Behavior of positive unbounded solutions for a class of nonhomogeneous \(p\)-Laplacian like equations (Q929664)

The authors study the behavior at the origin of positive radial solutions of a strongly nonlinear equation of the form \[ -\text{div}(A(| \nabla u| )\nabla u)=K(| x| )f(u) \text{ in } B_R(0)\backslash\{0\}, \tag{1} \] where \(B_R(0)\subset \mathbb R^N, N>1,\) is the ball centered at zero of radius \(R>0\); the function \(A\in C(0,\infty)\) is such that \(s\mapsto sA(s)\) is in \(C[0,\infty)\) and \(\lim_{s\rightarrow0}sA(| s| )=0,f\in C[0,\infty),\) and \(K\in C(0,R).\) Radial solutions to (1) satisfy \[ -(r^{N-1}\phi(u'))'=r^{N-1}K(r)f(u), \;r\in(0,R),\tag{2} \] where \(r=| x|\), \(x\in B_R(0)\backslash\{0\},\) denotes differentiation with respect to \(r\), and \(\phi(s)=sA(s).\) Several properties of the nonnegative solutions to (2) are established. An example is considered to illustrate the main results.