Positive solutions to the sublinear Lane-Emden equation are isolated

Abstract

We prove that on a smooth bounded set, the positive least energy solution of the Lane-Emden equation with sublinear power is isolated. As a corollary, we obtain that the first eigenvalue of the Dirichlet-Laplacian is not an accumulation point of the spectrum, on a smooth bounded set. Our results extend to a suitable class of Lipschitz domains, as well.