Existence and nonexistence of positive singular solutions for a class of semilinear elliptic systems
The authors study radial non-negative singular solutions of the Dirichlet boundary value problem for semilinear elliptic systems \[ \Delta u=| v|^{p-1}v,\qquad \Delta v=| u| ^{p-1}u,\tag{S} \] on the punctured unit ball in \(\mathbb{R}^N\) for \(p>0\), \(q>0\), and \(N\geq 3\). The authors divide the first quadrant of the \((p,q)\)-plane into different subsets including \[ {\mathcal H}_1^-=\{(p,q)\mid\max\{c_p,c_q\}>N-2\},\qquad {\mathcal H}_2^+=\{(p,q)\mid c_p+c_q < N-2\}, \] where \(c_p=2(p+1)/(pq-1)\), \(c_q=2(q+1)/(pq-1)\). For \((p,q)\in {\mathcal H}_1^-\), it is proved that a positive solution of (S) either is regular or converges to the origin like \(r^{2-N}\), where \(r=| x|\) for \(x\in\mathbb{R}^N\). Considering radial non-negative solutions, problem (S) is reduced to \[ -(r^{N-1}u')'= r^{N-1}v^p, \quad -(r^{N-1}v')'= r^{N-1}u^q, \quad u(1)= 0 = v(1).\tag{S\(_D\)} \] Then among other results, by using Pohozaev-identity type arguments, it is proved that if \((p,q)\) is in the closure of \({\mathcal H}_2^+\), then problem \((\text{S}_D)\) has no positive solution. By using Schauder's fixed point theorem, it is proved that if \((p,q)\) is in \({\mathcal H}_1^-\), then problem \((\text{S}_D)\) has infinitely many non-negative solutions.
scientific article; zbMATH DE number 1136747