Existence of nodal solutions for Lidstone eigenvalue problems. (English) Zbl 1136.34016

The authors study nodal solutions to the Lidstone eigenvalue problem \[ (-1)^m u^{(2m)}(x)= ra(x) f(u(x)),\qquad 0< x< 1, \]
\[ u^{(2i)}(0)= u^{(2i)}(1)= 0,\qquad i= 0,\dots, m-1. \] Sufficient conditions are obtained for the existence of solutions with exactly \(k-1\) simple zeros in \((0,1)\). These conditions use the eigenvalue \(\lambda_k\) of the corresponding linear problem. Illustrative examples are given.


34B15 Nonlinear boundary value problems for ordinary differential equations
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