Existence and global analytic bifurcation for singular biharmonic equation with Navier boundary condition

Abstract:

Let $\Omega$ be a bounded smooth domain in $\mathbb {R}^N$, $N\geq 2,$ and let $\rho$ denote the distance to the boundary function. Let $K$ be a positive function such that $K =O(\rho ^{-\beta })$ near $\partial \Omega$ for some $\beta \geq 0$. We consider the following fourth order singular elliptic problem (for $\alpha >0$): \begin{equation}\tag {$P$} \displaystyle \left \{\begin {array} {ll} & \Delta ^2 u = K(x)u^{-\alpha } \quad \mbox { in } \Omega , \\ &u> 0\quad \mbox { in } \Omega , \;\;u\vert _{\partial \Omega }=0, \Delta u\vert _{\partial \Omega } = 0. \end{array}\right . \end{equation} We show the existence of a unique $C^2(\overline {\Omega })$ solution to the above problem when $0<\alpha + \beta <2$. We also show the nonexistence of such $C^2$ solutions when $K \sim \rho ^{-\beta }$ near $\partial \Omega$ with $\alpha + \beta \geq 2$.

We then consider an associated bifurcation problem involving a nonlinearity of the type $K u^{-\alpha }+\lambda f(u)$ where $f$ is taken to be super-linear at infinity. We show the existence of a global (in $\lambda$) path-connected analytic branch of solutions to this bifurcation problem when again $\alpha +\beta <2$.