Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance (Q728092)

The authors study a class of Kirchhoff type problems with resonance \[ \begin{cases} -\left(a+b\int_\Omega |\nabla u|^2\, dx \right)\Delta u=\nu u^3+\lambda |u|^{q-1}u & \text{in } \Omega,\\ u=0 & \text{on } \partial\Omega \end{cases} \] where \(\Omega\subset \mathbb R^3\) is a bounded domain, \(a,b,\nu,\lambda>0\) and \(0<q<1.\) Using the minimizing method, it is obtained existence of positive ground state solutions for all \(0<\nu\leq b \nu_1\) and \(\lambda>0.\) Using the Nehari method, the authors obtain two positive solutions for all \(\nu> b\nu_1\) and \(0<\lambda<\bar\lambda,\) where \(\nu_1\) is the first eigenvalue of the considered problem, \(\bar\lambda\) is a positive constant and one of the two positive solutions is a ground state solution.