On the existence of positive eigenvalues for semilinear elliptic equation on all of \(\mathbb R^d\) (Q1876613)

The authors consider an equation of the form \(\Delta u(x)+\lambda g(x)f(u(x))= 0\) on \(\mathbb{R}^d\) \((d\geq 3)\), where \(f\) is a bounded positive locally Lipschitz function. Under quite weak conditions on \(g\) it is shown that, for all sufficiently large values of \(\lambda\), this equation has a positive solution \(u\) satisfying \(u(x)\to 0\) as \(\| x\|\to\infty\). This type of equation arises in population genetics.