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

Abstract

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 $\Omega \subset {\mathbb{R}}^N,N\ge 3$ 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 $\mu (x)\ge {\mu }_1>0$ 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.

This second version include added References

References in corpus (2)

• Existence and nonexistence of solutions for singular quadratic quasilinear equations
• Existence and multiplicity for elliptic problems with quadratic growth in the gradient