Positivity for a noncooperative system of elliptic equations in \({\mathbb{R}}^N\). (English) Zbl 0947.35050

The authors consider the system \[ -\Delta u=f-\mu a v, \qquad -\Delta v=b u \] in \(\mathbb{R}^n\), \(n\geq 3\), \(f\in L^{\frac{2n}{n-2}}\), \(\mu \in \mathbb{R}.\) The coefficients \(a(x), b(x)\) satisfy the decreasing conditions \((1+|x|)^{l_a}a(x), (1+|x|)^{l_b}b(x) \in L^\infty.\) The main result is: \(f\geq 0\) implies \(u\geq 0\) (unique solvability \(u, v \in W^{1,2}_0\) is also established) if \(l_a+ l_b>4\) and \(l_a, l_b\geq 0\) and \(l_a, l_b>4-n\) and \(|\mu|\) sufficiently small. As a corrolary the positivity preserving property of the biharmonic equation \[ \Delta v+\mu a v=f \] is established for \(n\geq 5\), \(l_a>4\) and sufficiently small \(|\mu|\). The proofs rely on optimal estimates for the Newtonian potential with weights and on corresponding 3G-type theorems.


35J45 Systems of elliptic equations, general (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)