RD-systems with chemotaxis: global existence, boundedness and blow-up of solutions

result shows that the blow-up is equivalent to the blow-up of the $L^r-$norms of the solutions for $r$ exceeding some critical value $r_c.$ Under very loose conditions we give the estimation of $r_c,$ relying on a variant of Gagliardo-Nirenberg inequality, and a kind of bootstrap method which is very similar to the Alikakos-Moser iteration procedure.

Multiple non-radially symmetrical nodal solutions for the Schrödinger system with positive quasilinear term

<p style='text-indent:20px;'>This paper is concerned with the following quasilinear Schrödinger system in the entire space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb R^{N}(N\geq3) $\end{document}</tex-math></inline-formula>:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\{\begin{aligned} &amp;-\Delta u+A(x)u+\frac{k}{2}\triangle(u^{2})u = \frac{2\alpha }{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta},\\ &amp;-\Delta v+Bv+\frac{k}{2}\triangle(v^{2})v = \frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v,\\ &amp; u(x)\to 0,\ \ v(x)\to 0\ \ \hbox{as}\ |x|\to \infty,\end{aligned}\right. $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ \alpha,\beta&gt;1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 2&lt;\alpha+\beta&lt;2^* = \frac{2N}{N-2} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ k &gt;0 $\end{document}</tex-math></inline-formula> is a parameter. By using the principle of symmetric criticality and the moser iteration, for any given integer <inline-formula><tex-math id="M5">\begin{document}$ \xi\geq2 $\end{document}</tex-math></inline-formula>, we construct a non-radially symmetrical nodal solution with its <inline-formula><tex-math id="M6">\begin{document}$ 2\xi $\end{document}</tex-math></inline-formula> nodal domains. Our results can be looked on as a generalization to results by Alves, Wang and Shen (Soliton solutions for a class of quasilinear Schrödinger equations with a parameter. J. Differ. Equ. 259 (2015) 318-343).</p>

Hölder Regularity of Quasiminimizers to Generalized Orlicz Functional on the Heisenberg Group

In this paper, we apply De Giorgi-Moser iteration to establish the Hölder regularity of quasiminimizers to generalized Orlicz functional on the Heisenberg group by using the Riesz potential, maximal function, Calderón-Zygmund decomposition, and covering Lemma on the context of the Heisenberg Group. The functional includes the p -Laplace functional on the Heisenberg group which has been studied and the variable exponential functional and the double phase growth functional on the Heisenberg group that have not been studied.

Traveling Waves in the Underdamped Frenkel–Kontorova Model

This paper studies a damped Frenkel–Kontorova model with periodic boundary condition. By using Nash–Moser iteration scheme, we prove that such model has a family of smooth traveling wave solutions.

A Global Curvature Pinching Result of the First Eigenvalue of the Laplacian on Riemannian Manifolds

The paper starts with a discussion involving the Sobolev constant on geodesic balls and then follows with a derivation of a lower bound for the first eigenvalue of the Laplacian on manifolds with small negative curvature. The derivation involves Moser iteration.