Asymptotic behavior of positive solutions of inhomogeneous semilinear elliptic equations
Let \(n\geq 3\) and \(\Delta=\sum^n_{i=1}\partial^2/\partial x^2_i\) be the Laplace operator. Let \(p>1\) and \(f\) be a continuous function in \(\mathbb{R}^n \setminus\{0\}\). The author studies the asymptotic behavior of positive solutions of the inhomogeneous elliptic equation \[ \Delta u+u^p+f(x)=0 \tag{P} \] and also establishes an existence result demonstrating the obtained asymptotic behavior. As the first step to understand the effect of \(f\) on the asymptotic behavior of solutions, the author investigates the asymptotic behavior of the positive radial solutions of the special equation \[ \Delta u+u^p+ d|x|^{-l} =0, \] near \(\infty\) for some \(d>0\) and \(l\geq 2p/(p-1)\).
scientific article; zbMATH DE number 1836652