Perturbation of second order nonlinear equation by delta-like potential

Abstract

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 $\left$ can be negative, but it is assumed to be bounded from below $\left\ge-m_0$. 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)|