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Elliptic equations with competing rapidly varying nonlinearities and boundary blow-up. (English) Zbl 1162.35036

The author considers boundary blow-up solutions for semilinear elliptic equation of the form \[ \Delta u + a u = b(x) f(u),\qquad x\in\Omega, \eqno{(*)} \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) with \(N\geq 2\), and \(a\in\mathbb{R}\) is a fixed parameter. The distinct feature of the study are two following assumptions:
(i) \(b\) is a smooth function on \(\overline{\Omega}\) which is positive in \(\Omega\) and may vanish on \(\partial\Omega\) (possibly at a very degenerate rate such as \(\exp(-[d(x,\partial\Omega)]^q)\) with \(q<0\));
(ii) \(f\) is locally Lipschitz continuous on \([0,\infty)\) with \(f(u)/u\) increasing for \(u>0\) and \(f(u)\) grows at \(\infty\) faster than any power \(u^p\) (\(p>1\)).
In this case, the author proves existence of a unique positive solution to equation (\(*\)) such that \[ \lim_{d(x,\partial\Omega)\to 0} u(x)=\infty. \] Moreover, she describes the asymptotic behaviour of such solution near the boundary \(\partial\Omega\). The form of obtained asymptotics depends significantly on the interplay between the vanishing rate of \(b\) at \(\partial\Omega\) and the growth of \(f\) at infinity.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J60 Nonlinear elliptic equations
35J67 Boundary values of solutions to elliptic equations and elliptic systems
35B40 Asymptotic behavior of solutions to PDEs
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