A perturbed nonlinear elliptic PDE with two Hardy-Sobolev critical exponents (Q2809265)

The authors consider the following perturbed PDE involving two Hardy-Sobolev critical exponents: NEWLINE\[NEWLINE \Delta u+ \lambda_1 \frac{u^{2^\ast (s_1)-1}}{|x|^{s_1}}+\lambda_2 \frac{u^{2^\ast (s_2)-1}}{|x|^{s_2}} +\lambda_3\frac{u^p}{|x|^{s_3}}=0 \text{ in } \Omega, NEWLINE\]NEWLINE NEWLINE\[NEWLINE u>0\text{ in } \Omega, \quad u=0 \text{ on } \partial \Omega. NEWLINE\]NEWLINE Here \(\Omega\subset\mathbb{R}^N\) (\(N\geq 3\)) is a bounded domain with \(0\in \partial \Omega\) and \(2^\ast(s_i)=\frac{2(N-s_i)}{N-2}\), \(i\in \{1,2,3\}\). The authors show the existence of ground state solutions under certain assumptions on the exponents \(s_i, p\) and parameters \(\lambda_i\). More precisely, suppose \(\partial\Omega\) is \(C^2\) at \(0\) and the mean curvature of \(\partial\Omega\) at \(0\) is negative, suppose \(0<s_2<s_1<2\), \(0\leq s_3<2\), \(\lambda_i\in \mathbb{R}\), \(\lambda_i\neq 0\), \(\lambda_2>0\) and \(1<p<2^\ast(s_3)-1\), suppose one of the following conditions is satisfiedNEWLINENEWLINE(1) \(\lambda_1>0\), \(\lambda_3>0\);NEWLINENEWLINE(2) \(\lambda_1>0\), \(\lambda_3<0\), \(p\leq 2^\ast(s_1)-1\);NEWLINENEWLINE(3) \(\lambda_1<0\), \(\lambda_3>0\), \(p\geq 2^\ast (s_1)-1\);NEWLINENEWLINE(4) \(\lambda_1<0\), \(\lambda_3<0\), \(p<2^\ast(s_2)-1\),NEWLINENEWLINEand furthermore, if \(\lambda_3<0\) suppose either \(p<\frac{N-2s_3}{N-2}\) or \(p\geq \frac{N-2s_3}{N-2}\) with \(|\lambda_3|\) sufficiently small. Then the equation has a ground state solution, which is a nontrivial critical point of the associated energy functional with minimal energy. The authors also apply a perturbation method to study the existence of positive solutions.NEWLINENEWLINEThe existence of a ground state solution is proved by using the Nehari manifold method [\textit{Z. Nehari}, Trans. Am. Math. Soc. 95, 101--123 (1960; Zbl 0097.29501)]. One of the key steps is to show the compactness of Palais-Smale sequences via the concentration compactness principle.