Conformally compact and complete smooth solutions to the Strominger system with non vanishing flux, non-trivial instanton and non-constant dilaton using the first Pontrjagin form of the (−)-connection on 6-dimensional non-Kähler nilmanifold are presented. In the conformally compact case the dilaton is determined by the real slices of the elliptic Weierstrass function. The dilaton of non-compact complete solutions is given by the fundamental solution of the Laplacian on R ⁴. All solutions satisfy the heterotic equations of motion up to the first order of α ^′.

Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp regularity for finite energy solutions of p-Laplace equations involving critical exponents and possible singularity on a sub-space of <!-- MATH $\mathbb{R}^n$ --> , which imply asymptotic behavior of the solutions at infinity. In addition, we find the best constant and extremals in the case of the considered Hardy-Sobolev inequality.