Regularity results for positive solutions of a semilinear elliptic system
The authors study regularity of positive weak solutions \((u,v)\) of the semilinear elliptic system \[ -\Delta u = v^p,\quad -\Delta v = u^q \quad \text{\;in\;} \Omega. \tag{1} \] Here, \(\Omega\subset \mathbb{R}^n\) is an open set, \(n \geq 3\), \(p\geq 1\), \(q\geq 1\), \(pq >1\). In order that \((u,v)\) is a positive weak solution, it is only required that \(v^p, u^q \in L^1 (\Omega)\), that \(u,v\geq 0\) and that (1) is satisfied in the distributional sense. Whether these solutions are in general regular or not depends on whether the exponents \(p,q\) are below or above a certain critical line. The authors prove that any positive weak solution \((u,v)\) to (1) is regular, provided that the exponents satisfy \[ \max \{ (p+1), (q+1) \} > \frac{n-2}{2}(pq-1). \] In the reverse case \(\max \{ (p+1), (q+1) \} \leq \frac{n-2}{2}(pq-1)\), they can still prove that \(u\in W^{1,r}_{\text{loc}} (\Omega)\), \(v\in W^{1,s}_{\text{loc}} (\Omega)\) for \(1<r< 2p(q+1)/(pq+2p+1)\), \(1<s< 2q(p+1)/(pq+2q+1)\). This improves the information, which one directly gets from the fact that \(\Delta u, \Delta v \in L^1 (\Omega)\). Moreover, applications to biharmonic equations are given.
scientific article; zbMATH DE number 1993711