# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

March 2005 , Volume 4 , Issue 1

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2005, 4(1): 1-8 doi: 10.3934/cpaa.2005.4.1 +[Abstract](2252) +[PDF](183.9KB)
Abstract:
In this paper, we study positive solutions of the following system of integral equations in $R^n$:

$u(x) = \int_{R^{n}} |x-y|^{\alpha -n} v(y)^q dy$, $v(x) = \int_{R^{n}} |x-y|^{\alpha -n} u(y)^p dy$

with $\frac{1}{q+1}+\frac{1}{p+1}=\frac{n-\alpha}{n}$. In our previous paper, under the natural integrability conditions $u \in L^{p+1} (R^n)$ and $v \in L^{q+1} (R^n)$, we prove that all the solutions are radially symmetric and monotone decreasing about some point. In this paper, we go further to study the regularity of the solutions. We show that the solutions are bounded, and hence continuous and smooth. We also prove that if $p = q$, then $u = v$, and they both must assume the standard form

$c(\frac{t}{t^2 + |x - x_o|^2})^{(n-\alpha)/2}$

with some constant $c = c(n, \alpha)$, and for some $t > 0$ and $x_o \in R^n$.

2005, 4(1): 9-22 doi: 10.3934/cpaa.2005.4.9 +[Abstract](2294) +[PDF](225.5KB)
Abstract:
In this paper, we analyse a family of stationary nonlinear equations with $p\& q$- Laplacian $-\Delta_p u -\Delta_q u=\lambda c(x,u)$ which have a wide spectrum of applications in many areas of science. We introduce a new type of variational principles corresponding to this family of equations. Using this formalism, we exhibit intervals for the scalar parameter $\lambda$ where there exist positive solutions of the considered problems. Furthermore, we prove, in another interval, the nonexistence of nontrivial solutions. These results are different from those of existence and nonexistence for stationary equations with single Laplacian.
2005, 4(1): 23-43 doi: 10.3934/cpaa.2005.4.23 +[Abstract](1512) +[PDF](261.5KB)
Abstract:
This work is concerned with the study of steady flows, around an obstacle, of slightly compressible viscoelastic fluids, for which the extra-stress tensor is given by a White-Metzner constitutive law. The existence and uniqueness of such flows are shown, when Newtonian viscosity is present (Jeffreys' model), and for small data.
2005, 4(1): 45-91 doi: 10.3934/cpaa.2005.4.45 +[Abstract](2028) +[PDF](363.9KB)
Abstract:
In this paper we give optimal (up to endpoint) maximal function type estimates for the solution of the linear initial value problem associated with the Kadomstev-Petviashvili I equation. These estimates enable us to obtain a well-posedness result for a modified Kadomstev-Petviashvili I equation.
2005, 4(1): 93-99 doi: 10.3934/cpaa.2005.4.93 +[Abstract](1457) +[PDF](176.1KB)
Abstract:
A generic semilinear equation in a star-shaped ring is considered. Any solution bounded between its boundary values is shown to be decreasing along rays starting from the origin, provided that a structural condition is satisfied. A corresponding property for the product between the solution and a (positive) power of $|x|$ is also investigated. Applications to the Emden-Fowler and the Liouville equation are developed.
2005, 4(1): 101-114 doi: 10.3934/cpaa.2005.4.101 +[Abstract](1288) +[PDF](223.0KB)
Abstract:
The paper extends the Faber--Krahn inequality for elliptic and periodic parabolic problems to random and general nonautonomous parabolic problems. Under proper assumptions, it also provides necessary and sufficient conditions for the Faber-Krahn inequality being equality.
2005, 4(1): 115-142 doi: 10.3934/cpaa.2005.4.115 +[Abstract](1837) +[PDF](295.0KB)
Abstract:
We consider in this article a general construction of trajectory attractors and global attractors of evolution equations with memory. In our approach, the corresponding dynamical system acts in the space of initial data of the Cauchy problem under study; we can note that, in previous studies, the so-called history space setting was introduced and the study of global attractors was made in an extended phase space.
As an application, we construct trajectory and global attractors for dissipative hyperbolic equations with linear memory. We also prove the existence of a global Lyapunov function for the dissipative hyperbolic equation with memory. The existence of such a Lyapunov function implies a regular structure for the trajectory and global attractors of the equation under consideration.
2005, 4(1): 143-164 doi: 10.3934/cpaa.2005.4.143 +[Abstract](1574) +[PDF](272.9KB)
Abstract:
The main purpose of this paper is to construct families of positive and changing-sign solutions for both the slightly subcritical and slightly supercritical equations

$-\Delta u+V(x)u=N(N-2)|u|^{\frac{4}{N-2}\pm\varepsilon}u$ in $\mathbb R^N,$

which blow-up and concentrate at different points of $\mathbb R^N$ as $\varepsilon$ goes to 0, under certain conditions on the potential $V.$

2005, 4(1): 165-174 doi: 10.3934/cpaa.2005.4.165 +[Abstract](1909) +[PDF](204.3KB)
Abstract:
We give an explicit estimate of the fractal dimension of the global attractor to the wave equation with nolinear damping. The nonlinearities are smooth functions of certain polynomial growth. As a by-product we estimate the dimension of the exponential attractor for the time $\tau$ solution operator provided that $\tau$ is sufficiently large. The main tool used in the proof is the so-called method of the trajectories.
2005, 4(1): 175-185 doi: 10.3934/cpaa.2005.4.175 +[Abstract](1457) +[PDF](209.9KB)
Abstract:
We consider a class of second-order Hamiltonian systems with a potential indefinite in sign. Applying the fibering approach we prove some existence and multiplicity results of periodic solutions with minimal period. We also give an answer to the problem of the existence of solutions with prescribed period $T$ which is greater than the first eigenvalue $\frac{2\pi}{\omega_n}$ of the corresponding linear problem.
2005, 4(1): 187-198 doi: 10.3934/cpaa.2005.4.197 +[Abstract](1957) +[PDF](199.0KB)
Abstract:
In this paper, we give an existence result for nonradial large solutions of the semilinear elliptic equation $\Delta u =p(x)f(u)$ in $R^N (N\ge 3)$, where $f$ is assumed to satisfy $(f_1)$ and $(f_2)$ below. The asymptotic behavior of the large solutions at infinity are also studied in the sublinear case that $f(u)$ behaves like $u^{\gamma}$ at $\infty$ for $\gamma \in (0, 1)$.
2005, 4(1): 199-207 doi: 10.3934/cpaa.2005.4.187 +[Abstract](1932) +[PDF](199.4KB)
Abstract:
Two results are proved in the paper. The first is a uniqueness theorem for viscosity solutions of Dirichlet boundary value problems for Bellman-Isaacs equations with just measurable lower order terms. The second is a proof that there always exist maximal and minimal viscosity solutions of Dirichlet boundary value problems for fully nonlinear, uniformly elliptic PDE that are measurable in the $x$-variable.

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