Symmetry results of large solutions for semilinear cooperative elliptic systems. (English) Zbl 1433.35074
Summary: In this paper, we consider symmetry of large solutions of the semilinear cooperative elliptic system \(\Delta u_i=g_i(u_1,\dots,u_M)\) in \(B_R\), \(u_i=+\infty\), on \(\partial B_R\), where \(u_i\in C^2(B_R)\), \(g_i:\mathbf{R}^M\rightarrow\mathbf{R}\) a \(C\) function, \(B_R\) is the open ball of center 0 and radius \(R>0\) in \(\mathbf{R}^N\), \(N \geq 3\), \(M\in\mathbf{N}\), \(i=1,2,\dots,M\). After introducing the weakly coupled, fully coupled hypotheses, and some assumptions for \(g\), we use the moving plane method to prove any large solution of the semilinear cooperative elliptic system is radially symmetric and radially increasing. In order to use the moving plane method for our problem, a strong maximum principle of the linear cooperative elliptic system is constructed. Furthermore, we also get the symmetry result of large solutions of the semilinear cooperative elliptic system in an annulus.
MSC:
| 35J57 | Boundary value problems for second-order elliptic systems |
| 35J61 | Semilinear elliptic equations |
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