On a second order boundary value problem with singular nonlinearity

Abstract

In this paper we investigate in a variational setting, the elliptic boundary value problem $-\Delta u={\textt{sign}u}/{|u|^{\alpha+1}}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is an open connected bounded subset of ${\mathbb R}^N$, and $\alpha> 0$. For the positive solution, which is checked as a minimum point of the formally associated functional $$ E(u)=\frac 12\int_\Omega|\nabla u|^2+\frac{1}{\alpha} \int_\Omega \frac1{|u|^\alpha}, $$ we prove dependence on the domain $\Omega$. Moreover, an approximative functional $E_\varepsilon$ is introduced, and an upper bound for the sequence of mountain pass points $u_\varepsilon$ of $E_\varepsilon$, as $\varepsilon\to 0$, is given. For the onedimensional case, all sign-changing solutions of $-u''={\text{sign}u}/{|u|^{\alpha+1}}$ are characterized by their nodal set as the mountain pass point and $n$-saddle points ($n> 1$) of the functional $E$.