Existence of large positive solutions of some nonlinear elliptic equations on singularly perturbed domains. (English) Zbl 0883.35040

The following semilinear elliptic problem \[ -\Delta u=u^p\text{ in }\Omega_\sigma,\;\Omega_\sigma\subset\mathbb{R}^m,\quad u>0 \text{ in }\Omega_\sigma,\quad u=0\text{ on }\partial\Omega_\sigma\tag{P} \] is considered. Here \(\Omega_\sigma\) (for \(\sigma>0\)) denotes a bounded Lipschitz domain which is referred to as a singularly perturbed one and \(1<p<(m+2)/(m-2)\) for \(m>2\), and \(1<p<\infty\) for \(m=2\). It is assumed that: (i) the domains \(\Omega_\sigma\) are \(\mathbb{Z}_2\) symmetric, (ii) the singular parts of \(\Omega_\sigma\) are of quite general form, (iii) the measure of \(\Omega_\sigma\) has at most polynomial growth with respect to \(\sigma^{-1}\). The main result says that for \(\sigma\) sufficiently small there exists a large solution of problem (P).
The existence is proved even when the condition of nondegeneracy does not hold. The latter is necessary if a degree argument is used for the proof of existence as in earlier papers of E. N. Dancer [Math. Ann. 285, No. 4, 647-669 (1989; Zbl 0699.35103) and Math. Z. 206, No. 4, 551-562 (1991; Zbl 0705.35043)]. The proof is based on a variational method to find a local minimizer for the energy \(\int_{\Omega_\sigma}|\nabla u|^2 dx\) under constraints \(\int_{\Omega_\sigma} u^{p+1} dx=1\) and \(u(x_1,x_2,\dots,x_m)=u(-x_1,x_2,\dots,x_m)\) whose local minimum value goes to \(\infty\) as \(\sigma\to 0\).


35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
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