The purpose of this paper is to study Liouville theorems for elliptic equations involving regional fractional Laplacian.The regional fractional Laplacian is the infinitesimal generator of the censored symmetric $2\alpha$-stable process in $\Omega$. Probability theory asserts that the censored $2\alpha$-stable process cannot approach the boundary when $\alpha\in(0,\frac12]$. We show the nonexistence of solutions bounded from above or bounded from below for Poisson equation$$(-\Delta)^\alpha_\Omega u= 1 \quad {\rm in}\ \, \, \Omega $$and nonexistence of solutions for the following Lane-Emden equation$$ \displaystyle (-\Delta)^\alpha_\Omega u=u^p\quad {\rm in}\ \, \Omega,\qquad u=0\quad {\rm on}\ \partial\Omega. $$

We investigate the existence of generalized solutions to coercive competing system driven by the $(p, q)$-Laplacian with unbounded perturbation corresponding to the leading term in the differential operator and with convection depending on the gradient. Some abstract principle leading to the existence of generalized solutions is also derived based on the Galerkin scheme.

Linear delayed integro-differential scalar equations with unbounded delays $$\dot{x}(t)+a_0(t)x(t)+ \sum_{k=1}^m a_k(t)x(h_k(t)) +\sum_{l=1}^n\int_{g_l(t)}^t G_l(t,s)x(s) ds+f(t)=0 $$are considered on a semi-axis $[t_0,\infty)$, $t_0\in\mathbb{R}$,where $x(t)=\varphi(t)$ if $t\leq t_0$ and$\varphi \colon (-\infty,t_0]\to\mathbb{R}$. It is assumed thatcoefficients$a_k\colon [t_0,\infty)\to \mathbb{R}$,delays$h_k, g_l\colon [t_0,\infty)\to{\mathbb{R}}$ satisfying$t_*\le h_k(t)\le t$, $t_*\le g_l(t)\le t$ for a $t_*\le t_0$,integral kernels $G_l\colon [t_0,\infty)\times [t_*,\infty)\to{\mathbb{R}}$and non-homogeneity $f\colon [t_0,\infty)\to{\mathbb{R}}$are Lebesgue measurable functions (of one or two variables respectively).The paper derives new explicit criteria on the boundedness and exponential stability of solutionswithout imposing the usual restrictions of all delays and coefficients being bounded.Investigation of exponential stability is reduced to a problem of theboundedness of all solutions to an auxiliary equation. Corollaries, specific for scalar equations, are deduced from the main results. Illustrative examples are considered as well.

We fill the two main remaining gaps in the full classification of non-degenerate planar traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under smooth perturbations. On one hand, we investigate the impact on the classification of allowing to restrict perturbations by further localization constraints, the so-called \emph{convective} stability analysis. On the other hand, in the multidimensional case, we extend the existing analysis to general perturbations, relaxing the artificial restriction, assumed so far, that initial perturbations are supported away from characteristic points. A striking feature of the latter analysis is that it involves infinite-dimensional families of genuinely multidimensional traveling waves that live near the perturbed planar traveling wave.

In this paper,we examine the spectrum of the eigenvalue problem associated to a cylindrical liquid bridge connecting two spheres of equal radius. We cover all possible types of spheres: concave, convex and concave-convex spheres. We also consider the special case that a support is a plane and the other one is a sphere. We find the Morse index in all the situations, and we determine its behavior with respect to the radius of the cylinder and the separation between the two spheres as well. As a consequence, we derive results of stability and instability of Plateau-Rayleigh type. Finally, a result of bifurcation is established proving that given a cylinder of fixed radius between two concave spheres, there is a critical separation between both spheres such that new axisymmetric capillary surfaces appear with the same contact angle as that for the initial cylinder.