Nonexistence of nontrivial solutions to Kirchhoff-like equations (Q6572987)

Denoting by \(*\) the usual convolution on \([0,1]\), the author consider the one-dimensional parameterized Kirchhoff equation \N\[\N-M \left ( (a * |u|^q )(1)\right ) u^{ \prime \prime}(t)= \lambda f(t,u(t)), \quad 1<t<1. \N\]\NHere, the coefficient function \(M\) is continuous, \(a \in L^1(0,1)\) is positive and \(q>0\). Making use of the Green's function \(G:[0,1] \times [0,1] \to [0, \infty)\) associated to the operator \(-u^{\prime \prime}\) subject to given boundary data, the main result of the paper provides sufficient conditions for the existence of some \(\lambda_0>0\) such that the above equation associated with the boundary data implied by \(G\), has no nontrivial positive solution for \(\lambda > \lambda_0\).