Existence of solutions for the critical elliptic system with inverse square potentials

Abstract

Let Ω ∋ 0 be an open bounded domain in R ^N (N ≥ 3) and $2*(s) = \frac{{2(N - s)}} {{N - 2}}$, 0 < s < 2. We consider the following elliptic system of two equations in H ¹₀ (Ω) × H ¹₀ (Ω):

$$- \Delta u - t\frac{u} {{\left| x \right|^2 }} = \frac{{2\alpha }} {{\alpha + \beta }}\frac{{\left| u \right|^{\alpha - 2} u\left| v \right|^\beta }} {{\left| x \right|^s }} + \lambda u, - \Delta v - t\frac{v} {{\left| x \right|^2 }} = \frac{{2\beta }} {{\alpha + \beta }}\frac{{\left| u \right|^\alpha \left| v \right|^{\beta - 2_v } }} {{\left| x \right|^s }} + \mu v,$$

where λ, µ > 0 and α, β > 1 satisfy α + β = 2*(s). Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions.