We consider the uniform attractors of a 3D non-autonomous Brinkman- Forchheimer equation with a singularly oscillating force

for ρ∈[0,1) and ε>0, and the averaged equation (corresponding to the limiting case $\epsilon =0$)

Given a certain translational compactness assumption for the external forces, we obtain the uniform boundedness of the uniform attractor $\mathcal{A}^{\varepsilon}$ of the first system in $\left(H_{0}^{1}(\Omega)\right)^{3}$, and prove that when ε tends to 0, the uniform attractor of the first system $\mathcal{A}^{\varepsilon}$ converges to the attractor $\mathcal{A}^{0}$ of the second system.

We investigate the Kirchhoff type elliptic problem

where both V and f are periodic in x, 0 belongs to a spectral gap of −Δ+V. Under suitable assumptions on V and f with more general conditions, we prove the existence of ground state solutions and infinitely many geometrically distinct solutions.

Let Ω be a smooth bounded domian in $\mathbb{R}^{2}$, $H_{0}^{1}(\Omega)$ be the standard Sobolev space, and $\lambda_{f}(\Omega)$ be the first weighted eigenvalue of the Laplacian, namely,

where f is a smooth positive function on Ω. In this paper, using blow-up analysis, we prove

for any 0≤α<λ_f(Ω). Furthermore, extremal functions for the above inequality exist when α>0 is chosen sufficiently small.

This paper is devoted to establishing the Adams-Onofri inequality with logarithmic weight for the second order radial Sobolev space defined on the unit ball in $\mathbb{R}^{4}$. By using this inequality we obtain the existence of solutions for mean field biharmonic equation with logarithmic weight.

This artical concerns the $C_{\mathrm{loc}}^{1, \alpha}$-regularity of weak solutions u to the degenerate subelliptic p-Laplacian equation

where $\mathcal{H}$ is the orthogonal complement of a Cartan subalgebra in SU(3) with the orthonormal basis composed of the vector fields X₁,...,X₆. When $1<p<2$, we prove that $\nabla_{\mathcal{H}} u \in C_{\mathrm{loc}}^{\alpha}$.

This work considers the initial boundary value problem for a viscoelastic wave equation with a nonlinear boundary source term. Under suitable assumptions, we prove the existence of global weak solutions using the Galerkin approximation. Then, we give a decay rate estimate of the energy by making use of the perturbed energy method.

In this paper we firstly prove the global well-posedness for the compressible Hall-magnetohydrodynamic system in a bounded domain when the initial data is small. On this basis, we continue to study the convergence of the corresponding equations with the well-prepared initial data as the Mach number tends to zero.