Existence of infinitely many solutions for a nonlocal elliptic PDE involving singularity

Positivity - In this article, we will prove the existence of infinitely many positive weak solutions to the following nonlocal elliptic PDE. $$begin{aligned} (-Delta )^s u&= frac{lambda...     

On continuous solutions of a generalized Cauchy-Riemann system with more than one singularity

We study the solvability of the Riemann-Hilbert problem for a generalized Cauchy-Riemann system with several singularities and reveal several new phenomenon. For the number of continuous solutions we shall show that it depends not only on the index but also on the location and type of the singularities; moreover, it does not depend continuously on the coefficients of the equation.     

On the asymptotic growth of positive solutions to a nonlocal elliptic blow-up system involving strong competition

For a competition-diffusion system involving the fractional Laplacian of the form

$?{(?\mathrm{\Delta })}^{s}u=u{v}^2,?{(?\mathrm{\Delta })}^{s}v=v{u}^2,u,v>0\text{in}{\mathbb{R}}^N,$

with $s\in (0,1)$, we prove that the maximal asymptotic growth rate for its entire solutions is 2s. Moreover, since we are able to construct symmetric solutions to the problem, when $N=2$ with prescribed growth arbitrarily close to the critical one, we can conclude that the asymptotic bound found is optimal. Finally, we prove existence of genuinely higher dimensional solutions, when $N\ge 3$. Such problems arise, for example, as blow-ups of fractional reaction-diffusion systems when the interspecific competition rate tends to infinity.     

On a Liouville-type comparison principle for solutions of semilinear elliptic partial differential inequalities

This Note is devoted to the study of a Liouville-type comparison principle for entire weak solutions of semilinear elliptic partial differential inequalities of the form $Lu+{|u|}^{q?1}u?Lv+{|v|}^{q?1}v$, where $q>0$ is a given number and L is a linear (possibly non-uniformly) elliptic partial differential operator of second order in divergent form given formally by the relation

$L=\sum _{i,j=1}^n\frac{?}{?x_i}[a_{ij}(x)\frac{?}{?x_j}].$

We assume that $n?2$, that the coefficients $a_{ij}(x)$, $i,j=1,\dots ,n$, are measurable bounded functions on ${\mathbb{R}}^n$ such that $a_{ij}(x)=a_{ji}(x)$, and that the corresponding quadratic form is non-negative. The results obtained in this work complete similar results on solutions of quasilinear elliptic partial differential inequalities announced in Kurta [C. R. Acad. Sci. Paris, Ser. I 336 (11) (2003) 897–900]. To cite this article: V.V. Kurta, C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Asymptotics of solutions of a linear singularity perturbed system with degeneracy

Asymptotic expansions with respect to a small parameter e are constructed for a fundamental system of solutions of a linear singularity perturbed system of equations of the form

On a Liouville-type theorem and the Fujita blow-up phenomenon

The main purpose of this paper is to obtain the well-known results of H.Fujita and K.Hayakawa on the nonexistence of nontrivial nonnegative global solutions for the Cauchy problem for the equation

with on the half-space as a consequence of a new Liouville theorem of elliptic type for solutions of () on . This new result is in turn a consequence of other new phenomena established for nonlinear evolution problems. In particular, we prove that the inequality

has no nontrivial solutions on when We also show that the inequality

has no nontrivial nonnegative solutions for , and it has no solutions on bounded below by a positive constant for 1.$">

     

Romberg's method for certain integrals involving a singularity

This paper deals with integrals where the integrand contains a weight function which becomes infinite in one or both ends of the integration interval. An evaluation technique based on Romberg's method is described, and several examples are discussed.     

On a quasilinear system involving the operator curl

This paper concerns a quasilinear system involving the operator curl. This system is an approximation of the anisotropic Ginzburg–Landau system which describes the Meissner state of type II superconductors. The existence of the weak solutions of the quasilinear system is proved by applying a variational method to a modified functional, and the C ^2+α regularity of the weak solutions H is established without assuming the boundedness of curl H.     

Fine singularity analysis of solutions to the Laplace equation

We present here a fine singularity analysis of solutions to the Laplace equation in special polygonal domains in the plane. We assume piecewise constant Neumann data on one component of the boundary. Our motivation is to study the so‐called Berg effect, which is explained in the introduction. Copyright © 2014 John Wiley & Sons, Ltd.     

A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation

We consider the semilinear parabolic equation _ut=Δu+u^p$u_t=\mathrm{\Delta }u+u^p$ on R^N${\mathbb{R}}^N$, where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all x∈R^N$x\in {\mathbb{R}}^N$ and t∈R$t\in \mathbb{R}$. Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all t?0$t?0$, then it necessarily converges to 0, as t→∞$t\to \infty$, uniformly with respect to x∈R^N$x\in {\mathbb{R}}^N$.     

Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.     

On the existence of more positive solutions of periodic bvps with singularity

We consider Periodic boundary value problems for ordinary second order differential equations of the form u′′=f(t,u,u′), Where f satisfies the (local) Carathéodory conditions and can have a singularity in the second variable.Writing our problem in an operator can be computed on. These sets are not convex, in general. Using the degree theory we get at least one fixed point of the operator at each such set which leads to the existence and localization of more solutions of the related Periodic boundary value problem. Our results are based on the generalized lower and upper functions method from Rach?nková and Tvrdý[15].     

On global existence of solutions to a cross-diffusion system

We consider a strongly coupled nonlinear parabolic system which arises in population dynamics in n-dimensional domains (n?1). We prove the global existence of classical solutions to the system for n<10.     

On the number of zeros of nonoscillatory solutions to half-linear ordinary differential equations involving a parameter

In this paper the following half-linear ordinary differential equation is considered:

where 0$"> is a constant, 0$"> is a parameter, and is a continuous function on , 0$">, and 0$"> for . The main purpose is to show that precise information about the number of zeros can be drawn for some special type of solutions of (H such that

It is shown that, if and if (H is strongly nonoscillatory, then there exists a sequence such that ,   as ; and with has exactly zeros in the interval and ; and with has exactly zeros in and . For the proof of the theorem, we make use of the generalized Prüfer transformation, which consists of the generalized sine and cosine functions.

     

On the summability of the formal solutions for some PDEs with irregular singularity

In this Note, we consider some classes of nonlinear partial differential equations with regular singularity with respect to t=0 and irregular one with respect to x=0. Our purpose is to establish a result which is similar to the k-summability property, known in the case of singular ordinary differential equations. We can prove that, except at most a countable set, the formal solution is Borel summable or k-summable with respect to x in all other directions. To cite this article: Z. Luo et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On inclination-flip homoclinic orbit associated to a saddle-node singularity

In this article, we study vector fields bifurcating through a saddlenode equilibrium with an unstable homiclinic orbit. Bifurcating diagrams for two-parameter perturbations of these vector fields are exhibited. It is proved that Smale's horseshoe dynamics, surrounding the bifurcating homoclinic orbit, exists for a large set of such perturbations.Partially supported by CNPq-Brazil and CONICIT-Venezuela.