In this paper, we study the following class of fractional Hénon problems involving exponents critical or supercritical\begin{equation*} \begin{cases}\displaystyle (-\Delta)^s u = \lambda |x|^{\mu} u +|x|^{\alpha}|u|^{ (p_{\alpha,s}^*+ \varepsilon) -1} u & \mbox{in } \Omega, \\ u = 0 & \mbox{in } \mathbb{R}^N\setminus\Omega, \end{cases}\end{equation*}where $p_{\alpha,s}^*= \frac{N + 2 \alpha + 2s}{N-2s}$ is the critical exponent for a nonlinearity with Hénon weight in nonlocal context, $\varepsilon \geq 0$, $\Omega$ is either a ball or an annulus in $\mathbb R^N$, $s \in (0,1)$ and $\mu, \alpha > -2s$.We used the Emden--Fowler transformation to make the one-dimensional reduction of problems and under appropriate hypotheses on the constant $\lambda$, we prove the existence of at least one non-trivial radial solution for these problems using the concentration compactness principle or Linking Theorem.

In this paper, we consider entire solutions of combustion reaction-diffusion equations in exterior domains. By constructing appropriate super- and sub-solutions, we show that any given transition front of homogeneous combustion reaction-diffusion equation can emanate an entire solution in exterior domains. In addition, we prove that the entire solution is a complete propagation in the exterior domain with some suitable spatial geometrical conditions.

We look for ground state solutions to the Schrödinger-type system\[\begin{cases} -\Delta u_j+\lambda_ju_j=\partial_jF(u)\\ \displaystyle \int_{\mathbb R^N}u_j^2\,dx=a_j^2\\ (\lambda_j,u_j)\in\mathbb R\times H^1 (\mathbb R^N)\end{cases}j\in\{1,\dots,M\}\]with $N,M \ge 1$, where $a=(a_1,\dots,a_M)\in ( 0,\infty ) ^M$ is prescribed and $(\lambda,u)=(\lambda_1,\dots,\lambda_M,u_1,\dots u_M)$ is the unknown. We provide generic assumptions about the nonlinearity $F$ which correspond to the $L^2$-subcritical and $L^2$-critical cases, i.e., when the energy is bounded from below for all or some values of $a$. Making use of a recent idea, we minimize the energy over the constraint $\{\left|u_j\right|_{L^2} \le a_j \text{ for all } j \}$ and then provide further assumptions that ensure $|u_j|_{L^2}=a_j$.

In this paper, the Cauchy problem for a Friedrichs system on a globally hyperbolic manifold with a timelike boundary is investigated. By imposing admissible boundary conditions, the existence and the uniqueness of strong solutions are shown. Furthermore, if the Friedrichs system is hyperbolic, the Cauchy problem is proved to be well-posed in the sense of Hadamard. Finally, examples of Friedrichs systems with admissible boundary conditions are provided.