Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent

Abstract

In this paper we shall consider the critical elliptic equation $-\triangle u + \lambda a(x) u = u^{(N+2)/(N-2)}, \ \ x\in \Bbb R^N, \\ u > 0, \quad \int_{\Bbb R^N} |\nabla u|^2 \, dx < + \infty, \quad\quad (0.1)$ where $\lambda >0, N > 4$ and a(x) is a real continuous, non negative function, not identically zero. By using a local Pohozaev identity, we show that problem (0.1) does not admit a family of solutions $u_\lambda$ which blows-up and concentrates as $\lambda \to +\infty$ at some zero point x ₀ of a(x) if the order of flatness of the function a(x) at x ₀ is $\beta\in[2,N-4)$ and $N \geq 7$