Consider the semilinear elliptic system $-\Delta u=f(x,u,v)$, $-\Delta v=g(x,u,v)$, $x\in\Omega$, complemented by the nonlinear boundary conditions $\partial_{\nu}u =\tilde f(y,u,v)$, $\partial_{\nu}v =\tilde g(y,u,v)$, $y\in\partial\Omega$, where $\Omega$ is a smooth bounded domain in $\mathbb R^N$ and $\partial_\nu$ denotes the derivative with respect to the outer unit normal $\nu$. We show that any positive very weak solution of this problem belongs to $L^\infty$ provided the functions $f,g,\tilde f,\tilde g$ satisfy suitable polynomial growth conditions. In addition, all positive solutions are uniformly bounded provided the right-hand sides are bounded in $L^1$. We also prove that our growth conditions are optimal. Finally, we show that our results remain true for problems involving nonlocal nonlinearities and we use our a priori estimates to prove the existence of positive solutions.

In this paper we analyze the existence, uniqueness or multiplicity and stability of positive solutions to some nonlinear heterogeneous problems with nonlocal reaction term and linear diffusion. We employ mainly bifurcation and sub-supersolutions methods.

We prove global existence from $L^2$ initial data for a nonlinear Dirac equation known as the Thirring model [12]. Local existence in $H^s$ for $s>0$, and global existence for $s>\frac{1}{2}$, has recently been proven by Selberg and Tesfahun in [9] where they used $X^{s, b}$ spaces together with a type of null form estimate. In contrast, motivated by the recent work of Machihara, Nakanishi, and Tsugawa, [7] we first prove local existence in $L^2$ by using null coordinates, where the time of existence depends on the profile of the initial data. To extend this to a global existence result we need to rule out concentration of $L^2$ norm, or charge, at a point. This is done by decomposing the solution into an approximately linear component and a component with improved integrability. We then prove global existence for all $s\geqslant 0$.

We study bilinear $L^2$ Fourier restriction estimates which are related to the 2d wave equation in the sense that we restrict to subsets of thickened null cones. In an earlier paper we studied the corresponding 3d problem, obtaining several refinements of the Klainerman-Machedon-type estimates. The latter are bilinear generalizations of the $L^4$ estimate of Strichartz for the 3d wave equation. In 2d there is no $L^4$ estimate for solutions of the wave equation, but, as we show here, one can nevertheless obtain $L^2$ bilinear estimates for thickened null cones, which can be viewed as analogs of the 3d Klainerman-Machedon estimates. We then prove a number of refinements of these estimates. The application we have in mind is the Maxwell-Dirac system.

We consider the Hamilton-Jacobi equation $H(x,Du)=0$ in $\mathbb R^n$, with $H$ not enjoying any convexity properties in the second variable. Our aim is to establish existence and nonexistence theorems for viscosity solutions of associated Dirichlet problems, find representation formulae and prove comparison principles. Our analysis is based on the introduction of a metric intrinsically related to the $0$--sublevels of the Hamiltonian, given by an inf-sup game theoretic formula. We also study the case where the equation is critical; i.e., $H(x,Du)= - \varepsilon$ does not admit any viscosity subsolution, for $\varepsilon >0$.

Based on the construction of Bourgain and Pavlović [1] we show that the solutions to the Cauchy problem for the three-dimensional incompressible magneto-hydrodynamics (MHD) system can develop different types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$. In particular the magnetic field can develop norm inflation in a short time even when the velocity remains small and vice versa. Efforts are made to present a very expository development of the ingenious construction of Bourgain and Pavlović in [1].

Let $D:=\{(x,y);\ x^2+y^2<1\}\subset\mathbb R^2$ and let $f\in C^2$. We study sign-changing solutions to $$ \Delta u+\lambda f(u)=0\ \ \textrm{in}\ \ D,\quad u=0\ \ \textrm{on}\ \ \partial D $$ under the conditions $f(0)=0$ and $f'(0)=1$ from the viewpoint of bifurcation theory. We show that this problem has infinitely many bifurcation points from which unbounded continua of sign-changing solutions emanate, where the eigenfunctions corresponding to each bifurcation point are nonradially symmetric. When $f(u)$ is of Allen-Cahn type (e.g., $f(u)=u-u^3$), then we show that the maximal continuum emanating from the second eigenvalue is homeomorphic to $\mathbb R\times S^1$. It is well-known that each eigenvalue is a bifurcation point. We show that the closure of the maximal continuum emanating from each bifurcation point is locally homeomorphic to a curve or disk.

In this paper we study the following system of nonlinear Schrödinger equations: $$ \begin{cases}-\Delta u +u = f(x,u)+\lambda v, & x\in \mathbb R^N,\\ -\Delta v +v =g(x,v)+\lambda u, & x\in \mathbb R^N.\end{cases} $$ Under some assumptions on $f$ and $g$, we obtain the existence of positive ground and bound states of the coupled system for $\lambda \in (0, 1)$. More importantly, we will give more precise descriptions of the limit behavior and energy estimates of the bound states as $\lambda$ changes.