On zeros of solutions of higher order homogeneous linear differential equations

In this article, we investigate the exponent of convergence of zeros of solutions for some higher-order homogeneous linear differential equation, and prove that if A_k−1 is the dominant coefficient, then every transcendental solution f(z) of equation

f(k)+A_k-1 f(k-1)+?+A₀ f=0$f^{(k)}+A_{k-1}\hspace{0.17em}{f}^{(k-1)}+?+A_0\hspace{0.17em}f=0$

satisfies λ(f) = ∞, where λ(f) denotes the exponent of convergence of zeros of the meromorphic function f(z).     

Morse theory for indefinite nonlinear elliptic problems

Using the heat flow as a deformation, a Morse theory for the solutions of the nonlinear elliptic equation:

−Δu−λu=a₊(x)|u|^q−1u−a₋(x)|u|^p−1u+h(x,u)$-\mathrm{\Delta }u-\lambda u=a_{+}(x){|u|}^{q-1}u-a_{-}(x){|u|}^{p-1}u+h(x,u)$

Global bifurcation for nonlinear equations

Using a bifurcation result on noncompact branches of solutions in an abstract setting, we establish the existence of global bifurcation for the following nonlinear equation

−div(a|∇u|^p−2∇u)−_μ0b|u|^p−2u=q(λ,x,u,∇u)$-\text{div}\left(a{\left|\nabla u\right|}^{p-2}\nabla u\right)-{\mu }_{0}b{\left|u\right|}^{p-2}u=q(\lambda ,x,u,\nabla u)$