2015/01/08 by R. Dhanya, Abhishek Sarkar, Rajendran, Dhanya +1
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Nonlinear Differential Equations Analysis #Advanced Mathematical Modeling in Engineering
paper · pdf · doi:10.48550/arxiv.1501.01793
We consider the equation Δ2 u=g(x,u) ≥ 0 in the sense of distribution in Ω'=Ω∖ \0\ where u and -Δu≥ 0. Then it is known that u solves Δ2 u=g(x,u)+αδ0-βΔδ0, for some non-negative constants α and β. In this paper we study the existence of singular solutions to Δ2 u= a(x) f(u)+αδ0-βΔδ0 in a domain Ω⊂ ℝ4, a is a non-negative measurable function in some Lebesgue space. If Δ2 u=a(x)f(u) in Ω', then we find the growth of the nonlinearity f that determines α and β to be 0. In case when α=β=0, we will establish regularity results when f(t)≤ C eγt, for some C, γ>0. This paper extends the work of Soranzo (1997) where the author finds the barrier function in higher dimensions (N≥ 5) with a specific weight function a(x)=|x|σ. Later we discuss its analogous generalization for the polyharmonic operator.