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 \[{cases}-div(φ_p(∇ y))=μ a(x)g(y)+α(x)φ_p(y^+)+β(x)φ_p(y^-), & x∈ B,\\ y=0, & x∈∂ B,{cases}\] where \(μ≠ 0\) is a parameter, \(a\) is a positive continuous function defined on \(B̄\), \(α,β∈ C(B̄,R)\), \(φ_p(s)=|s|ᵖ⁻²s\) is the \(p\)-Laplacian operator, \(y^+=max\{y,0\}\), \(y^-=-min\{y,0\}\), \(g∈ C(R^+,R^+)\), \(B\) is a unit open ball of \(R^N\). 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.