Volume 5, Issue 2

We consider the following third order three-point boundary value problem on a half-line: x '''( t )+ q ( t ) f ( t , x ( t ), x '( t ), x ''( t )) = 0, t ∈ (0,+∞), x '(0) = A , x (η) = B , x ''(+∞) = C , where η ∈ (0,+∞), but fixed, and f : [0,+∞) × ℝ 3 → ℝ satisfies Nagumo's condition. We apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory, to establish existence theory for at least one unbounded solution, and at least three unbounded solutions. To demonstrate the usefulness of our results, we illustrate two examples.

In this paper, we study the problem of removable isolated singularities for elliptic equations with variable exponents. We give a sufficient condition for removability of the isolated singular point for the equations in W1,p(x)(Ω)${W^{1,p(x)}(\Omega )}$.

The aim of this paper is to obtain the existence of solution for the fractional p -Laplacian Dirichlet problem with mixed derivatives t D T α (| 0 D t α u ( t )| p -2 0 D t α u ( t )) = f ( t , u ( t )), t ∈ [0, T ], u (0) = u ( T ) = 0, where 1/ p < α < 1, 1 < p < ∞ and f : [0, T ] × ℝ → ℝ is a Carathéodory function which satisfies some growth conditions. We obtain the existence of nontrivial solutions by using the direct method in variational methods and mountain pass theorem.

We present some solvable subclasses of the class of nonlinear second-order difference equations of the form A x n +1 + B x n +1 x n + C x n x n -1 + G x n +1 x n -1 + D x n + E x n -1 + F = 0, n ∈ ℕ 0 , where the parameters A , B , C , D , E , F , G and the initial values x -1 , x 0 are real numbers. This difference equation is a natural extension of the nonhomogeneous linear second-order difference equation with constant coefficients as well as of the bilinear difference equation with constant coefficients.

In this paper, we are interested in a general equation that has finite speed of propagation compatible with Einstein's theory of special relativity. This equation without external force fields has been derived recently by means of optimal transportation theory. We first provide an argument to incorporate the external force fields. Then, we are concerned with comparison and maximum principles for this equation. We consider both stationary and evolutionary problems. We show that the former satisfies a comparison principle and a strong maximum principle while the latter fulfils weaker ones. The key technique is a transformation that matches well with the gradient flow structure of the equation.

We consider a quasilinear elliptic equation with right-hand side measure, which does not satisfy the usual coercivity assumption. We prove an existence result in the line of the Fredholm alternative. For this purpose we develop a variant of degree theory suited to this setting.