Eigenvalue problems for a quasilinear elliptic equation on \(\mathbb R^n\) (Q2368407)

The authors study the following quasilinear elliptic problem: \[ -\Delta_pu(x) =\lambda g(x)|u|^{p-2}\cdot u,\;x\in\mathbb R^N\tag{1} \] \[ \lim_{|x|\to+\infty} u(x)=0\tag{2} \] where \(\lambda\in\mathbb R\). Under suitable assumptions on \(g\) (in particular, being bounded, changing sign and being negative and away from zero at infinity), the authors prove the existence of a simple, isolated, positive principal eigenvalue for (1)--(2) using Lyusternik-Schnirelman theory.