Existence and shape of solutions for a class of elliptic systems on the critical hyperbola

We study the existence and shape of least energy solutions to the linear perturbation of Lane-Emden system: (Formula presented.) where (Formula presented.) is a smooth bounded domain, p, q lie on the critical hyperbola (Formula presented.) (Formula presented.) Using the dual variational method, we show the existence of a nontrivial least energy solution (Formula presented.) of this system (Theorem 1.1). As the system is critical so the dual energy functional does not satisfy the Palais-Smale condition. Using higher order Cherrier type inequality, we compensate the lack of the Palais-Smale condition. Thereafter, we establish that for (Formula presented.) sufficiently large, the maxima of (Formula presented.) and (Formula presented.) occur at unique points (Formula presented.) and (Formula presented.) respectively, on the boundary of domain Ω (Theorem 1.2), which complements earlier works in the subcritical case.