Existence of High Energy-Positive Solutions for a Class of Elliptic Equations in the Hyperbolic Space

We study the existence of positive solutions for following class scalar field problem on hyperbolic space: $$\begin{aligned} -\Delta _{{\mathbb {B}}^N} u - \lambda = a(x) |u|^{p-1} \, u\,\,\text {in}\,{\mathbb {B}}^{N}, \quad \in H^{1}{({\mathbb {B}}^{N})}, \end{aligned}$$ where $${\mathbb {B}}^N$$ denotes space, $$1<p<2^*-1:=\frac{N+2}{N-2}$$ , if $$N \geqslant 3; 1<p<+\infty $$ 2,\;\lambda < \frac{(N-1)^2}{4}$$ and $$0< a\in L^\infty ({\mathbb {B}}^N).$$ prove a solution by introducing min–max procedure in spirit Bahri–Li space using series new estimates involving interacting bubbles.