We prove existence and nonexistence results for positive solutions to the subelliptic Brezis-Nirenberg type problem with Hardy potential $$- \Delta_{\mathbb G} u -\mu \frac{\psi^2}{d^2} u =u^{2^*-1}+\lambda u\,\,\, \mbox{in}\,\,\, \Omega,\quad u=0\,\,\, \mbox{on}\,\, \partial \Omega, $$ extending to the Stratified setting well-known Euclidean results by Jannelli [J. Diff. Equ. 156, 1999]. Here, $\Delta_{\mathbb G}$ is a sub-Laplacian on an arbitrary Carnot group ${\mathbb G}$, $\Omega$ is a bounded domain of ${\mathbb G}$, $0\in \Omega$, $d$ is the $\Delta_{\mathbb G}$-gauge, $\psi:=|\Delta_{\mathbb G} d|$, where $\Delta_{\mathbb G}$ is the horizontal gradient associated to $\Delta_{\mathbb G}$, $0 \leq \mu < \bar \mu$, where $\bar \mu =\left ( \frac{Q-2}{2} \right )^2$ is the best Hardy constant on ${\mathbb G}$ and $\lambda\in {\mathbb R}$. The main difficulty in this abstract framework is the lack of knowledge of the ground state solutions to the limit problem $$- \Delta_{\mathbb G} u -\mu \dfrac{\psi^2}{d^2} u =u^{2^*-1}\,\, \mbox{on}\,\, {\mathbb G}, $$ whose explicit expression is not known, except for the case when $\mu=0$ and ${\mathbb G}$ is a group of Iwasawa-type. So, a preliminary refined analysis of qualitative properties of solutions to the above problem on the whole space is required, which has independent interest. In particular, Lorentz regularity and a priori decay estimates are obtained.

Our aim of this paper is to study qualitative properties of isolated singular solutions to Choquard equation \begin{equation}\label{eq 0.1} -\Delta u+ u =I_\alpha[u^p] u^q+k\delta_0\quad {\rm in}\ \, \mathcal{D}'(\mathbb{R}^N), \tag{0.1} \end{equation} where $p, \, q\ge 1$, $N\ge2$, $\alpha\in(0,N)$, $k > 0$, $\delta_0$ is the Dirac mass concentrated at the origin and $I_\alpha[u^p](x)=\int_{\mathbb{R}^N} \frac{u(y)^p}{|x-y|^{N-\alpha}}\, dy.$ Multiple properties of very weak solutions of (0.1) are considered: (i) to obtain the existence of minimal solutions and extremal solutions for $N=2$, which are derived in [8] when $N\geq3$; (ii) to analyze the stability of minimal solutions and the semi stability of extremal solutions; (iii) to derive a second solution by the Mountain Pass theorem when $q=p-1$ and $N=2,3$; (iv) to obtain the radial symmetry of the positive singular solutions by the method of moving planes.

The objective of this paper is to construct Riemann $k$-wave solutions of the general form of first-order quasilinear hyperbolic systems of partial differential equations geometrically. To this end, we adapt and combine elements of two approaches to the construction of Riemann $k$-waves, namely, the symmetry reduction method and the generalized method of characteristics. We formulate a geometrical setting for the general form of the $k$-wave problem and discuss in detail the conditions for the existence of $k$-wave solutions. An auxiliary result concerning the Frobenius theorem is established. We use it to obtain formulae describing the $k$-wave solutions in closed form. Our theoretical considerations are illustrated by examples of hydrodynamic type systems including the Brownian motion equation.

In this work, we study the following class of systems of coupled nonlinear fractional Schrödinger equations, \begin{equation*} \left \{ \begin{array}{l} (-\Delta)^s u_1 + \lambda_1 u_1 = \mu_1 |u_1|^{2p-2}u_1 + \beta |u_2|^{p} |u_1|^{p-2}u_1 \quad\text{in }\mathbb{R}^N, \\[3pt] (-\Delta)^s u_2 + \lambda_2 u_2 = \mu_2 |u_2|^{2p-2}u_2 + \beta |u_1|^{p}|u_2|^{p-2}u_2 \quad\text{in }\mathbb{R}^N, \end{array} \right. \end{equation*} where $ u_1,\, u_2\in W^{s,2}(\mathbb{R}^N)$, with $ N = 1,\, 2,\, 3$; $\lambda_j,\,\mu_j>0$, $j = 1,2$, $\beta\in \mathbb{R}$, $p\geq 2$ and $ \frac{p-1}{2p}N < s < 1$. Precisely, we prove the existence of positive radial bound and ground state solutions provided the parameters $p, \beta, \lambda_j,\mu_j$, ($j = 1,\, 2$) satisfy appropriate conditions. We also study the previous system with $m$ equations, $$ (-\Delta)^s u_j + \lambda_j u_j = \mu_j |u_j|^{2p-2}u_j + \sum_{\substack{k = 1\\k\neq j}}^m\beta_{jk} |u_k|^p|u_j|^{p-2}u_j,\quad u_j\in W^{s,2}(\mathbb{R}^N) $$ where $ j = 1,\ldots,m\ge 3$, $\lambda_j,\, \mu_j > 0$, the coupling parameters $\beta_{jk} = \beta_{kj}\in \mathbb{R}$ for $j,k = 1,\ldots,m$, $j\neq k$. For this system, we prove similar results as for $m = 2$, depending on the values of the parameters $p, \beta_{jk}, \lambda_j,\mu_j$, (for $j,k = 1,\ldots,m$, $j\neq k$).

We are concerned with differential inequalities of the form $$ -[d_{M_2}(y_0,y)]^{2\rho_2}\Delta_{M_1} u-[d_{M_1}(x_0,x)]^{2\rho_1}\Delta_{M_2}u\geq V |u|^p\mbox { in } M_1\times M_2, $$ where $M_i$ ($i=1,2$) are complete noncompact Riemannian manifolds, $(x_0,y_0)\in M_1\times M_2$ is fixed, $d_{M_1}(x_0,\cdot)$ is the distance function on $M_1$, $d_{M_2}(y_0,\cdot)$ is the distance function on $M_2$, $\Delta_{M_i}$ is the Laplace-Beltrami operator on $M_i$, $V=V(x,y) > 0$ is a measurable function, and $p>1$. Namely, we establish necessary conditions for existence of nontrivial weak solutions to the considered problem. The obtained conditions depend on the parameters of the problem as well as the geometry of the manifolds $M_i$. Next, we discuss some special cases of potential functions $V$. The proof of our main result is based on the nonlinear capacity method and a result due to Bianchi and Setti (2018) about the construction of cut-off functions with controlled gradient and Laplacian, under certain assumptions on the Ricci curvatures of the manifolds.