Sign-definiteness of q-eigenfunctions for a super-linear p-Laplacian eigenvalue problem

Abstract

We consider the following q-eigenvalue problem for the p-Laplacian

$$\left\{\begin{array}{ll}-{\rm div}\big( |\nabla u|^{p-2}\nabla u\big) = \lambda \|u\|_{L^{q}(\Omega)}^{p-q}|u|^{q-2}u \quad \quad\, {\rm in} \,\,\,\, \Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,\,{\rm on } \,\,\,\, \partial\Omega,\end{array}\right.$$

where ${\lambda\in\mathbb{R},}$ p > 1, Ω is a bounded and smooth domain of ${\mathbb{R}^{N},}$ N > 1, ${1\leq q < p^{\star}}$, ${p^{\star}=\frac{Np}{N-p}}$ if p < N and ${p^{\star}=\infty}$ if ${p\geq N.}$ Let λ _q denote the first q-eigenvalue. We prove that in the super-linear case, ${p < q < p^{\star},}$ there exists ${\epsilon_{q}>0}$ such that if ${\lambda\in(\lambda_{q},\lambda _{q}+\epsilon_{q})}$ is a q-eigenvalue, then any corresponding q-eigenfunction does not change sign in Ω. As a consequence of this result we obtain, in the super-linear case, the isolatedness of λ _q for those Ω such that the Lane–Emden problem

$$\left\{\begin{array}{ll}-{\rm div}\big(|\nabla u|^{p-2}\nabla u\big) = |u|^{q-2}u \qquad\quad\quad\quad \,\,{\rm in}\,\,\,\Omega\\ \quad\quad\quad \quad \quad \quad u = 0 \quad\qquad\qquad \quad\quad \,{\rm on } \,\,\, \partial\Omega,\end{array}\right.$$

has exactly one positive solution.