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

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 $Ω$ is a smooth bounded domain, $μ, c\in L^\infty(Ω)$, $h\in L^r(Ω)$ for some $r > n/2$ and $μ,c,h > {\hskip -3.5mm} {\atop \neq} 0$. Our main motivation is to study the "noncoercive" case. Namely, unlike in previous work on the subject, we do not assume $μ$ to be positive everywhere in $Ω$. In space dimensions up to $n=5$, we establish uniform a priori estimates for weak solutions of ($P_λ$) when $λ>0$ is bounded away from $0$. This is proved under the assumption that the supports of $μ$ and $c$ intersect, a condition that we show to be actually necessary, and in some cases we further assume that $μ$ 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 ${\cal C}$ of solutions, such that the projection of ${\cal C}$ onto the $λ$-axis is an interval of the form $[0,a]$ for some $a>0$ and that the continuum ${\cal C}$ bifurcates from infinity to the right of the axis $λ=0$. In particular, for each $λ>0$ small enough, problem $(P_λ)$ has at least two distinct solutions.