Normalized Solutions for Kirchhoff Equations with Exponential Nonlinearity and Singular Weights

This paper concerns the existence of normalized solutions for the following N-Kirchhoff equation {-(a epsilon(N) + b epsilon(2N) integral(N)(R)|del u|(N) dx)div (|del u|(N-2)del u) = lambda|u|(N-2)u +epsilon(-(N-1))f(epsilon u)/|x|(beta), x is an element of R-N, u is an element of W-1,W-N(R-N), integral(N)(R)|u|(N) dx=rho, where N >= 2, a, b > 0, 0 < beta < N, rho > 0, lambda is an element of R is a Lagrange multiplier, epsilon > 0 and f is an element of C(R) behaves like exp(alpha t(N/N-1)) with some alpha > 0 as t ->infinity. Under some suitable assumptions, the existence of normalized ground state solutions is studied by restricting the discussion on Nehari-Pohozaev manifold and using the singular Trudinger-Moser inequality. As by-products, the regularity and exponential decay estimates are obtained. Moreover, another solution is discussed by the mountain pass theorem. In addition, the asymptotic behavior of solutions is also investigated as epsilon -> 0. The main novelty of this paper is that we consider the Kirchhoff problem in the borderline case and the nonlinearity is singular.