Ground state sign-changing solutions for Kirchhoff type problems in bounded domains (Q288742)

The authors study the existence of sign-changing solutions for non-local equations NEWLINE\[NEWLINE -\left(a+b\int_\Omega |\nabla u|^2 \;dx\right) \Delta u =f(u)NEWLINE\]NEWLINE with zero boundary conditions in bounded, smooth domains \(\Omega \subset \mathbb{R}^N\), in dimensions \(N=1, 2, 3\). Here \(a\) and \(b\) are positive constants. Compared to the existing literature, the authors are able to relax the conditions on the function \(f(u)\) and also to clarify a question regarding the energy levels of sign changing solutions, as they are obtained as critical points of the energy functional NEWLINE\[NEWLINE\Phi_b(u) = \frac{a}{2}\int_\Omega |\nabla u|^2 \;dx +\frac{b}{4} \left(\int_\Omega |\nabla u|^2 \;dx\right)^2-\int_\Omega F(u) \;dxNEWLINE\]NEWLINE with \(F(u)\) an antiderivative of \(f(u)\).