A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrodinger equations

The classical critical Trudinger-Moser inequality in R^(2)under the constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for anyτ>0,it holds that sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))and 4πis sharp.However,if we consider the less restrictive constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1,where V(x)is nonnegative and vanishes on an open set in R^(2),it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x)makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))when V is nonnegative and vanishes on an open set in R^(2).As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u)in R^(2),(0.1)where V(x)≥0 and vanishes on an open set of R^(2)and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential)has been the standard assumption when one deals with the existence of solutions to the above Schrodinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.

Interactions of ion acoustic multi-soliton and rogue wave with Bohm quantum potential in degenerate plasma

This work investigates the interactions among solitons and their consequences in the production of rogue waves in an unmagnetized plasmas composing non-relativistic as well as relativistic degenerate electrons and positrons, and inertial non-relativistic helium ions. The extended Poincare′–Lighthill–Kuo（PLK） method is employed to derive the two-sided Korteweg–de Vries（KdV） equations with their corresponding phase shifts. The nonlinear Schrodinger equation（NLSE） is obtained from the modified Kd V（mKdV） equation, which allows one to study the properties of the rogue waves. It is found that the Fermi temperature and quantum mechanical effects become pronounced due to the quantum diffraction of electrons and positrons in the plasmas. The densities and temperatures of the helium ions, degenerate electrons and positrons, and quantum parameters strongly modify the electrostatic ion acoustic resonances and their corresponding phase shifts due to the interactions among solitons and produce rogue waves in the plasma.

The Existence of Ground State Solutions for Schrödinger-Kirchhoff Equations Involving the Potential without a Positive Lower Bound

In this paper, we study the following Schrödinger-Kirchhoff equation where V(x) ≥ 0 and vanishes on an open set of R2 and f has critical exponential growth. By using a version of Trudinger-Moser inequality and variational methods, we obtain the existence of ground state solutions for this problem.

Egg-derived tyrosine phosphatase as a Potential Biomarker for Muscle Ageing and Degeneration in Drosophila melanogaster

Ageing is associated with declined activity of behaviors, physiology and metabolic processes （Arking, 2006）. In- vestigations in model organisms have indicated the exis- tence of ＂functional senescence＂, the progressive decline of biological functions with age and the decline in the activity may vary from tissue to tissue. Consequently, studies per- taining to the key organs/tissues whose functions deterio- rate/fail with age have led to the development of tissue specific ageing biomarkers （Grotewiel et al., 2005; Demontis et al., 2013）.