A priori estimates and bifurcation of solutions for a noncoercive elliptic equation with critical growth in the gradient

Abstract

We study nonnegative solutions of the boundary value problem $$-Δu = λc(x)u + μ(x)|\nabla u|^2 + h(x),\quad u\in H^1_0(Ω)\cap L^\infty(Ω), \leqno(P_λ)$$ where $\Omega$ is a smooth bounded domain, $\mu ,c\in L^{\infty }(\Omega )$, $h\in L^r(\Omega )$ for some $r>n/2$ and $\mu ,c,h>\genfrac{}{}{0ex}{}{}{\ne }0$. Our main motivation is to study the "noncoercive" case. Namely, unlike in previous work on the subject, we do not assume $\mu$ to be positive everywhere in $\Omega$. In space dimensions up to $n=5$, we establish uniform a priori estimates for weak solutions of ($P_{\lambda }$) when $\lambda >0$ is bounded away from $0$. This is proved under the assumption that the supports of $\mu$ and $c$ intersect, a condition that we show to be actually necessary, and in some cases we further assume that $\mu$ is uniformly positive on the support of $c$ and/or some other conditions. As a consequence of our a priori estimates, assuming that ($P_0$) has a solution, we deduce the existence of a continuum $\mathcal{C}$ of solutions, such that the projection of $\mathcal{C}$ onto the $\lambda$-axis is an interval of the form $[0,a]$ for some $a>0$ and that the continuum $\mathcal{C}$ bifurcates from infinity to the right of the axis $\lambda =0$. In particular, for each $\lambda >0$ small enough, problem $(P_{\lambda })$ has at least two distinct solutions.

Cited by in corpus (1)

• Uniform bounds via regularity estimates for elliptic PDE with critical growth in the gradient