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This class of differential operators includes the usual Laplace, m-Laplace, and k-Hessian operators in the space of radial functions. The equation has a singular positive solution u * (r) under certain conditions on α, β, γ, and p. A generalized Joseph-Lundgren exponent, which we denote by p * JL , is obtained. We study the intersection numbers between u(r, ρ) and u * (r) and between u(r, ρ0) and u(r, ρ1), and see that p * JL plays an important role. We also determine the bifurcation diagram of the problem ⎧ ⎪ ⎨ ⎪ ⎩ r −(γ−1)

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We investigate radial solutions for the problem \[ { − Δ U = λ + δ | ∇ U | 2 1 − U , U > 0 a m p ; in   B , U = 0 a m p ; on   ∂ B , \begin {cases} \displaystyle -\Delta U=\frac {\lambda +\delta |\nabla U|^2}{1-U},\; U>0 & \text {in}\ B,\\ U=0 & \text {on}\ \partial B, \end {cases} \] where B ⊂ R N B\subset \mathbb {R}^N ( N ≥ 2 ) (N\geq 2) denotes the open unit ball and λ , δ > 0 \lambda , \delta >0 are real numbers. Two classes of solutions are considered in this work: (i) regular solutions, which satisfy 0 > U > 1 0>U>1 in B B , and (ii) rupture solutions, which satisfy U ( 0 ) = 1 U(0)=1 , and thus make the equation singular at the origin. Bifurcation with respect to parameter λ > 0 \lambda >0 is also discussed.