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In this paper, we prove the existence of a nontrivial solution for the following boundary value problem    −div ω(x)|∇u(x)| N −2 ∇u(x) = f (x, u), in B;

In present paper, we study the limit behavior of normalized ground states for the following mass critical Kirchhoff equation2 and Q is the unique positive radially symmetric solution of equation −2∆uWe consider the existence of constraint minimizers for the associated energy functional involving the parameter a. The minimizer corresponds to the normalized ground state of above problem, and it exists if and only if a > 0. Moreover, when V (x) attains its flattest global minimum at an inner point or only at the boundary of Ω, we analyze the fine limit profiles of the minimizers as a ց 0, including mass concentration at an inner point or near the boundary of Ω. In particular, we further establish the local uniqueness of the minimizer if it is concentrated at a unique inner point.

In this paper, we consider the following nonlinear Choquard equation driven by fractional Laplacianwhere V (x) is a nonnegative continuous potential function, 0 < s < 1, N > 2s, (N − 4s) + < α < N and λ is a positive parameter. By variational methods, we prove the existence of least energy solution which localizes near the bottom of potential well int V −1 (0) as λ large enough.

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