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 ′ ′ + ( λ 2 + b ( x ) ) u = f ( u ) in R , u ( x ) → 0 as | x | → ∞ . Here, λ > 0 , b ( x ) satisfies m 1 μ e − μ | x | ≤ | b ( x ) | ≤ m 2 μ e − μ | x | , and f is a locally Lipschitz function with f ( 0 ) = 0 that is supposed as general condition as possible. Then there exists γ ( ≥ λ ) that is explicitly determined from f , and we prove the following. If m 1 > γ , then there exist no non-trivial solutions for large μ . If m 2 < λ , then there exists at least a positive solution for large μ . If γ < m 1 < m 2 < λ and b ( − x ) = b ( x ) , then there exist at least two positive solutions for large μ . In the proofs, we use a shooting method from ± ∞ .