We prove the existence of positive solutions to boundary-value problems of the form \begin{align*} \begin{gathered} (\phi(u'))'+f(t,u)=0,\quad t\in(0,1)\\ \theta(u(0))=\beta \theta(u'(0)), \quad \theta(u(1))=-\delta \theta(u'(1)), \quad \beta,\delta\ge 0, \end{gathered} \end{align*} where $\phi$ and $\theta$ are odd increasing homeomorphisms of the real line. We also prove the existence of positive solutions to related systems. Our approach is via a priori estimates and Leray-Schauder degree.

We study a degenerate parabolic equation derived from the kinetic theory using Rényi-Tsallis entropy. If the exponent is critical, we have the threshold mass for the blowup of the solution and also the finiteness of type II blowup points. These results extend some facts on the Smoluchowski-Poisson equation associated with the Boltzmann entropy in two space dimensions and actually, we use mass quantization of the blowup family of stationary solutions for the proof. In this first paper, we show local in time existence of the weak solution.

In this paper, we study an infinite horizon optimal control problem arising in economic dynamics. We establish convergence of approximate solutions to the golden-rule stock (turnpike) and show the existence of overtaking optimal solutions.

In this paper, a generalization of the well-known Pohožaev identity and a non-existence result for some Dirichlet problems with the p-Laplacian are obtained. As a preliminary, we will give some necessary technical results regarding the differential calculus in Sobolev spaces and the differentiability properties of superposition operators between Sobolev spaces (in the Marcus-Mizel direction).

We investigate the existence of solutions to the scalar differential inclusion $$ \qquad (D(x(t))\Phi(x'(t)))' \in G(t,x(t),x'(t)) \ \ \mbox{a.e. } t\in I=[0,T], $$ where $D(x)$ is a positive and continuous function, $G(t,x,x')$ is a Carathéodory multifunction and the increasing homeomorphism $\Phi$ can have a bounded domain of the type $(-a,a)$ or it can be the p-Laplacian operator. Using fixed-point techniques combined, in some cases, with the method of lower and upper solutions, we prove the existence of solutions satisfying various boundary conditions.

We consider the stochastic Navier-Stokes equations forced by a multiplicative white noise on a bounded domain in space dimensions two and three. We establish the local existence and uniqueness of strong or pathwise solutions when the initial data takes values in $H^1$. In the two-dimensional case, we show that these solutions exist for all time. The proof is based on finite-dimensional approximations, decomposition into high and low modes and pairwise comparison techniques.