The aim of this note is to present the first qualitative global bifurcation diagram of the equation $-\Delta u = \mu |x|^{2\alpha} e^{u}$. To this end, we introduce the notion of domains of first/second kind for singular mean field equations and base our approach on a suitable spectral analysis. In particular, we treat also non-radial solutions and non-symmetric domains and show that the shape of the branch of solutions still resembles the well-known one of the model regular radial case on the disk. Some work is devoted also to the asymptotic profile for $\mu \to - \infty.$

We prove the well posedness of a class of non linear and non local mixed hyperbolic--parabolic systems in bounded domains, with Dirichlet boundary conditions. In view of control problems, stability estimates on the dependence of solutions on data and parameters are also provided. These equations appear in models devoted to population dynamics or to epidemiology, for instance.

We investigate localized solutions and their concentration behaviors of singularly perturbed Schrödinger-Poisson system \begin{equation*} \left\{ \begin{aligned} & -\varepsilon^{2}\Delta v+V(x)v+\phi v=P(x)f(v), \ x\in\mathbb{R}^{3}\\ & -\varepsilon^{2}\Delta \phi=v^{2}, \ x\in\mathbb{R}^{3} . \end{aligned} \right. \end{equation*} Here, $V, P\in C^{1}(\mathbb{R}^{3},\mathbb{R})$. Without the Ambrosetti-Rabinowitz type condition and $f$ only belongs to $C(\mathbb{R},\mathbb{R})$, by using the penalization techniques in [31] and the Nehari manifold method in [40,41], we study multiple localized solutions of high topological type for above system.

We are concerned with the Schrödinger-Poisson system with the nonlinearity involving multiple weighted critical exponents. The weighted critical exponent may be the Hénon-Sobolev, the Sobolev or the Hardy-Sobolev critical exponent. Moreover, the potentials contained in the Schrödinger-Poisson system are weights of power type, which may be singular at origin or unbounded at infinity. The existence of ground state solutions for the Schrödinger-Poisson system is obtained.

We propose a fine analysis of the various possible long time behaviors of the solutions of the replicator-mutator equation with so-called Kingman's house of cards mutations. In particular, we give what is to our knowledge the first concentration result for this model.