A semilinear problem with a gradient term in the nonlinearity (Q2062696)

The author considers the following semilinear problem with a gradient term in the linearity on the unit ball \(B=B\left(0,1\right)\subset\mathbb{R} ^{N},\) \[ -\Delta u=\lambda\frac{\left(1+\left\vert \bigtriangledown u\right\vert ^{q}\right)}{\left(1-u\right)^{p}}\text{ in }B, \] \[ u>0~~\text{ in }B, \] \[ u=0\text{ on }\partial B, \] where \(\lambda>0,~p>0\) and \(q\geq0.\) In general, this problem can have different regimes depending on the parameters \(p,\) \(q\) and \(N.\) Here, in the radial case, the author proves that the problem has infinitely many radial solutions for \(2\leq N<2\frac{6-q+2\sqrt{8-2q}}{\left( 2-q\right) ^{2}} +1~\)and \(\lambda=\frac{N-1}{2},~\)and has a unique radial solution for \(N>2\frac{6-q+2\sqrt{8-2q}}{\left( 2-q\right) ^{2}}+1\) and \(0<\lambda <\frac{N-1}{2}.\)