Existence and uniqueness of positive solution for \(p\)-Laplacian Kirchhoff-Schrödinger-type equation (Q1629339)

Summary: We study the existence and uniqueness of positive solution for the following \(p\)-Laplacian-Kirchhoff-Schrödinger-type equation: \(\left\{- \left(a + b \int_\Omega | \nabla u|^p\right) \Delta_p u + \lambda v \left(x\right) | u| ^{p - 2} u = h f \left(u\right) - \mu g \left(u\right), \text{ in } \Omega, \quad u > 0, \text{ in } \Omega,\quad u = 0, \text{ on } \partial \Omega\right\}\), where \(\Omega \subset R^N\) \((N \geq 3)\), \(\lambda, \mu \geq 0\) are parameters, \(v(x)\), \(f(u)\), \(g(u)\) and \(h\) are under some suitable assumptions. For the purpose of overcoming the difficulty caused by the appearance of the Schrödinger term and general singularity, we use the variational method and some mathematical skills to obtain the existence and uniqueness of the solution to this problem.