In this paper, we study the existence of multiple positive solutions for a class of fractional Schrödinger-Poisson systems involving sign-changing potential and critical nonlinearities on an unbounded domain. By means of the Nehari manifold and Ljusternik-Schnirelmann category, we show how the coefficient $g(x)$ of the critical nonlinearity affects the number of positive solutions, and present a novel relationship between the number of positive solutions and the category of the global maximum set of $g(x)$. The new feature which distinguishes this paper from other related works lies in the fact that we focus on the case of $1 < q < 2$, which has not been presented in the literature.

This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\mathbb R^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\mathbb R^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time behavior of globally defined solutions. The zero solution $u\equiv 0$ is shown to be asymptotically stable in $L^{q_c}(\mathbb R^{n})$ -- it is the only self-similar solution which is initially small in $L^{q_c}(\mathbb R^{n})$. Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.

In this paper, we study the existence and multiplicity of weak solutions for the following asymmetric nonlinear Choquard problem on fractional Laplacian: \begin{align*} (-\Delta)^s u & = -\lambda|u|^{q-2}u + au + b \Big ( \int_{\Omega} \frac{(u^{+}(y))^{2^{*}_{\mu ,s}}} {|x-y|^ \mu}\, dy \Big ) (u^{+})^{2^{*}_{\mu ,s}-2}u \quad\text{in} \; \Omega,\\ u & = 0\quad \text{in} \; \mathbb{R}^{N}\backslash\Omega, \end{align*} where $\Omega$ is open bounded domain of $\mathbb{R}^{N}$ with $C^2$ boundary, $N > 2s$ and $s \in (0,1)$. Here, $(-\Delta)^s$ is the fractional Laplace operator, $\lambda > 0$ is a real parameter, $q \in (1, 2)$, $a > 0$ and $b > 0$ are given constants, $0 < \mu < \min\{N,4s\}$ and $2^{*}_{\mu ,s} = \frac{2N-\mu}{N-2s}$ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and the notation $u^{+} = \max \{u, 0\}$. We prove that the above problem has at least three nontrivial solutions using the Mountain pass Lemma and Linking theorem.

In this paper, we are concerned with the following eigenvalue problem \begin{equation*} (-\Delta)^su+\lambda g(x)u=\alpha u,\ \ u\in H^s(\mathbb{R}^N),\ N\geq3, \end{equation*} where $s\in(0,1),\,\alpha,\lambda\in\mathbb{R}$ and \begin{equation*} g(x)\equiv0\ \text{on}\ \bar{\Omega},\ \ g(x)\in(0,1]\ \text{on}\ \mathbb{R}^N\backslash\bar{\Omega} \ \text{and}\ \lim_{|x|\rightarrow\infty}g(x)=1 \end{equation*} for some bounded open set $\Omega\subset\mathbb{R}^N$. We discuss the existence and some properties of the first two eigenvalues for this problem, which extend some classical results for semilinear Schrödinger equations to the nonlocal fractional setting.