Using the mountain pass theorem combined with the Ekeland variational principle, we obtain at least two distinct, non-trivial weak solutions for a class of p(x)-Kirchhoff type equations with combined nonlinearities. We also show that the similar results can be obtained in the case when the domain has cylindrical symmetry.

The present work focuses on geometrically exact finite elements for highly slender beams. It aims at the proposal of novel formulations of Kirchhoff-Love type, a detailed review of existing formulations of Kirchhoff-Love and SimoReissner type as well as a careful evaluation and comparison of the proposed and existing formulations. Two different rotation interpolation schemes with strong or weak...

This paper deals with the following degenerate fractional Kirchhoff-type system magnetic fields and critical growth: $$\begin{aligned} \left\{ \begin{array}{lll} -\mathfrak {M}(\Vert u\Vert _{s,A}^2)[(-\Delta )^s_Au+u] = G_u(|x|,|u|^2,|v|^2) \\ \quad +\left( \mathcal {I}_\mu *|u|^{p^*}\right) |u|^{p^*-2}u \ &{}\text{ in }\,\,\mathbb {R}^N,\\ \mathfrak v\Vert _{s,A})[(-\Delta )^s_Av+v] G_v(|x|,|...

We present the analogue, for an arbitrary complex reductive group G, of the elliptic integrable systems of Sklyanin. The Sklyanin integrable systems were originally constructed on symplectic leaves, of a quadratic Poisson structure, on a loop group of type A. The phase space, of our integrable systems, is a group-like analogue of the Hitchin system over an elliptic curve Σ. The phase space is t...

1 I n t r o d u c t i o n Elliptic curves can be applied to public-key cryptosystems, and as such several schemes have been proposed [3, 4, 5, 6, 9, 11]. There are two typical elliptic curve cryptosystems: E1Gamal-type scheme [4, 11] and RSA-type schemes [3, 5, 6]. The security of the EIGamal-type elliptic curve cryptosystem is based on the difficulty of solving a discrete logarithm over ellipt...

wt t (t, x)− K (‖wx (t, ·)‖ β Lr (R))a(wx (t, x))wxx (t, x) = 0, (2) w(0, x) = 8(x), wt (0, x) = 9(x), where K is an arbitrary function, sufficiently smooth and taking only positive values; and a = a(s) behaves like |s|p−2 near s = 0. The detailed assumptions on K , r , β, and a are given in (3), (4) and Condition 1 below. For K = K (s) = c1 + c2s (c1, c2 > 0) and p = r = β = 2, we get the famo...