Nonlinear second-order multivalued boundary value problems

In this paper we study nonlinear second-order differential inclusions involving the ordinary vector $p$-Laplacian, a multivalued maximal monotone operator and nonlinear multivalued boundary conditions. Our framework is general and unifying and incorporates gradient systems, evolutionary variational inequalities and the classical boundary value problems, namely the Dirichlet, the Neumann and the periodic problems. Using notions and techniques from the nonlinear operator theory and from multivalued analysis, we obtain solutions for both the `convex' and `nonconvex' problems. Finally, we present the cases of special interest, which fit into our framework, illustrating the generality of our results.Comment: 26 page

Existence and multiplicity results for resonant fractional boundary value problems

We study a Dirichlet-type boundary value problem for a pseudo-differential equation driven by the fractional Laplacian, with a non-linear reaction term which is resonant at infinity between two non-principal eigenvalues: for such equation we prove existence of a non-trivial solution. Under further assumptions on the behavior of the reaction at zero, we detect at least three non-trivial solutions (one positive, one negative, and one of undetermined sign). All results are based on the properties of weighted fractional eigenvalues, and on Morse theory

Two-loop soft anomalous dimensions and NNLL resummation for heavy quark production

I present results for two-loop soft anomalous dimensions for heavy quark production which control soft-gluon resummation at next-to-next-to-leading-logarithm (NNLL) accuracy. I derive an explicit expression for the exact result and study it numerically for top quark production via e+ e- -> t tbar, and I construct a surprisingly simple but very accurate approximation. I show that the two-loop soft anomalous dimensions with massive quarks display a simple proportionality relation to the one-loop result only in the limit of vanishing quark mass. I also discuss the extension of the calculation to single top and top pair production in hadron colliders.Comment: 10 pages, 6 figures; improved form of the analytical result; equation adde

Modified brane cosmologies with induced gravity, arbitrary matter content and a Gauss-Bonnet term in the bulk

We extend the covariant analysis of the brane cosmological evolution in order to take into account, apart from a general matter content and an induced-gravity term on the brane, a Gauss-Bonnet term in the bulk. The gravitational effect of the bulk matter on the brane evolution can be described in terms of the total bulk mass as measured by a bulk observer at the location of the brane. This mass appears in the effective Friedmann equation through a term characterized as generalized dark radiation that induces mirage effects in the evolution. We discuss the normal and self-accelerating branches of the combined system. We also derive the Raychaudhuri equation that can be used in order to determine if the cosmological evolution is accelerating.Comment: 12 pages, no figures, RevTex 4.0; (v2) new references are added; (v3,v4) minor changes, acknowledgment is included; to appear in Phys. Rev.

Nonlinear singular problems with indefinite potential term

We consider a nonlinear Dirichlet problem driven by a nonhomogeneous differential operator plus an indefinite potential. In the reaction we have the competing effects of a singular term and of concave and convex nonlinearities. In this paper the concave term is parametric. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $\lambda$ varies. This work continues our research published in arXiv:2004.12583, where $\xi \equiv 0$ and in the reaction the parametric term is the singular one.Comment: arXiv admin note: text overlap with arXiv:2004.1258

Double-phase problems with reaction of arbitrary growth

We consider a parametric nonlinear nonhomogeneous elliptic equation, driven by the sum of two differential operators having different structure. The associated energy functional has unbalanced growth and we do not impose any global growth conditions to the reaction term, whose behavior is prescribed only near the origin. Using truncation and comparison techniques and Morse theory, we show that the problem has multiple solutions in the case of high perturbations. We also show that if a symmetry condition is imposed to the reaction term, then we can generate a sequence of distinct nodal solutions with smaller and smaller energies

On a class of parametric (p,2)(p,2)-equations

We consider parametric equations driven by the sum of a $p$-Laplacian and a Laplace operator (the so-called $(p,2)$-equations). We study the existence and multiplicity of solutions when the parameter $\lambda>0$ is near the principal eigenvalue $\hat{\lambda}_1(p)>0$ of $(-\Delta_p,W^{1,p}_{0}(\Omega))$. We prove multiplicity results with precise sign information when the near resonance occurs from above and from below of $\hat{\lambda}_1(p)>0$