Singular limits in Liouville-type equations

We consider the boundary value problem in a bounded, smooth domain in with homogeneous Dirichlet boundary conditions. Here 0,k(x) $$ " align="middle" border="0"> is a non-negative, not identically zero function. We find conditions under which there exists a solution which blows up at exactly m points as and satisfies . In particular, we find that if , 0 $" align="middle" border="0"> and is not simply connected then such a solution exists for any given Received: 11 February 2004, Accepted: 17 August 2004, Published online: 22 December 2004     

Singular limits for Liouville-type equations on the flat two-torus

We consider the problem $$\begin{aligned} -\Delta u=\varepsilon ^{2}e^{u}- \frac{1}{|\Omega |}\int _\Omega \varepsilon ^{2} e^{u}+ {4\pi N\over |\Omega |} - 4 \pi N\delta _p, \quad \text{ in} {\Omega }, \quad \int _\Omega u=0 \end{aligned}$$ in a flat two-torus $\Omega $ with periodic boundary conditions, where $\varepsilon >0,\,|\Omega |$ is the area of the $\Omega $ , $N>0$ and $\delta _p$ is a Dirac mass at $p\in \Omega $ . We prove that if $1\le m<N+1$ then there exists a family of solutions $\{u_\varepsilon \}_{\varepsilon }$ such that $\varepsilon ^{2}e^{u_\varepsilon }\rightharpoonup 8\pi \sum _{i=1}^m\delta _{q_i}$ as $\varepsilon \rightarrow 0$ in measure sense for some different points $q_{1}, \ldots , q_{m}$ . Furthermore, points $q_i$ , $i=1,\dots ,m$ are different from $p$ .     

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.     

Nonlocal higher order evolution equations

In this article, we study the asymptotic behaviour of solutions to the nonlocal operator u _t (x, t) = (?1) ^n?1 (J * Id ? 1) ⁿ (u(x, t)), x ∈ ? ^N , which is the nonlocal analogous to the higher order local evolution equation v _t = (?1) ^n?1(Δ) ⁿ v. We prove that the solutions of the nonlocal problem converge to the solution of the higher order problem with the right-hand side given by powers of the Laplacian when the kernel J is rescaled in an appropriate way. Moreover, we prove that solutions to both equations have the same asymptotic decay rate as t goes to infinity.     

Evolution equations of higher order in banach spaces

The present paper deals with a minimal extension of the classical semigroup theory for equations of any order in Banach spaces with closed densely defined linear operators as coefficients. We do not ask anymore from our operators than in the case of first-order equations, i.e., Semigroups. We present here a generalization of Myadera-Phillips--Feller theorem, of Hille theorem and some other results. The method is quite general. We focus our attention on a particular operator solution (main propagator or abstract initial value Green function) and we assume some properties about it. From this we can obtain all needed information about complementary operator solutions, among others.     

Oscillation of higher order delay differential equations

A sufficient condition was obtained for oscillation of all solutions of theodd-order delay differential equation $$x^{(n)} (t) + \sum\limits_{i = 1}^m {p_i (t)} x(t - \sigma _{_i } ) = 0,$$ wherep _i (t) are non-negative real valued continuous function in [T ∞] for someT≥0 and σ_i,∈(0, ∞)(i = 1,2,…,m). In particular, forp _i (t) =p _i ∈(0, ∞) andn > 1 the result reduces to $$\frac{1}{m}\left( {\sum\limits_{i = 1}^m {(p_i \sigma _i^m )^{1/2} } } \right)^2 > (n - 2)!\frac{{(n)^n }}{e},$$ implies that every solution of (*) oscillates. This result supplements forn > 1 to a similar result proved by Ladaset al [J. Diff. Equn.,42 (1982) 134–152] which was proved for the casen = 1.     

Uniqueness properties of higher order dispersive equations

In this paper we study unique continuation properties of solutions to higher (fifth) order nonlinear dispersive models. The aim is to show that if the difference of two solutions of the equations, u₁−u₂, decays sufficiently fast at infinity at two different times, then u₁≡u₂.     

Anisotropic equations in weighted Sobolev spaces of higher order

We consider the strongly nonlinear boundary value problem,

Liouville-type results for semilinear elliptic equations in unbounded domains

This paper is devoted to the study of some class of semilinear elliptic equations in the whole space: