Nonexistence of positive solutions of semilinear elliptic systems in \(\mathbb{R}^ N\). (English) Zbl 0848.35034

The author studies nonexistence results for the elliptic system \[ - \Delta u= f(x, u, v),\quad - \Delta v= g(x, u, v)\quad\text{in} \quad \mathbb{R}^N, \] where \(N\geq 3\) and \(f\) and \(g\) are nonnegative continuous functions. The main results are related to right-hand sides of the following type:
(i) \(f(u, v)\geq a_{11} u^k+ a_{12} v^p\), \(g(u, v)\geq a_{21} u^q+ a_{22} v^s\) for \(u, v\geq 0\) with \(f(0, 0)= g(0, 0)= 0\) and constants \(a_{ij}> 0\) and \(k, p, q, s> 1\).
(ii) \(f(u, v)= u^k v^p\), \(g(u, v)= u^q v^s\) with \(0\leq k\), \(s\leq 1\) and \(p, q> 1\).
(iii) \(f(x, v)= a(|x|) v^p\), \(g(x, u)= b(|x|) u^q\), where \(p, q> 1\) and the functions \(a, b\in C^1([0, \infty))\) satisfy \(a(r)\), \(b(r)> 0\), \((a(r) r^\gamma)'\), \((b(r) r^\gamma)'\geq 0\) for \(r> 0\), and \(\lim_{r\to \infty} a(r) r^\gamma= \lim_{r\to \infty} b(r) r^\gamma= \infty\) with \(\gamma= (2(p+ 1)(q+ 1)- N(pq- 1))/(p+ q+ 2)\).
Assume that the constants \(k\), \(p\), \(q\), \(s\) satisfy \[ {N- 2\over 2}\leq \max\{ {1\over k- 1},\;{p+ 1\over pq- 1},\;{1\over s- 1},\;{q+ 1\over pq- 1}\} \] in the case (i) and \[ {N\over 2}< \min \{{(p+ 1) (q+ 1)+ k(q+ 1)- sk\over pq- (1- s) (1- k)}, {(p+ 1) (q+ 1)+ k(q+ 1)+ s(p+ 1)\over pq+ k(q+ 1)+ s(p+ 1)- 1}\} \] in the case (ii). Then it is proved that the elliptic system has no positive classical solutions in the first case and no positive radial solutions of class \(C^2(\mathbb{R}^N)\) in the remaining cases (ii) and (iii).
In the last section, the author gives some nonexistence results for positive solutions of the above system in a smooth bounded domain with boundary data \(u= v= 0\). Here it is not assumed that the system has Lagrangian structure.


35J45 Systems of elliptic equations, general (MSC2000)
35J60 Nonlinear elliptic equations