We are concerned with the existence and nonexistence of positive weak solutions to $$ -\Delta_m u + \frac{\lambda}{|x|^\gamma}u^{m-1} \geq (K*u^p)u^q \quad\mbox{ in }\mathbb R^N\setminus \overline B_1, $$ where $m > 1$, $N \geq 1$, $\lambda,p > 0$ and $q, \gamma\in\mathbb R$. We assume that $K$ is a positive and continuous function on $\mathbb R^N\setminus \overline B_1$ such that $L(R):=\min_{|x|=R} K(x)$ is decreasing in a neighborhood of infinity. Also, $K\ast u^p$ denotes the standard convolution operation. We obtain conditions on $N,m,\gamma,p$, and $q$ such that the above problem has no solutions. For potentials of slow decay (corresponding to the case $\gamma < m$) and functions $K(x)$ that behave like $|x|^{\alpha-N} (\log |x|)^\beta$ at infinity, these conditions are shown to be essentially optimal.

In this paper, we investigate a class of nonhomogeneous fractional Schrödinger-Poisson systems with subcritical or critical nonlocal term, and a nonlinearity involving critical and subcritical growth. Meanwhile, we assume that the potential function can change sign, which is also a novelty of our results. By using a fixed point theorem, we can find a nontrivial weak solution in a reflexive Banach semilattice. Furthermore, in the last part of the article, we apply the fixed point theorem to handle a class of Schrödinger-Poisson systems with inconsistent fractional exponents and obtain an existence result.

We study Streater's model on Brownian particle gas in statistical mechanics on bounded domains in ${\bf R}^n$. If $n\leq 2$, there is a solution global in time, provided that the entropy production is bounded.

We consider singular problems associated with the weighted anisotropic $p$-Laplace operator $$ H_{p,w}u=\text{div}(w(x)(H(\nabla u))^{p-1}\nabla H(\nabla u)), $$ where $H$ is a Finsler-Minkowski norm and the weight $w$ belongs to a class of $p$-admissible weights, which may vanish or blow up near the origin. We discuss existence and regularity properties of weak solutions for the mixed and exponential singular nonlinearities. In particular, the existence result for the purely singular problem leads us to the validity of a weighted anisotropic Sobolev inequality with an extremal.

A geometric heat flow called the symphonic map flow was introduced in [25]. We present the finite-time blow-up phenomenon of rotationally symmetric solutions to the symphonic map flow.

In the present paper, we prove the existence of solutions $(\lambda, u)\in \mathbb R\times H^1(\mathbb R^N)$ to the following elliptic equations with potential $$\displaystyle -\Delta u+(V(x)+\lambda)u=g(u)\;\hbox{in}\;\mathbb R^N, $$ satisfying the normalization constraint $$ \int_{\mathbb R^N}u^2=a>0, $$ which is deduced by searching for solitary wave solution to the time-dependent nonlinear Schrödinger equations. Besides the importance in the applications, not negligible reasons of our interest for such problems with potential $V(x)$ are their stimulating and challenging mathematical difficulties. We develop an interesting way based on iteration and give a new proof of the so-called ``sub-additive inequality", which can simplify the standard process in the traditional sense. Under some mild assumption on the potential $V(x)$ and some other suitable assumptions on $g$, we can obtain the existence of ground state solution for prescribed $a > 0$.