Principal eigenvalues for some quasilinear elliptic equations on $\mathbb{R}^N$

We improve some previous results for the principal eigenvalue of the $p$-Laplacian defined on $\mathbb{R}^N,$ study regularity of the corresponding eigenfunctions and give an existence result of the type below the first eigenvalue (or between the first eigenvalues) for a certain perturbed problem. Based on our approach for the equation we deduce existence, uniqueness and simplicity of positive principal eigenvalues for the $p$-Laplacian system $$ \begin{align} &{-\Delta}_{p} u = \lambda a(x) |u|^{p-2}u + \lambda b(x) |u|^{\alpha - 1} u |v|^{\beta +1}, \quad x \in \mathbb{R}^N, \\ &{-\Delta}_{q} v = \lambda b(x) |u|^{\alpha + 1} |v|^{\beta -1}v + \lambda d(x) |v|^{q - 2}v, \quad x \in \mathbb{R}^N, \\ & 0 <u, 0<v, \text{ in } \quad \mathbb{R}^N, \lim_{|x| \rightarrow +\infty} u(x) = \lim_{|x| \rightarrow +\infty} v(x) = 0. \end{align} $$ We also establish the regularity of the corresponding eigenfunctions.