In this paper, we consider the semilinear elliptic problem $$ -\triangle u+ u=|u|^{p-2}u+ \mu f(x), \quad u \in H^1(\Bbb R^N), \quad N>2. \tag"$(*)_\mu$" $$ For $p> 2$, we show that there exists a positive constant $\mu ^*>0$ such that $(*)_\mu$ possesses a minimal positive solution if $\mu \in (0, \mu ^*)$ and no positive solutions if $\mu > \mu^*$. Furthermore, if $p< \frac{2N}{N-2}$, then $(*)_\mu$ possesses at least two positive solutions for $\mu \in (0, \mu^{*})$, a unique positive solution if $\mu =\mu^*$ and there exists a constant $\mu _{*} >0 $ such that when $ \mu\in (0, \mu_{*})$, problem $(*)_\mu$ possesses at least three solutions. We also obtain some bifurcation results of the solutions at $\mu =0$ and $\mu=\mu^*$.

This paper is concerned with the behavior near an isolated singularity of positive radial solutions of the semilinear elliptic system $$ \begin{cases} -\Delta u=\vert v\vert^{p-1}v \\ -\Delta v=\vert u\vert^{q-1}u, \end{cases} \text{in}\ \ \Omega^*\subset\Bbb R^N, $$ where $\Omega^*$ denotes the punctured unit ball in $\Bbb R^N\ (N\ge 3)$ and $(p,q)$ lies in the region between the two critical hyperbolas $\mathcal{H}_1$ and $\mathcal{H}_2$ defined below.

Existence and nonexistence results for a nonlinear system which contains two Hessian operators are proved using the Mountain Pass Theorem and the Pucci-Serrin identity.

We prove a global-in-time existence result of a weak solution for a magnetohydrodynamic (MHD) problem set in a bounded domain of $\mathbb{R}^3$. The fluid is supposed to be incompressible but with an unhomogeneous density, viscosity and electrical conductivity. The displacement currents are neglected in the time-dependent Maxwell equations. The model describes in particular the flow of two immiscible fluids in presence of a magnetic field

This paper is devoted to study the so-called phase-field model when the classical Fourier law is replaced by the Gurtin-Pipkin constitutive assumption. The resulting system of partial differential equations is investigated in a quite general setting. A hyperbolic equation is coupled with a parabolic variational inequality, the state variables being temperature and non-conserved order parameter. By including initial and boundary conditions, the existence and uniqueness of strong solutions is shown along with regularity results ensuring the global boundedness of both the unknowns.

A phase-field model based on the Gurtin-Pipkin heat flux law is considered. This model consists in a Volterra integrodifferential equation of hyperbolic type coupled with a nonlinear parabolic equation. The system is then associated with a set of initial and Neumann boundary conditions. The resulting problem was already studied by the authors who proved existence and uniqueness of a smooth solution. A~careful and detailed investigation on weak solutions is the goal of this paper, going from the aspects of the approximation to the proof of continuous dependence estimates. In addition, a sufficient condition for the boundedness of the phase variable is given.