The effect of the shape of the domain on the existence of solutions of an equation involving the critical Sobolev exponent. (English) Zbl 0838.35042
In a bounded regular domain \(\Omega\) of \(\mathbb{R}^n\) the semilinear elliptic problem
\[
- \Delta u= u^p,\;u> 0\quad\text{in }\Omega,\;u= 0\quad\text{on }\partial\Omega\tag{\(*\)}
\]
is studied, where \(p= (n+ 2)/(n- 2)\) is the critical Sobolev exponent. In a previous paper, W. W. Ding [J. Partial Differ. Equations 2, No. 4, 83-88 (1989; Zbl 0694.35067)] constructed a nontrivial solution of \((*)\) even in a contractible domain. The present paper gives a generalization of this result. The main theorem states that, if \(\Omega\) is a domain such that there exists a ball \(B_R(x)\), \(x\in \Omega\), and a subset \(\Sigma\) of \(\mathbb{R}^n\) with \(\partial\Sigma\) disjoint from \(\partial B_R(x)\), satisfying \(\Omega\cap B_R(x)= B_R(x)\backslash \Sigma\), then there exists a solution of \((*)\), provided \(\text{meas}(\Sigma)\) is small and \(\text{dist}(\partial\Sigma, \partial B_R(x))\) is large enough. This theorem applies for example to domains, where \(\Omega\) is a ball in \(\mathbb{R}^n\) with a small hole. In the last section, also a multiplicity result is proved.
Reviewer: K.Pflüger (Berlin)
MSC:
35J65 | Nonlinear boundary value problems for linear elliptic equations |
35J20 | Variational methods for second-order elliptic equations |