## 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 $$\sigma\to 0$$.

### MSC:

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

### Citations:

Zbl 0699.35103; Zbl 0705.35043
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### References:

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