This paper is concerned with the following Choquard equation with perturbation:-Δu+V(x)u=(1/|x|α∗|u|p)|u|p-2u+g(x),u∈H1(RN), whereN≥3,α∈(0,N), and2-(α/N)<p<(2N-α)/(N-2). This kind of equations is well known as the Choquard or nonlinear Schrödinger-Newton equation. Under some assumptions for the functionsV(x), we prove the existence of multiple positive solutions of the equation. Moreover, we also show that these results still hold for more generalized Choquard equation with perturbation.

In this paper, we concern with the following semilinear Kirchhoff type equation \begin{equation*} \begin{cases} -\left(\varepsilon^{2}a+b\varepsilon\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+V(x)u=K(x)f(u)+Q(x)|u|^{p-2}u, &x\in\mathbb{R}^{3},\\ u\in H^{1}(\mathbb{R}^{3}), u>0 &x\in\mathbb{R}^{3}, \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $a, b$ are positive constants, $V, K$ and $Q$ are positive bounded functions and $p\in(4,6]$, $f$ is a continuous superlinear and subcritical nonlinearity. On the one hand, for subcritical case, i.e., $p\in(4,6)$, we prove that there are three families of semiclassical positive solutions for $\varepsilon>0$ small, one is concentrating on the set of minima of $V$, the rest of two families of solutions are concentrating on the sets of maxima of $K$ and $Q$ respectively. On the other hand, we also prove the multiplicity and concentration of positive solutions for critical case($p=6$). The novelty is that we prove some new concentration phenomena for the positive solutions.

Online reviews are an important part of product information and have important effects on consumers’ purchasing decisions. Some sellers try to manipulate the market by inducing online reviews. In this study, a signal game model based on Bayesian conditional probability is constructed to analyze the preconditions, decision-making process, and effect on market demand and profit of this behavior. The results show that first, when consumer sensitivity to rebates reaches a certain threshold, low-quality sellers will adopt a conditional rebate strategy to induce consumers to give positive reviews. Second, the optimal rebate cost (${\beta }^{*}$) is obtained, where ${\beta }^{*}$ increases with the product price (p), but it is not necessarily monotonic in consumers’ sensitivity to rebates ($\rho$) or the proportion of high-quality products ($\alpha$). Third, the conditional rebate strategy can only work in a market dominated by low-quality goods. Using the conditional rebate strategy in a market dominated by high-quality goods will not bring benefits to low-quality sellers but will harm their profits. This study proposes that some developing online markets have collusive behaviors owing to a lack of regulations and laws, as well as consumers’ concern for small interests. Ensuring the orderly development of online markets will require joint efforts by platform enterprises, government agencies, and consumers.