Symmetry properties of solutions of semilinear elliptic equations in the plane

Abstract

If g is a nondecreasing nonnegative continuous function we prove that any solution of −Δu+g(u)=0 in a half plane which blows-up locally on the boundary, in a fairly general way, depends only on the normal variable. We extend this result to problems in the complement of a disk. Our main application concerns the exponential nonlinearity g(u)=eau, or power–like growths of g at infinity. Our method is based upon a combination of the Kelvin transform and moving plane method.