Perturbed \(p\)-Laplacian in \(\mathbb{R}^ N\): Bifurcation from the principal eigenvalue (Q1353674)

The authors consider the perturbed bifurcation problem in \(\mathbb{R}^N\), \[ - div (a(x,u){}\nabla u{}^{p-2} \nabla u) = \lambda g (x) {}u{}^{p-2} u + f(\lambda, x, u), \eqno (1) \] and the associated problem with homogeneous principal part, \[ - div (a_{0} (x){}\nabla u{}^{p-2} \nabla u) = \lambda g (x) {}u{}^{p-2} u + f(\lambda, x, u), \eqno (2) \] where \(1 < p < N\) ; \(a\) and \(a_{0}\) are both positive in \(\mathbb{R}^N\) and may be singular or degenerate at infinity, no growth restriction on \(a (x, \cdot)\) is postulated, and both \(f\) and \(g\) may change sign. The authors prove that the principal eigenvalue \(\lambda_{1}\) of the homogeneous eigenvalue problem \(- div (a_{0} (x){}\nabla u{}^{p-2} \nabla u) = \lambda g (x) {}u{}^{p-2} u\) is a bifurcation point for (1) and (2). \noindent This paper extends earlier results from the same authors.