This paper is concerned with the existence of cylindrically symmetric traveling fronts for bistable fractional Allen-Cahn equation in $\mathbb{R}^{3}$. By constructing a sequence of pyramidal traveling fronts and taking its limit, we obtain a traveling front with cylindrically symmetric type.

It is well-established that there are two main methods to show existence of critical points for coercive functionals: either one focuses on proving the existence of global minimizers through the direct method and weak lower semicontinuity; or else, one relies on the Palais-Smale condition and goes directly to showing the existence of such critical, non-minimizer points. Both essentially require the convexity of the principal part of the underlying functional. In this contribution, we would like to explore the possibility of being dispensed with such convexity condition in such a way that neither of the two methods is applicable. We will show that, in most regular, coercive cases, associated Euler-Lagrange equations of optimality admit (weak) solutions in spite of non-convexity. Our results are even valid for vector problems of PDEs. Important restrictions to be taken into account in particular situations like positivity for scalar equations, or injectivity for systems, would require further insight.

In this paper, we study the Kirchhoff type problems involving fractional $p$&$q$ problem with critical Sobolev-Hardy exponents and sign-changing weight functions. By using the fractional version of concentration-compactness principle together with Krasnoselskii's genus, we obtain the multiplicity of solutions for this kind problem. The main feature and difficulty of our equations arise in the fact that the Kirchhoff term $M$ could vanish at zero, that is, the problem is degenerate.

We propose a theoretical model of a non-local dispersive-dissipative equation which contains as a particular case a large class of non-local PDE's arising from stratified flows. Within this fairly general framework, we study the spatial behavior of solutions proving some sharp pointwise and averaged decay properties as well as some pointwise grow properties.