×

Existence of trivial and nontrivial solutions of a fourth-order ordinary differential equation. (English) Zbl 1058.34016

The article is devoted to the scalar nonlinear boundary value problem \[ u^{iv}+ Au''+ Bu+ f(x,u)= 0,\quad u(0)= u(L)= u''(0)= u''(L)= 0, \] with continuous nonlinearity \(f\) such that \(f(x,0)\equiv 0\). Sufficient conditions are given for the existence of at least two nontrivial solutions. The case of \(f(x,u)\equiv u^3\) is studied separately. The proofs are based on variational methods. The considered problems are connected with parabolic differential equations.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
35K35 Initial-boundary value problems for higher-order parabolic equations
PDFBibTeX XMLCite
Full Text: EuDML EMIS