Abstract
This paper is devoted to using perturbation and variational techniques to derive some sufficient conditions for the existence of multiple positive solutions in the sense of distributions to a singular second-order dynamic equation with homogeneous Dirichlet boundary conditions, which includes those problems related to the negative exponent Emden-Fowler equation.
Highlights
The Emden-Fowler equation, uΔΔ t q t uα σ t 0, t ∈ 0, 1 T, 1.1 arises in the study of gas dynamics and fluids mechanics, and in the study of relativistic mechanics, nuclear physics, and chemically reacting system see, e.g., 1 and the references therein for the continuous model
We are interested in a broad class of singular problem that includes those related with 1.1 and the more general equation uΔΔ t q t uα σ t g t, u σ t, t ∈ 0, 1 T
The sufficient conditions for the existence of multiple positive solutions obtained in this paper are applied to a great class of bounded time scales such as finite union of disjoint closed intervals, some convergent sequences and their limit points, or Cantor sets among others
Summary
The Emden-Fowler equation, uΔΔ t q t uα σ t 0, t ∈ 0, 1 T, 1.1 arises in the study of gas dynamics and fluids mechanics, and in the study of relativistic mechanics, nuclear physics, and chemically reacting system see, e.g., 1 and the references therein for the continuous model. The sufficient conditions for the existence of multiple positive solutions obtained in this paper are applied to a great class of bounded time scales such as finite union of disjoint closed intervals, some convergent sequences and their limit points, or Cantor sets among others. We will deduce sufficient conditions for the existence of solutions in the sense of distributions to P , where F f g and f, g : D × 0, ∞ → R satisfy C and PM , f satisfies Cc , and g satisfies the following condition. Propositions 2.3 and 2.4 lead to the following sufficient condition for the existence of at least one solution in the sense of distributions to problem P.
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