Combined effects of changing-sign potential and critical nonlinearities in Kirchhoff type problems (Q2826966)

The authors study the existence and multiplicity of positive solutions to a class of Kirchhoff type problems involving sign-changing potential and critical growth terms: NEWLINE\[NEWLINE \begin{cases} -\Big(a+ b\int_\Omega|\nabla u|^2dx\Big)\Delta u =|u|^4u+\mu|x|^{\alpha-2}u+\lambda f(x)|u|^{q-2}u & \quad \text{in}\;\Omega, \\ u=0 & \quad\text{on}\;\partial\Omega, \end{cases} NEWLINE\]NEWLINE where \(\Omega\subset \mathbb{R}^3\) is a smooth and bounded domain, \(a, b>0,\) \(0<\alpha<1,\) \(1<q<2,\) \(\lambda>0\) is a positive real number, and \(0<\mu<a\mu_1\) where \(\mu_1\) is the first eigenvalue of \(-\Delta u=\mu|x|^{\alpha-2}u\) under Dirichlet boundary condition. The functions \(f\in C(\overline{\Omega})\) is a sign-changing potential satisfying \(f^+=\max\{f, 0\}\neq0.\)NEWLINENEWLINEExistence and multiplicity of nonzero non-negative solutions are obtained by means of the concentration compactness principle and the Nehari manifolds method.