## A symmetry result for cooperative elliptic systems with singularities.(English)Zbl 1447.35148

The authors investigate symmetry results for the solutions of an elliptic system of equations possessing a cooperative structure in a domain that may have “holes” or “cuts” or, more in general, “small vacancies” along which the solutions can become singular. More precisely, they consider a convex open set $$\Omega \subseteq \mathbb{R}^n$$ of class $$C^\infty$$ which is bounded and symmetric with respect to the hyperplane $$\{x_1=0\}$$ and a closed set $$\Gamma \subseteq \Omega \cap \{x_1=0\}$$ which consists of a point if $$n=2$$ or verifying $$\mathrm{Cap}_2(\Gamma)=0$$ if $$n\geq 3$$. Then they analyse a second-order cooperative elliptic system $\left\{ \begin{array}{ll} -\Delta u_i =f_i(u_1,\dots,u_m) & \mathrm{in}\ \Omega \setminus \Gamma\, ,\\ u_i>0 & \mathrm{in}\ \Omega \setminus \Gamma\, ,\\ u_i \equiv 0 & \mathrm{on}\ \partial \Omega\, , \end{array} \right.$ where $$f_1,\dots, f_m \in \mathrm{Lip}(\mathbb{R}^m)$$. Under suitable assumptions, they show that the solutions $$u_1, \dots u_m$$ are symmetric with respect to the hyperplane $$\{x_1=0\}$$ and increasing in the $$x_1$$-direction in $$\Omega \cap \{x_1<0\}$$ and that for every $$i \in {1,\dots,m}$$ one has $\frac{\partial u_i}{\partial x_1}(x)$ for every $$x \in \Omega \cap \{x_1<0\}$$.

### MSC:

 35J57 Boundary value problems for second-order elliptic systems 35B06 Symmetries, invariants, etc. in context of PDEs 31B30 Biharmonic and polyharmonic equations and functions in higher dimensions
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