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

We are concerned with the linear problem \(\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+\frac{\kappa }{|x|^2} x\cdot \nabla u =\lambda K(|x|) u, & x\in \mathbb {R}^N,\\ u(x)>0, & x\in \mathbb {R}^N,\\[2ex] u(x)\rightarrow 0, & |x|\rightarrow \infty , \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u+\frac{\kappa }{{|x|}^2}x·\mathrm{\nabla }u=\lambda K(|x|)u,\hfill & x\in {\mathbb{R}}^N,\hfill \\ u(x)>0,\hfill & x\in {\mathbb{R}}^N,\hfill \\ [2ex]u(x)\to 0,\hfill & |x|\to \infty ,\hfill \end{array}\right)\end{array}$$ where \(\lambda \) $\lambda$ is a positive parameter, \(\kappa \in [0,N-2)\) $\kappa \in [0,N-2)$ , \(N> 2\) $N>2$ and \(K:\mathbb {R}^N \rightarrow (0,\infty )\) $K:{\mathbb{R}}^N\to (0,\infty )$ is continuous and satisfies certain decay assumptions. We obtain the existence of the principal eigenvalue \(\lambda _1^{\text {rad}}\) ${\lambda }_1^{\text{rad}}$ and the corresponding positive eigenfunction \(\varphi _1\) ${\phi }_1$ satisfies \(\lim \nolimits _{|x|\rightarrow \infty }\varphi _1(|x|)=\frac{c}{|x|^{N-2-\kappa }}\) ${lim}_{|x|\to \infty }{\phi }_1(|x|)=\frac{c}{{|x|}^{N-2-\kappa }}$ for some \(c>0\) $c>0$ . As applications, we also study the existence of connected component of positive solutions for nonlinear infinite semipositone elliptic problems by bifurcation techniques.