A Counterexample to the Nodal Domain Conjecture and a Related Semilinear Equation

Abstract

ABSTRACT. In this paper we first establish a nonuniqueness result for a semilinear Dirichlet problem of which the nonlinearity is of super-critical growth. We then apply this result to construct a Schrödinger operator on a domain O such that the second eigenfunctions of this operator (with zero Dirichlet boundary data) have their nodal sets completely contained in the interior of the domain f2.

Spectral and related properties about the Emden-Fowler equation ??u=?e u on circular domains

“…of (2.13) and hence i n never changes on the arc TkTk+ l (excluding end points). Further, owing to [6], i = i R holds in this case. Thus we finally reach the result of this section.…”

“…As far as nonpositive e.v.s are concerned, owing to [6] we may only consider the radial version of (2.12):…”

“…Further, since an eigenfunction corresponding to a nonpositive eigenvalue of (1.1) must be radial [6], we have i = i R = k on the arc TkTk+ 1.…”

On the complex structure of positive solutions to Matukuma-type equations

“…first considered by Lin and Ni [14] and further investigated by Bamón, del Pino and Flores [1], Flores [6] and recently Campos [2]. In this problem the behavior of the non-linearity is also super-critical for small r > 0 and sub-critical for large r and the analogous of some of the solutions we obtain for (1.1) has been obtained for (1.8).…”

On S-shaped bifurcation curves for a class of perturbed semilinear equations

“…The next remarkable results regarding (1.1) are due to B. Gidas, W.-M. Ni and L. Nirenberg [10], and C-S. Lin and W-M. Ni [16].…”