A scalar field equation on hyperbolic space with indefinite sign nonlinearity

Abstract

In this article, we study threshold phenomena for the semilinear double-power elliptic equation $$-\hat{I}{”}_{{\mathbb{B}}^N}u-\hat{I}»u=|u{|}^{p-1}u-|u{|}^{q-1}u,u\in H^1({\mathbb{B}}^N),$$ on the hyperbolic space ${\mathbb{B}}^N$ for $N\ge 3$. For parameters $1<p\le 2^{\ast }-1$ (though we occasionally allow for supercritical exponents) and $q>0$, we seek to identify the optimal spectral regimes for $\hat{I}»\in \mathbb{R}$ that delineate the existence and non-existence of positive-energy solutions. We achieve a complete resolution of these thresholds across all exponent configurations: $p<q$, $0<q<1<p$, and $1<q<p$. Our results demonstrate that the boundary separating these regimes is governed by an explicit critical spectral parameter, which depends on $p$, $q$, and $N$ in the regime where $p<q$, but depends solely on $N$ in the remaining cases.

There are mistakes in the barrier argument. We will rectify the mistakes as soon as possible and reupload