Perturbation of second order nonlinear equation by delta-like potential

We consider boundary value problems for one-dimensional second order quasilinear equation on bounded and unbounded intervals $I$ of the real axis. The equation perturbed by the delta-shaped potential $\varepsilon^{-1}Q\left(\varepsilon^{-1}x\right)$, where $Q(\xi)$ is a compactly supported function, $0<\varepsilon\ll1$. The mean value of $\left<Q\right>$ can be negative, but it is assumed to be bounded from below $\left<Q\right>\ge-m_0$. The number $m_0$ is defined in terms of coefficients of the equation. We study the convergence rate of the solution of the perturbed problem $ u^\varepsilon $ to the solution of the limit problem $ u_0 $ as the parameter $ \varepsilon $ tends to zero. In the case of a bounded interval $I$, the estimate of the form $|u^\varepsilon(x)-u_0(x)|<C\varepsilon$ is established. As the interval $I$ is unbounded, we prove a weaker estimate $|u^\varepsilon(x)-u_0(x) / <C\varepsilon^{1/2}$. The estimates are proved by using original cut-off functions as trial functions. For simplicity, the proof of the existence of solutions to perturbed and limiting problems are made by the method of contracting mappings. The disadvantage of this approach, as it is known, is the smallness of the nonlinearities in the equation. We consider the cases of the Dirichlet, Neumann and Robin condition.