This paper is motivated by a gauged Schrödinger equation in dimension 2 including the so-called Chern-Simons term. The study of radial stationary states leads to the nonlocal problem: $-\Delta u\left(x\right)+\left(\omega +\frac{h^2\left(\right|x\left|\right)}{{\left|x\right|}^2}+{\int }_{\left|x\right|}^{+\infty }\frac{h\left(s\right)}{s}{u}^2\left(s\right)ds\right)u\left(x\right)={\left|u\left(x\right)\right|}^{p-1}u\left(x\right)$, where $h\left(r\right)=\frac{1}{2}{\int }_0^{r}s{u}^2\left(s\right)ds$. This problem is the Euler-Lagrange equation of a certain energy functional. In this paper the study of the global behavior of such functional is completed. We show that for $p\in (1,3)$, the functional may be bounded from below or not, depending on $\omega$. Quite surprisingly, the threshold value for $\omega$ is explicit. From...

The existence of at least one non-decreasing sequence of positive eigenvalues for the problem driven by both $p(·)$-Harmonic and $p(·)$-biharmonic operators $$\begin{array}{c}{\Delta }_{p\left(x\right)}^{2}u-{\Delta }_{p\left(x\right)}u=\lambda w\left(x\right){\left|u\right|}^{q\left(x\right)-2}u\text{in}\Omega ,\\ u\in W^{2,p(·)}\left(\Omega \right)\cap W_0^{1,p(·)}\left(\Omega \right),\end{array}$$ is proved by applying a local minimization and the theory of the generalized Lebesgue-Sobolev spaces $L^{p(·)}\left(\Omega \right)$ and $W^{m,p(·)}\left(\Omega \right)$.

This paper is devoted to a study of harmonic mappings $\phi$ of a harmonic space $\tilde{E}$ on a harmonic space $E$ which are related to a family of harmonic mappings of $\tilde{E}$ into $\tilde{E}$. In this way balayage in $E$ may be reduced to balayage in $E$. In particular, a subset $A$ of $E$ is polar if and only if ${\phi }^{-1}\left(A\right)$ is polar. Similar result for thinness. These considerations are applied to the heat equation and the Laplace equation.

This article studies the abelian analytic torsion on a closed, oriented, Sasakian three-manifold and identifies this quantity as a specific multiple of the natural unit symplectic volume form on the moduli space of flat abelian connections. This identification computes the analytic torsion explicitly in terms of Seifert data.