EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY

doi:10.1016/S0252-9602(16)30070-4

Abstract

Abstract:

This paper is concerned with the harmonic equation (P_?ε):△u=0, u>0 in Bⁿ and (∂u)/(∂ν)+(n-2)/(2)u=(n-2)/(2)Ku^(n)/(n-2)?ε on S^n-1 where Bⁿ is the unit ball in Rⁿ, n≥4 with Euclidean metric g0, ∂Bⁿ=S^n-1 is its boundary, K is a function on S^n-1 and ε is a small positive parameter. We construct solutions of the subcritical equation (P_-ε) which blow up at one critical point of K. We give also a sufficient condition on the function K to ensure the nonexistence of solutions for (P_-ε) which blow up at one point. Finally, we prove a nonexistence result of single peaked solutions for the supercritical equation (P_+ε).

Key words: variational problem, critical points, harmonic equation, mean curvature, critical exponent

CLC Number:

35J20

Kamal OULD BOUH. EXISTENCE AND NONEXISTENCE OF SOLUTIONS FOR A HARMONIC EQUATION WITH CRITICAL NONLINEARITY[J].Acta mathematica scientia,Series B, 2016, 36(5): 1305-1316.

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