Existence and non-existence of positive solutions of the scalar field system in R
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Authors: | Henghui Zou |
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Institution: | 1. Department of Mathematics, University of Alabama at Birmingham, 35294, Birmingham, AL, USA
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Abstract: | Letn > 3 andΩ be either the entire spaceR n or a Euclidean ball in R n . Consider the following boundary value problem (I) $$\{ _{\Delta v - u + u^q = 0,}^{\Delta u - v + v^p = 0,} u,v > 0, x \in \Omega $$ with homogeneous Dirichlet boundary data (replaced byu, v → 0 as ¦x¦ → ∞ when Ω=R n ), where p > 1 and q > 1. In this paper, we investigate the question of existence and non-existence of solutions of (I) and prove that (I) admits a solution if and only if $$\frac{1}{{p + 1}} + \frac{1}{{q + 1}} > \frac{{n - 2}}{n}$$ . The existence on a ball and onR n are established by a variational approach and an approximation argument respectively. The Pohozaev identity is used to show non-existence onR n . |
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