By means of variational methods we prove the existence of a positive, homoclinic solution to an equation of the kind $u''=au-bu^p$, where $p>1$, and both coefficients $a(x)$, $b(x)$ are positive and asymptotically constant. Our main result requires a control from above on the ratios $M/\alpha$ and $\beta/\nu$, where $M=\sup a$, $\alpha=a(\infty)$, $\nu=\inf b$, $\beta=b(\infty)$.
Andrea Gavioli. Luís Sanchez. "Positive homoclinic solutions to some Schrödinger type equations." Differential Integral Equations 29 (7/8) 665 - 682, July/August 2016. https://doi.org/10.57262/die/1462298680