Concentration of solutions for the scalar curvature equation on \(\mathbb{R}^n\)
The paper deals with the scalar curvature equation \[ \begin{cases}-\Delta u=K(y)u^{2^\star-1-j},\qquad & y\in{\mathbb{R}}^N, \\ u>0, & y\in{\mathbb{R}}^N, \\ u\to 0, & \text{as\;}\left|y\right|\to 0,\end{cases} \] with \(2^\star={{2N}\over{N-2}}\) and \(N\geq 3\). The author establishes the existence of infinitely many solutions if there is a sequence \((z_j)\in({\mathbb{R}}^{N})^{\mathbb{N}}\) with \(\left|z_j\right|\to\infty\) such that \(K\) has strict local maxima at those \(z_j\). A perturbation result is also derived, and it is observed that the case \(N=3\) allows slightly sharper versions of the results.
scientific article; zbMATH DE number 1474622
scientific article