We consider the Neumann problem for the equation $-\Delta u-\lambda u={Q\left(x\right)\left|u\right|}^{2*-2}u$, u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues ${\lambda }_{k-1}$ and ${\lambda }_k$. Applying a min-max principle based on topological linking we prove the existence of a solution.

In this paper we investigate the solvability of some Neumann problems involving the critical Sobolev and Hardy exponents.

We obtain a precise decay estimate of the energy of the solutions to the initial boundary value problem for the wave equation with nonlinear internal and boundary feedbacks. We show that a judicious choice of the feedbacks leads to fast energy decay.

We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When $V\left(x\right)$ has multiple critical points, (1.1) has a wide variety of positive solutions for small $\epsilon$ and the number of positive solutions increases to $\infty$ as $\epsilon \to 0$. We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V\left(x\right)$. Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

We investigate the origin of deterministic chaos in the Belousov–Zhabotinsky (BZ) reaction carried out in closed and unstirred reactors (CURs). In detail, we develop a model on the idea that hydrodynamic instabilities play a driving role in the transition to chaotic dynamics. A set of partial differential equations were derived by coupling the two variable Oregonator–diffusion system to the Navier–Stokes equations. This approach allows us to shed light on the correlation between chemical oscillations...

Oscillation theorems are established for forced second order mixed-nonlinear elliptic differential equations ⎧ ${div\left(A\left(x\right)\right|\left|\nabla y\right||}^{p-1}{\nabla y)+⟨b\left(x\right),|\left|\nabla y\right||}^{p-1}\nabla y⟩+C(x,y)=e\left(x\right)$, ⎨ ⎩ $C(x,y)={c\left(x\right)\left|y\right|}^{p-1}y+{\sum }_{i=1}^m{c}_i\left(x\right){\left|y\right|}^{p_i-1}y$$$under quite general conditions. These results are extensions of the recent results of Sun and Wong, [J. Math. Anal. Appl. 334 (2007)] and Zheng, Wang and Han [Appl. Math. Lett. 22 (2009)] for forced second order ordinary differential equations with...

In this paper, several oscillation criteria are established for some nonlinear impulsive functional parabolic equations with several delays subject to boundary conditions. We shall mainly use the divergence theorem and some corresponding impulsive delayed differential inequalities.

We establish new results on convergence, in strong topologies, of solutions of the parabolic-parabolic Keller-Segel system in the plane to the corresponding solutions of the parabolic-elliptic model, as a physical parameter goes to zero. Our main tools are suitable space-time estimates, implying the global existence of slowly decaying (in general, nonintegrable) solutions for these models, under a natural smallness assumption.

This review is dedicated to recent results on the 2d parabolic-elliptic Patlak-Keller-Segel model, and on its variant in higher dimensions where the diffusion is of critical porous medium type. Both of these models have a critical mass $M_c$ such that the solutions exist globally in time if the mass is less than $M_c$ and above which there are solutions which blowup in finite time. The main tools, in particular the free energy, and the idea of the methods are set out. A number of open questions are also...

We study the existence of solutions for a p-biharmonic problem with a critical Sobolev exponent and Navier boundary conditions, using variational arguments. We establish the existence of a precise interval of parameters for which our problem admits a nontrivial solution.