Asymptotically Linear Euclidean Bosonic Equations

We investigate the following nonlinear bosonic equation on Euclidean space arising in string theory and cosmology: P $-\mathrm{\Delta }{e}^{-c\mathrm{\Delta }}u+mu=f(x,u),x\in {\mathbb{R}}^n,$ \( -\Delta e^{-c\Delta }u+mu=f(x,u),\quad x\in {\mathbb{R}}^{n}, \) where $n\ge 3$ $n\ge 3$ , $m>0$ $m>0$ , $c>0$ $c>0$ and $\frac{f(x,u)}{u}$ $\frac{f(x,u)}{u}$ tends to a positive function $h(x)$ $h(x)$ independent of $u$ $u$ as $u\to +\mathrm{\infty }$ $u\rightarrow +\infty $ , $e^{-c\mathrm{\Delta }}$ $e^{-c\Delta }$ is given by a power series with $\mathrm{\Delta }$ $\Delta $ is the Euclidean Laplace operator. Here, the nonlinear term $f(x,u)$ $f(x,u)$ does not satisfy the usual condition: AR $0\le F(x,u):={\int }_0^{u}f(x,t)dt\le \frac{1}{2+\theta }f(x,u)u,$ \( 0\le F(x,u):=\int _{0}^{u}f(x,t)\,dt\le \frac{1}{2+\theta }f(x,u)u, \) for $\theta >0$ $\theta >0$ and $|u|$ $|u|$ is large, which is important in using the mountain pass theorem, see Alves et al. (J. Differ. Equ. 323:229-252, 2022) and Corrêa et al. (J. Differ. Equ. 363:491-517, 2023). This paper is devoted to discuss how to use the mountain pass theorem to obtain the existence of nontrivial solution to problem (P) without the (AR) condition.