We consider a relation between a mixed problem for a class of quasilinear wave equations with small parameter $\epsilon$ and a reduced problem of a parabolic type. By constructing the stable set the global existence of solutions can be discussed. It is shown that the solution $u_{\epsilon}$ of the mixed problem converges, uniformly on any finite time interval, to the solution $u$ of the parabolic equation in an appropriate Hilbert space as $\epsilon \rightarrow 0$. Several $\epsilon$ weighted energy estimates will be obtained in order to evaluate the difference norm of $u_{\epsilon}-u$

Recently significant developments have been made in the understanding of the theory and solutions of the fourth Painlev\'e equation given by \begin{equation} ww'' = \tfrac 12(w')^2+\tfrac 32w^4 + 4zw^3 + 2(z^2-\alpha)w^2 + \beta, \tag{1} \end{equation} where $\alpha$ and $\beta$ are arbitrary constants. All exact solutions of (1) are thought to belong to one of three solution hierarchies. In two of these hierarchies solutions may be determined in terms of the complementary error and parabolic cylinder functions whilst the third hierarchy contains rational solutions which may be expressed as ratios of polynomials in $z$. It is known that the fourth Painlevé equation (1) can arise as a symmetry reduction of a number of significant partial differential equations and here we consider the application of solutions of (1) to the nonlinear Schrödinger, potential nonlinear Schrödinger, spherical Boussinesq and $2+1$-dimensional dispersive long wave equations. Consequently we generate a number of new exact solutions of these equations. The fourth Painlevé equation (1) is also related to problems that arise in the consideration of two-dimensional quantum gravity and this aspect is also discussed

This article is devoted to the study of nonnegative solutions of the semi-linear elliptic equation $$ \Delta u - u^q + u^p = 0 \ \ \text{ in } \; \mathbb{R}^N, $$ where $0 < q < 1 < p < {N+2 \over N-2}$ and $N\geq 3$. We prove the existence of solutions in $H^1(\mathbb{R}^N)$ and that these solutions are compactly supported. Moreover, we show that any solution with the property that $\{x\in \mathbb{R}^N : u(x)>0\}$ is connected, is supported by a ball and is radial. Then we prove that such a solution is unique up to translations. Any other solution is the sum of a finite number of translations of this solution, in such a way that the interior of their supports are mutually disjoint. Existence is proved by a variational argument. The compactness of the support is obtained by comparison. To prove the radial symmetry of solutions we use the Moving Planes device and then, using techniques of ordinary differential equations, we show uniqueness.

In this paper we consider the equation $\Delta^2 u = \lambda |u|^{q-2} u + |u|^{2^*-2} u\equiv f(u)$ in a smooth bounded domain $\Omega\subset\mathbf{R}{N}$ with boundary conditions either $u|_{\partial\Omega} =\frac{\partial u}{\partial n}|_{\partial \Omega}=0$ or $u|_{\partial\Omega}=\Delta u|_{\partial \Omega}=0$, where $N>4$, $ 1< q <2, \,\lambda>0$ and $2^* = 2N/(N-4)$ We prove the existence of $\lambda_0$ such that for $0<\lambda<\lambda_0$ the above problems have infinitely many solutions. For the problem with the second boundary conditions, we prove the existence of a positive solution also in the supercritical case, i.e., when we have an exponent larger than $2^*$. Moreover, in the critical case, we show the existence of at least two positive solutions.

We prove some results about existence, uniqueness and qualitative behavior of positive solutions to equations of the type $$ -\Delta u=a(x/|x|){u\over |x|^2}+f(x,u)\qquad\;\;\hbox{in }\;\mathbf{R}^n\setminus\{0\}\;,\tag 0.1 $$ depending on the behavior of the function $a$ of the angular variable $x/|x|$. Our main results concern the critical nonlinearity $f(s)=s^{(n+2)/(n-2)}$. The proofs are based on variational arguments and the moving plane method.

We describe the influence of Coulomb friction $\mu$ sgn $ \dot x$ on the behavior of the linear oscillator given by $\ddot x+x=\varphi(t)$, where $\varphi$ is continuous and almost periodic. Depending on $\varphi$, we characterize the range of $\mu>0$ such that nontrivial almost periodic motions exist. We also show that dissipation caused by Coulomb friction may be too weak to ensure uniqueness of such motions, a phenomenon which appears already in case $\varphi$ is $2k\pi$-periodic with $k>1$. Nevertheless, we get a rather complete picture of the asymptotic behavior of such a system, but also have some interesting open questions, for example concerning the shape of the almost periodic solutions.

Hopf bifurcations with symmetry are typically treated in a two-step process: First the center manifold theorem is used to reduce the equations to a finite dimensional system, and then the resulting ordinary differential equations are reduced to Birkhoff normal form. The latter step involves transformations where the frequency appears in the denominator. In this paper, we are interested in situations where the frequency at the bifurcation point is small, and it would be of interest to consider bifurcated solutions with amplitudes on the same order as the frequency. In this case, the Birkhoff normal form breaks down, and the bifurcation equations become significantly more complex. We consider the Hopf bifurcation on the hexagonal lattice in this context. A possible application arises in the two-layer Bénard problem.