Symmetry results for cooperative elliptic systems via linearization (Q2843442)

The authors consider the following semilinear system NEWLINE\[NEWLINE \begin{cases} \Delta U=F(|x|, U) &\text{ in }\Omega\\ U=0 &\text{ on } \partial\Omega,\end{cases} \tag{P} NEWLINE\]NEWLINE where \(\Omega\subset\mathbb{R}^N\) is a bounded domain and \(F=(f_1, f_2,\dots, f_m)\) with \(m, N\geq 2.\) They prove that if the nonlinearity is convex and satisfies a full coupling condition along the solution and the domain \(\Omega\) is a ball or an annulus then the solutions with Morse index \(j\leq N\) are foliated Schwarz symmetric.