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

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.