the variational argument can also be applied to a corresponding fourth order \(p\)-Laplacian problem or a more general semi-linear fourth order problem.

34B08 Parameter dependent boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
58E50 Applications of variational problems in infinite-dimensional spaces to the sciences
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