Existence of positive radial solutions for singular elliptic systems (Q5943378)

The authors consider the system of \(m\) semilinear singular elliptic equations \(\Delta u+ p(r) f(u)= 0\), \(0< A< r< B\), with one of the following three sets of bondary conditions (\(u= 0\) on \(r= A\) and \(r= B\)) or (\(u= 0\) on \(r= A\) and \({\partial u\over\partial r}= 0\) on \(r= B\)) or (\({\partial u\over\partial r}= 0\) on \(r= A\) and \(u= 0\) on \(r= B\)), with \(p(r) f(r)= (p_1(r) f_1(r),\dots, p_m(r) f(r))\), \(m\geq 1\), and \(r= \sqrt{x^2_1+\cdots+ x^2_n}\in (A,B)\), \(n\geq 1\). They assume the existence of \(\omega> 1\) such that \(p(r)\) satisfies either \[ 0< \int^B_A \Biggl(\int^r_A {ds\over s^{n-1}} \int^B_r {ds\over s^{n-1}}\Biggr) p^\omega(r) dr< +\infty\qquad\text{or} \] \[ 0< \int^B_A \Biggl(\int^r_A {ds\over s^{n-1}}\Biggr) p^\omega(r) dr< +\infty\qquad\text{or} \] \[ 0< \int^B_A \Biggl(\int^B_r {ds\over s^{n-1}}\Biggr) p^\omega(r) dr< +\infty \] and \(\lim_{|u|\to 0^+} f(u)= \infty\), \(\lim_{|u|\to+\infty} {f(u)\over u}= 0\). They prove the existence of at least one positive radial solution on the annulus \(\{x\in\mathbb{R}^n: A< r< B\}\). Actually, they use a change of variables to transform the original problem into an equivalent one-dimensional one and next, they use perturbation techniques, Schauder's fixed-point theorem and a dominated convergence theorem. The main result is to be compared with those of the second author and \textit{H. Liu} [Ann. Pol. Math. 71, No. 1, 19-29 (1999; Zbl 0928.34021)] and \textit{X. Liu} [Nonlinear Anal., Theory Methods Appl. 27, No. 10, 1147-1164 (1996; Zbl 0860.34010)]. Note, that their conditions allow a singularity to appear at \(r= A\) or \(r=B\) and \(u= 0\). This is an important improvement in the literature as far as singular boundary value problems for elliptic systems are considered.