Existence of Many Nonequivalent Nonradial Positive Solutions of Semilinear Elliptic Equations on Three-Dimensional Annuli

Abstract

We consider a semilinear elliptic equation,Δu+up=0 onΩR≡{x∈Rn∣R−1<|x|<R+1} with zero Dirichlet boundary condition, where 1<p<∞ forn=2, 1<p<(n+2)/(n−2) forn>2. We prove that, when the space dimension n is three, the number of nonequivalent nonradial positive solutions of the equation goes to ∞ asR→∞. The same result has been known forn=2 andn⩾4; in those cases, the result was obtained by showing that the minimal energy solutions in various symmetry classes have different energy levels. As we will show in this paper, this is not true ifn=3. This makes the casen=3 highly exceptional, and explains why past attempts failed in this case. In this paper we will prove the above result by considering local—rather than global—minimizers in some symmetry classes.