Radial Positive Solutions for Semilinear Elliptic Problems with Linear Gradient Term in $$\mathbb {R}^N$$

We are concerned with the linear problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\frac{\kappa }{|x|^2} x\cdot \nabla u =\lambda K(|x|) u, & x\in \mathbb {R}^N,\\ u(x)>0, & x\in \mathbb {R}^N,\\[2ex] u(x)\rightarrow 0, & |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$

where \(\lambda \) is a positive parameter, \(\kappa \in [0,N-2)\), \(N> 2\) and \(K:\mathbb {R}^N \rightarrow (0,\infty )\) is continuous and satisfies certain decay assumptions. We obtain the existence of the principal eigenvalue \(\lambda _1^{\text {rad}}\) and the corresponding positive eigenfunction \(\varphi _1\) satisfies \(\lim \nolimits _{|x|\rightarrow \infty }\varphi _1(|x|)=\frac{c}{|x|^{N-2-\kappa }}\) for some \(c>0\). As applications, we also study the existence of connected component of positive solutions for nonlinear infinite semipositone elliptic problems by bifurcation techniques.