Non-trivial solutions for nonlocal elliptic problems of Kirchhoff-type (Q313707)

The authors are concerned with the Kirchoff-type equation NEWLINE\[NEWLINE K\left(\int_a^b\mid u'(x)\mid^2dx\right)u''=\lambda f(x,u),\quad x\in(a,b) NEWLINE\]NEWLINE with homogeneous Dirichlet boundary conditions. \(\lambda\) is a positive real parameter, \(K: [0,+\infty)\to\mathbb{R}\) is continuous and lower bounded while the nonlinearity \(f\) is \(L^1\)-Carathéodory. Using variational techniques, the authors prove the existence of a weak solution in the Sobolev space \(W_0^{1,2}\) for some values of the parameter \(\lambda\).