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

TL;DR

This paper investigates solutions to a noncoercive elliptic boundary value problem with critical gradient growth, establishing uniform a priori estimates and demonstrating bifurcation phenomena, including multiple solutions for small positive parameters.

Contribution

It provides the first uniform a priori estimates for noncoercive elliptic equations with critical gradient growth in dimensions up to five, under minimal assumptions on the coefficients.

Findings

Established uniform a priori bounds for solutions when λ>0

Proved existence of a solution continuum bifurcating from infinity

Showed multiple solutions exist for small positive λ

Abstract

We study nonnegative solutions of the boundary value problem $$-\Delta u = \lambda c(x)u + \mu(x)|\nabla u|^2 + h(x),\quad u\in H^1_0(\Omega)\cap L^\infty(\Omega), \leqno(P_\lambda)$$ where $\Omega$ is a smooth bounded domain, $\mu, c\in L^\infty(\Omega)$, $h\in L^r(\Omega)$ for some $r > n/2$ and $\mu,c,h > {\hskip -3.5mm} {\atop \neq} 0$. Our main motivation is to study the "noncoercive" case. Namely, unlike in previous work on the subject, we do not assume $\mu$ 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 ${\cal C}$ of solutions, such that the projection of ${\cal C}$ onto the $\lambda$-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 $\lambda=0$. In particular, for each $\lambda>0$ small enough, problem $(P_\lambda)$ has at least two distinct solutions.