Asymptotically Linear Euclidean Bosonic Equations

Abstract

We investigate the following nonlinear bosonic equation on Euclidean space arising in string theory and cosmology:

$$ -\Delta e^{-c\Delta }u+mu=f(x,u),\quad x\in {\mathbb{R}}^{n}, $$

(P)

where \(n\ge 3\), \(m>0\), \(c>0\) and \(\frac{f(x,u)}{u}\) tends to a positive function \(h(x)\) independent of \(u\) as \(u\rightarrow +\infty \), \(e^{-c\Delta }\) is given by a power series with \(\Delta \) is the Euclidean Laplace operator. Here, the nonlinear term \(f(x,u)\) does not satisfy the usual condition:

$$ 0\le F(x,u):=\int _{0}^{u}f(x,t)\,dt\le \frac{1}{2+\theta }f(x,u)u, $$

(AR)

for \(\theta >0\) and \(|u|\) 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.