In this article we continue with the research initiated in our previous work on singular Liouville equations quantized singularity. The main goal of is to prove that as long bubbling solutions violate spherical Harnack inequality near a source, first derivatives coefficient functions must tend zero.

A fluid of N smooth, hard spheres is considered as a model for normal (elastic collision) and granular (inelastic collision) fluids. The potential energy is discontinuous for hard spheres so that the pairwise forces are singular and the usual forms of Newtonian and Hamiltonian mechanics do not apply. Nevertheless, particle trajectories in the N particle phase space are well defined and the gene...

Abstract The Sturm–Liouville differential equation is one of interesting problems which has been studied by researchers during recent decades. We study the existence a solution for partial fractional using α - ψ -contractive mappings. Also, we give an illustrative example. By -multifunctions, prove solutions inclusion version problem. Finally providing another example and some figures, try to i...

p−1 p p−1 . This equation is a special case of a general half-linear second order differential equation (r(t)Φ(x′))′ + c(t)Φ(x) = 0, (2) where Φ(x) := |x| sgn x, p > 1, and r, c are continuous functions, r(t) > 0 (in the studied equation (1) we have r(t) ≡ 1). Let us recall that similarly as in the linear case, which is a special case of (2) for p = 2 and equation (2) then reduces to the linear...

Abstract The fractional wave equation is presented as a generalization of the wave equation when arbitrary fractional order derivatives are involved. We have considered variable dielectric environments for the wave propagation phenomena. The Jumarie’s modified Riemann-Liouville derivative has been introduced and the solutions of the fractional Riccati differential equation have been applied to ...