A shooting approach for some semilinear scalar field equation with a Dirac-like potential in one-dimension

We study the following semilinear scalar field equation in one-dimension $-u^{\prime \prime }+({\lambda }^2+b\left(x\right))u=f\left(u\right)\text{in}R,u\left(x\right)\to 0\text{as}\left|x\right|\to \infty .$Here, $\lambda >0$, $b\left(x\right)$ satisfies $m_1\mu e^{-\mu \left|x\right|}\le \left|b\left(x\right)\right|\le m_2\mu e^{-\mu \left|x\right|}$, and $f$ is a locally Lipschitz function with $f\left(0\right)=0$ that is supposed as general condition as possible. Then there exists $\gamma (\ge \lambda )$ that is explicitly determined from $f$, and we prove the following. If $m_1>\gamma$, then there exist no non-trivial solutions for large $\mu$. If $m_2<\lambda$, then there exists at least a positive solution for large $\mu$. If $\gamma <m_1<m_2<\lambda$ and $b(-x)=b\left(x\right)$, then there exist at least two positive solutions for large $\mu$. In the proofs, we use a shooting method from $\pm \infty$.