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

作者：Souplet Philippe^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2015, 121: 412-423.

DOI：10.1016/j.na.2015.02.005

• Elliptic equation
• Critical growth in the gradient
• A priori estimates
• Existence
• Multiplicity
• Bifurcation
• L-infinity(p) spaces
• Weighted Sobolev inequalities
• Hardy inequalities

We study nonnegative solutions of the boundary value problem
-Delta u=lambda c(x)u + mu(x)vertical bar del u vertical bar(2) + h(x), u is an element of H-0(1) (Omega) boolean AND L-infinity(Omega), (P-lambda)
where Omega is a smooth bounded domain of R-n, mu, c is an element of L-infinity(Omega), h is an element of L-r (Omega) for some r > n/2 and mu, c, h not greater than or equal to 0. Our main motivation is to study the "semidefinite'' case. Namely, unlike in previous work on the subject, we do not assume mu to be uniformly positive in Omega, nor even positive everywhere.
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 C of solutions, such that the projection of C onto the lambda-axis is an interval of the form [0, a] for some a > 0 and that the continuum C bifurcates from infinity to the right of the axis. = 0. In particular, for each lambda > 0 small enough, problem (P-lambda) has at least two distinct solutions.

• 出版日期2015-7