The Principal Eigenvalue for a Class of Singular Quasilinear Elliptic Operators and Applications

Abstract

We characterize the principal eigenvalue associated to the singular quasilinear elliptic operator $-\Delta u - \mu (x) \frac{|\nabla u|^q}{u^{q-1}}$ in a bounded smooth domain $\Omega \subset \mathrm{I\!R}^N$ with zero Dirichlet boundary conditions. Here, $1<q\le 2$ and $0\le \mu \in L^\infty (\Omega )$. As applications we derive some existence of solutions results (as well as uniqueness, nonexistence and homogenization results) to a problem whose model is

$$ \left\{ \begin{array}{l} \displaystyle -\Delta u = \lambda u + \mu (x) \frac{|\nabla u|^q}{|u|^{q-1}}+f(x) \quad \text { in }\,\, \Omega , \\ u= 0\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\quad \,\!\!\!\!\!\!\!\!\!\! \text { on} \,\, \partial \Omega , \end{array} \right. $$

where $\lambda \in \mathrm{I\!R}$ and $f\in L^p(\Omega )$ for some $p>\frac{N}{2}$.

Dedicado a Amin Kaidi por su 70^o cumpleaños.

Research supported by PGC2018-096422-B-I00 (MCIU/AEI/FEDER, UE), Junta de Andalucía FQM-194 (first author) and FQM-116, Programa de Contratos Predoctorales del Plan Propio de la Universidad de Granada (second author). First author also thanks the support from CDTIME.