Continuum of solutions for an elliptic problem with critical growth in the gradient

TL;DR

This paper studies the solution structure of a nonlinear elliptic boundary value problem with critical gradient growth, revealing conditions for existence, uniqueness, and bifurcation of solutions depending on parameters and the existence of solutions at zero parameter.

Contribution

It explicitly characterizes when solutions exist and form a continuum, and describes bifurcation phenomena depending on the existence of solutions at zero parameter.

Findings

Solutions form a continuum depending on parameter 

Existence of solutions at  determines continuum crossing or bifurcation

Multiple solutions exist for small positive  under strengthened conditions

Abstract

We consider the boundary value problem \begin{equation*} - \Delta u = \lambda c(x)u+ \mu(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(\Omega) \cap L^{\infty}(\Omega) \eqno{(P_{\lambda})} \end{equation*} where $\Omega \subset \R^N, N \geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\gneqq 0$, $c,h$ belong to $L^p(\Omega)$ for some $p > N/2$ and that $\mu \in L^{\infty}(\Omega).$ We explicit a condition which guarantees the existence of a unique solution of $(P_{\lambda})$ when $\lambda <0$ and we show that these solutions belong to a continuum. The behaviour of the continuum depends in an essential way on the existence of a solution of $(P_0)$. It crosses the axis $\lambda =0$ if $(P_0)$ has a solution, otherwise if bifurcates from infinity at the left of the axis $\lambda =0$. Assuming that $(P_0)$ has a solution and strenghtening our assumptions to $\mu(x)\geq \mu_1>0$ and $h\gneqq 0$, we show that the continuum bifurcates from infinity on the right of the axis $\lambda =0$ and this implies, in particular, the existence of two solutions for any $\lambda >0$ sufficiently small.