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

We consider the boundary value problem \begin{equation*} - Δu = λc(x)u+ μ(x) |\nabla u|^2 + h(x), \quad u \in H^1_0(Ω) \cap L^{\infty}(Ω) \eqno{(P_λ)} \end{equation*} where $Ω\subset \R^N, N \geq 3$ is a bounded domain with smooth boundary. It is assumed that $c\gneqq 0$, $c,h$ belong to $L^p(Ω)$ for some $p > N/2$ and that $μ\in L^{\infty}(Ω).$ We explicit a condition which guarantees the existence of a unique solution of $(P_λ)$ when $λ<0$ 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 $λ=0$ if $(P_0)$ has a solution, otherwise if bifurcates from infinity at the left of the axis $λ=0$. Assuming that $(P_0)$ has a solution and strenghtening our assumptions to $μ(x)\geq μ_1>0$ and $h\gneqq 0$, we show that the continuum bifurcates from infinity on the right of the axis $λ=0$ and this implies, in particular, the existence of two solutions for any $λ>0$ sufficiently small.

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