Existence and Asymptotic Behavior of Positive Solutions for Half-quasilinear Elliptic Problem

We investigate the existence of positive solutions for the half-quasilinear elliptic problem \[\begin{cases}-\operatorname{div}(\varphi_p(\nabla y))=\mu a(x)g(y)+\alpha(x)\varphi_p(y^+)+\beta(x)\varphi_p(y^-), & x\in B,\\ y=0, & x\in\partial B,\end{cases}\] where \(\mu\ne 0\) is a parameter, \(a\) is a positive continuous function defined on \(\overline{B}\), \(\alpha,\beta\in C(\overline{B},\mathbb{R})\), \(\varphi_p(s)=|s|^{p-2}s\) is the \(p\)-Laplacian operator, \(y^+=\max\{y,0\}\), \(y^-=-\min\{y,0\}\), \(g\in C(\mathbb{R}^+,\mathbb{R}^+)\), \(B\) is a unit open ball of \(\mathbb{R}^N\). Under some suitable assumptions on nonlinear term \(g\), we get the global structure and the existence of positive radial solutions according to different asymptotic behaviors nonlinearity near zero or infinity.