THE HENON EQUATION WITH A CRITICAL EXPONENT UNDER THE NEUMANN BOUNDARY CONDITION

Abstract

For n >= 3 and p = (n+2)/(n- 2), we consider the Henon equation with the homogeneous Neumann boundary condition -Delta u + u = vertical bar x vertical bar(alpha) u(p), u > 0 in Omega, partial derivative u/partial derivative v = 0 on partial derivative Omega, where Omega subset of B(0,1) subset of R-n, n >= 3, alpha >= 0 and partial derivative*Omega partial derivative Omega boolean AND partial derivative B(0, 1) not equal empty set. It is well known that for alpha = 0, there exists a least energy solution of the problem. We are concerned on the existence of a least energy solution for alpha > 0 and its asymptotic behavior as the parameter a approaches from below to a threshold alpha(0) is an element of (0, infinity] col for existence of a least energy solution.