We are concerned with the following Lane–Emden system: (0.1) \(\begin{cases}-\Delta u_{1}=|u_{2}|^{p-1} u_{2} & \text{in}\ D,\cr -\Delta u_{2}=|u_{1}|^{q-1} u_{1} & \text{in}\ D,\cr u_{1}=u_{2}=0 & \text{on} \ \partial D,\end{cases}\) where D is a bounded smooth domain in ℝN with N ≥ 4. We focus on the supercritical regime, characterized by the exponent pair (p, q) ∈ (1, ∞) × (1, ∞) satisfying \({1 \over {p + 1}} + {1 \over {q + 1}} < {{N - 2} \over N}\) . Our main objective is to establish solutions with layers concentrating along one or several k-dimensional sub-manifolds of ∂D. This concentration phenomenon arises when \({1 \over {p + 1}} + {1 \over {q + 1}} \) approaches \({n-2 \over {n}}\) , where n:= N − k with 1 ≤ k ≤ N − 3 and \({{n - 2} \over n} < {1 \over {p + 1}} + {1 \over {q + 1}} < {{N - 2} \over N}\) .
Our methodology involves transforming the original problem into a lower-dimensional weighted system, enabling us to carry out the reduction framework and apply the blow-up analysis techniques. Particularly significant is the role played by the exponent pair (p0, q0), which corresponds to the limit of (p, q) and lies on the critical hyperbola \({n \over {{p_0} + 1}} + {n \over {{q_0} + 1}} = n - 2\) . Notably, the choice of the smaller exponent, denoted as p0, has a profound influence on the behavior of the solutions, with \({p_0} = {n \over {n - 2}}\) serving as a critical threshold.
What distinguishes this paper is its treatment of two distinct ranges of p0, each of which is contained in \({p_0} \geq {n \over {n - 2}}\) and \({p_0} < {n \over {n - 2}}\) respectively. These two ranges involve entirely different coupling mechanisms, necessitating diverse treatment approaches. This challenge represents the primary obstacle we address in this study and constitutes a novel element of our research. This is likely to be the inaugural work presenting solutions concentrated on higher dimensional sets for the Lane–Emden systems.
