Nonexistence of nontrivial solutions to Kirchhoff-like equations

Abstract

Subject to given boundary data, nonexistence of solution to the one-dimensional Kirchhoff-like equation − M ( ( a ∗ | u | q ) ( 1 ) ) u ( t ) = λ f ( t , u ( t ) ) ,   0 > t > 1 \begin{equation*} -M\Big (\big (a*|u|^q\big )(1)\Big )u(t)=\lambda f\big (t,u(t)\big ),\ 0>t>1 \end{equation*} is considered. In particular, a condition is provided on the parameter λ \lambda such that for each λ > λ 0 \lambda >\lambda _0 , where λ 0 \lambda _0 is defined in terms of initial data, the boundary value problem has no nontrivial positive solution.