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Proving the existence of eigenvalues and eigenvectors by Weierstrass's theorem
Jean Van Schaftingen
I propose a proof of the existence of the existence of eigenvectors and eigenvalues in the spirit of Argand's proof of the fundamental theorem of algebra. The proof only relies on…
Stationary solutions of the nonlinear Schrödinger equation with fast-decay potentials concentrating around local maxima
Jonathan Di Cosmo, Jean Van Schaftingen
We study positive bound states for the equation where is a real parameter, a…
Existence, stability and oscillation properties of slow decay positive solutions of supercritical elliptic equations with Hardy potential
Vitaly Moroz, Jean Van Schaftingen
We prove the existence of a family of slow decay positive solutions of a supercritical elliptic equation with Hardy potential in the entire space and study stability and oscillatio…
Finding critical points whose polarization is also a critical point
Marco Squassina, Jean Van Schaftingen
We show that near any given minimizing sequence of paths for the mountain pass lemma, there exists a critical point whose polarization is also a critical point. This is motivated b…
Limiting Sobolev inequalities for vector fields and canceling linear differential operators
Jean Van Schaftingen
The estimate [\lVert D^{k-1}u\rVert_{L^{n/(n-1)}} \le \lVert A(D)u \rVert_{L^1} ] is shown to hold if and only if (A(D)) is elliptic and canceling. Here (A(D)) is a homogeneous lin…