Existence of solution for Kirchhoff model problems with singular nonlinearity

Abstract

We study the fourth order Kirchhoff equation Delta(2)u (a + integral(Omega) vertical bar Delta(u)vertical bar(2))(gamma)Delta u = f (u) in Omega with -Delta u > 0 and u > 0 in Omega, and Delta u = u = 0 on partial derivative Omega, where f (t) = alpha 1/t(theta) + lambda t(q) + mu t + g (t) for t >= 0, g has subcritical growth, alpha > 0, lambda > 0, mu >= 0, 0 < theta < 1, 0 < q < 1, gamma >= 0, a > 0, b >= 0. We use the Galerkin projection method to show the existence of solution under some boundedness restriction on alpha, lambda, mu. In some cases we study the behavior of the norm of the solution u as lambda -> 0 and as lambda -> infinity. Similar issues are addressed for the equation (a + b integral(Omega) vertical bar Delta u vertical bar(2))(gamma) Delta(2)u - rho Delta u = f (u), rho >= 0.