Existence and phase separation of entire solutions to a pure critical competitive elliptic system

By M. Clapp, Angela Pistoia. 2017

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We establish the existence of a positive fully nontrivial solution (u, v) to the weakly coupled elliptic system $$\begin{aligned} {\left\{ \begin{array}{ll} -\,\Delta u=\mu _{1}|u|^{{2}^{*}-2}u+\lambda \alpha |u|^{\alpha -2}|v|^{\beta }u,\\ -\,\Delta v=\mu _{2}|v|^{{2}^{*}-2}v+\lambda \beta |u|^{\alpha } |v|^{\beta {-2} }v,\\ u,v\in D^{1,2}({\mathbb {R}}^{N}), \end{array}\right. } \end{aligned}$$-Δu=μ1|u|2∗-2u+λα|u|α-2|v|βu,-Δv=μ2|v|2∗-2v+λβ|u|α|v|β-2v,u,v∈D1,2(RN),where $$N\ge 4,$$N≥4,$$2^{*}:=\frac{2N}{N-2}$$2∗:=2NN-2 is the critical Sobolev exponent, $$\alpha ,\beta \in (1,2],$$α,β∈(1,2],$$\alpha +\beta =2^{*},$$α+β=2∗,$$\mu _{1},\mu _{2}>0,$$μ1,μ2>0, and $$\lambda <0.$$λ<0. We show that these solutions exhibit phase separation as $$\lambda \rightarrow -\,\infty ,$$λ→-∞, and we give a precise description of their limit domains. If $$\mu _{1}=\mu _{2}$$μ1=μ2 and $$\alpha =\beta $$α=β, we prove that the system has infinitely many fully nontrivial solutions, which are not conformally equivalent.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/809ab191dc6a1083296efda2c7190267dc0d45dd