The effect of the shape of the domain on the existence of solutions of an equation involving the critical Sobolev exponent (Q1906172)

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, \textit{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.