Narrow Results

We study the multiplicity of positive solutions of a Brezis–Nirenberg-type problem on a region of the three-dimensional sphere, which is invariant by the natural torus action. In the paper by Brezis and Peletier, the case in which the region is invariant by the $\mathrm{\text{SO}}(3)$-action is considered, namely, when the region is a spherical cap. We prove that the number of positive solutions increases as the parameter of the equation tends to $-\infty$, giving an answer to a particular case of an open problem proposed in the above referred paper.

We study the following $1$-Yamabe equation on connected finite graphs

This paper deals with the construction of solutions to autonomous semilinear elliptic equations considered in entire space. In the absence of space dependence or explicit geometries of the ambient space, the point is to unveil internal mechanisms of the equation that trigger the presence of families of solutions with interesting concentration patterns. We discuss the connection between minimal surface theory and entire solutions of the Allen-Cahn equation. In particular, for dimensions 9 or higher, we build an example that provides a negative answer to a celebrated question by De Giorgi for this problem. We will also discuss related results for the (actually more delicate) standing wave problem in nonlinear Schrödinger equations and for sign-changing solutions of the Yamabe equation.