Article
Ufa Mathematical Journal
Volume 10, Number 2, pp. 31-43
Perturbation of second order nonlinear equations by
delta-like potential
Gadylshin T.R., Mukminov F.Kh.
DOI:10.13108/2018-10-2-31
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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)|