In this paper we will study the existence of one, two and three weak solutions for a class of variable exponent elliptic equations under appropriate growth conditions on the nonlinearity. Our technical approach is based on the existence theorems obtained by G. Bonanno.

In this paper, we study the following three-dimensional Schrödinger equation with combined Hartree-type and power-type nonlinearities

with 1<p<5. Using standard variational arguments, the existence of ground state solutions is obtained. And then we prove that when p≥3, the standing wave solution $e^{i \omega t} u_{\omega}(x)$ is strongly unstable for the frequency ω>0.

This paper studied a chemotaxis fluid coupled system which models the so-called “chemotactic Boycott effect”. Under certain inhomogeneous boundary conditions, we proved the existence of nontrivial steady state which depends only on one variable of such system in [0,1]³. A positive lower bound of the oxygen concentration c was also obtained.

We consider the following Schrödinger-Newton system with negative critical nonlocal term

where a and f satisfy some certain conditions. By using the variational method and analytical techniques, we obtain the existence of positive ground state solutions which improves the recent results in the literature.

In this paper, we study mainly the long-time behavior of the stochastic plate equations of kirchhoff type with nonlinear damping on unbounded domains. Due to the lack of compactness of Sobolev embeddings on unbounded domains, pullback asymptotic compactness of cocycle associated with the system is proved by the tail-estimates method and splitting technique.

In this paper, Lie symmetry analysis method is applied to one type of mathematical physics equations named the (2+1)-dimensional fractional Hirota-Maccari system. All Lie symmetries and the corresponding conserved vectors for the system are obtained. The one-dimensional optimal system is utilized to reduce the aimed equations with Riemann-Liouville fractional derivative to the (1+1)-dimensional fractional partial differential equations with Erdélyi-Kober fractional derivative.

In this paper, we study the nonlinear Choquard equation

on a Cayley graph of a discrete group of polynomial growth with the homogeneous dimension N≥1, where $\alpha \in(0, N)$, $p>\frac{N+\alpha}{N}$, λ is a positive parameter and R_α stands for the Green’s function of the discrete fractional Laplacian, which has no singularity at the origin but has same asymptotics as the Riesz potential at infinity. Under some assumptions on a(x), we establish the existence and asymptotic behavior of ground state solutions for the nonlinear Choquard equation by the method of Nehari manifold.