We consider the following nonlinearly perturbed version of the elliptic-parabolic system of Keller-Segel type: ∂tu−∆u +∇ · (u∇v) = 0, t > 0, x ∈ R, −∆v + v − v = u, t > 0, x ∈ R, u(0, x) = u0(x) ≥ 0, x ∈ R, where 1 < p < ∞. It has already been shown that the system admits a positive solution for a small nonnegative initial data in L1(R2) ∩ L2(R2) which corresponds to the local minimum of the as...

&lt;p style='text-indent:20px;'&gt;We examine the following degenerate elliptic system:&lt;/p&gt;&lt;p style='text-indent:20px;'&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ -\Delta_{s} u \! = v^p, \quad v\! u^\theta, \;\; u, v&amp;gt;0 \;\;\mbox{in }\; \mathbb{R}^N \mathbb{R}^{N_1}\times \mathbb{R}^{N_2}, \quad\mbox{where}\;\; s \geq 0\;\; \mbox{and} \;\;p...

A system of two discrete nonlinear Schrödinger equations of the Ablowitz-Ladik type with a saturable nonlinearity is shown to admit a doubly periodic wave, whose long wave limit is also derived. As a by-product, several new solutions of the elliptic type are provided for NLS-type discrete and continuous systems.

In this paper, we study the existence and regularity of positive solution for an elliptic system on a bounded and regular domain. The non linearities in this equation are functions of Caratheodory type satisfying some exponential growth conditions.

We analyze a diffuse interface type approximation, known as the diffuse domain approach, of a linear coupled bulk-surface elliptic PDE system. The well-posedness of the diffuse domain approximation is shown using weighted Sobolev spaces and we prove that the solution to the diffuse domain approximation converges weakly to the solution of the coupled bulk-surface elliptic system as the approxima...

Many equations can be expressed as a cubic nonlinear Schrödinger (NLS) equation with additional terms, such as the Davey-Stewartson (DS) system [1]. As it is the case for the NLS equation, the solutions of the DS system are invariant under the pseudo-conformal transformation. For the elliptic NLS, this invariance plays a key role in understanding the blow-up profile of solutions, whereas in the...

In the note, we prove that additive Schwarz type preconditioner is applicable to the nested finite dimensional subspaces generated by Hermite cubic splines for the second order elliptic problem. It is shown that the preconditioned system is well conditioned.

We demonstrate that the symmetric elliptic polynomials $E_\lambda(x)$ originally discovered in study of generalized Noumi-Shiraishi functions are eigenfunctions Ruijsenaars-Schneider (eRS) Hamiltonians act on mother function variable $y_i$ (substitute Young-diagram $\lambda$). This means they dual eRS system. At same time, their orthogonal complements Schur scalar product, $P_R(x)$ reduction Ko...