Elliptic equations with critical Sobolev exponents in dimension 3
The author studies the following problem: \[ \begin{cases} \Delta u+au= u^{2^*-1} \quad &\text{in }\Omega\\ u>0\quad &\text{in }\Omega\\ u=0\quad &\text{on }\partial \Omega,\end{cases} \tag{1} \] where \(a\) is smooth function in \(\Omega\), \(2^*={2n \over n-2}\) is critical for the embeddings of \(H^1_0(\Omega)\) into \(L^q(\Omega)\), and \(\Omega\) is a smooth bounded domain of \(\mathbb{R}^n\), \(n\geq 3\). He is mainly interested in the existence of minimizing solutions for an elliptic equation of the form (1).
scientific article; zbMATH DE number 1785826