Nehari manifold approach for superlinear double phase problems with variable exponents

In this paper we consider quasilinear elliptic equations driven by the variable exponent double phase operator with superlinear right-hand sides. Under very general assumptions on the nonlinearity, we prove a multiplicity result for such problems whereby we show the existence of a positive solution, a negative one and a solution with changing sign. The sign-changing solution is obtained via the Nehari manifold approach and, in addition, we can also give information on its nodal domains

The AMBRE Project: Parameterisation of FGK-type stars from the ESO:HARPS archived spectra

The AMBRE project is a collaboration between the European Southern Observatory (ESO) and the Observatoire de la Cote d'Azur (OCA). It has been established to determine the stellar atmospheric parameters (effective temperature, surface gravity, global metallicities and abundance of alpha-elements over iron) of the archived spectra of four ESO spectrographs. The analysis of the ESO:HARPS archived spectra is presented. The sample being analysed (AMBRE:HARPS) covers the period from 2003 to 2010 and is comprised of 126688 scientific spectra corresponding to 17218 different stars. For the analysis of the spectral sample, the automated pipeline developed for the analysis of the AMBRE:FEROS archived spectra has been adapted to the characteristics of the HARPS spectra. Within the pipeline, the stellar parameters are determined by the MATISSE algorithm, developed at OCA for the analysis of large samples of stellar spectra in the framework of galactic archaeology. In the present application, MATISSE uses the AMBRE grid of synthetic spectra, which covers FGKM-type stars for a range of gravities and metallicities. We first determined the radial velocity and its associated error for the ~15% of the AMBRE:HARPS spectra, for which this velocity had not been derived by the ESO:HARPS reduction pipeline. The stellar atmospheric parameters and the associated chemical index [alpha/Fe] with their associated errors have then been estimated for all the spectra of the AMBRE:HARPS archived sample. Based on quality criteria, we accepted and delivered the parameterisation of ~71% of the total sample to ESO. These spectra correspond to ~10706 stars; each are observed between one and several hundred times. This automatic parameterisation of the AMBRE:HARPS spectra shows that the large majority of these stars are cool main-sequence dwarfs with metallicities greater than -0.5 dex

Parallel dynamics of the fully connected Blume-Emery-Griffiths neural network

The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied at zero temperature for arbitrary using a probabilistic approach. A recursive scheme is found determining the complete time evolution of the order parameters, taking into account all feedback correlations. It is based upon the evolution of the distribution of the local field, the structure of which is determined in detail. As an illustrative example, explicit analytic formula are given for the first few time steps of the dynamics. Furthermore, equilibrium fixed-point equations are derived and compared with the thermodynamic approach. The analytic results find excellent confirmation in extensive numerical simulations.Comment: 22 pages, 12 figure

On logarithmic double phase problems

In this paper we introduce a new logarithmic double phase type operator of the form\begin{align*}\mathcal{G}u:=-\operatorname{div}\left(|\nabla u|^{p(x)-2}\nabla u+\mu(x)\left[\log(e+|\nabla u|)+\frac{|\nabla u|}{q(x)(e+|\nabla u|)}\right]|\nabla u|^{q(x)-2} \nabla u \right),\end{align*}where $\Omega\subseteq\mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partial\Omega$, $p,q\in C(\overline{\Omega})$ with $1<p(x)\leq q(x)$ for all $x\in\overline{\Omega}$ and $\mu\in L^1(\Omega)$. First, we prove that the logarithmic Musielak-Orlicz Sobolev spaces $W^{1,\mathcal{H}_{\log}}(\Omega)$ and $W^{1, \mathcal{H}_{\log}}_0(\Omega)$ with $\mathcal{H}_{\log}(x,t)=t^{p(x)}+\mu(x)t^{q(x)}\log(e+t)$ for $(x,t)\in \overline{\Omega}\times [0,\infty)$ are separable, reflexive Banach spaces and $W^{1,\mathcal{H}_{\log}}_0(\Omega)$ can be equipped with an equivalent norm. We also prove several embedding results for these spaces and the closedness of these spaces under truncations. In addition we show the density of smooth functions in $W^{1,\mathcal{H}_{\log}}(\Omega)$ even in the case of an unbounded domain by supposing Nekvinda's decay condition on $p(\cdot)$. The second part is devoted to the properties of the operator and it turns out that it is bounded, continuous, strictly monotone, of type (S$_+$), coercive and a homeomorphism. As a result of independent interest we also present a new version of Young's inequality for the product of a power-law and a logarithm. In the last part of this work we consider equations driven by our new operator with superlinear right-hand sides. We prove multiplicity results for this type of equation, in particular about sign-changing solutions, by making use of a suitable variation of the corresponding Nehari manifold together with the quantitative deformation lemma and the Poincar\'e-Miranda existence theorem

Least energy sign-changing solution for degenerate Kirchhoff double phase problems

In this paper we study the following nonlocal Dirichlet equation of double phase type \begin{align*} -\psi \left [ \int_\Omega \left ( \frac{|\nabla u |^p}{p} + \mu(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x\right] \mathcal{G}(u) = f(x,u)\quad \text{in } \Omega, \quad u = 0\quad \text{on } \partial\Omega, \end{align*} where $\mathcal{G}$ is the double phase operator given by \begin{align*} \mathcal{G}(u)=\operatorname{div} \left(|\nabla u|^{p-2}\nabla u + \mu(x) |\nabla u|^{q-2}\nabla u \right)\quad u\in W^{1,\mathcal{H}}_0(\Omega), \end{align*} $\Omega\subseteq \mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partial\Omega$, $1<p<N$, $p<q<p^*=\frac{Np}{N-p}$, $0 \leq \mu(\cdot)\in L^\infty(\Omega)$, $\psi(s) = a_0 + b_0 s^{\vartheta-1}$ for $s\in\mathbb{R}$, with $a_0 \geq 0$, $b_0>0$ and $\vartheta \geq 1$, and $f\colon\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carath\'{e}odory function that grows superlinearly and subcritically. We prove the existence of two constant sign solutions (one is positive, the other one negative) and of a sign-changing solution which has exactly two nodal domains and which turns out to be a least energy sign-changing solution of the problem above. Our proofs are based on variational tools in combination with the quantitative deformation lemma and the Poincar\'{e}-Miranda existence theorem

The Blume-Emery-Griffiths neural network: dynamics for arbitrary temperature

The parallel dynamics of the fully connected Blume-Emery-Griffiths neural network model is studied for arbitrary temperature. By employing a probabilistic signal-to-noise approach, a recursive scheme is found determining the time evolution of the distribution of the local fields and, hence, the evolution of the order parameters. A comparison of this approach is made with the generating functional method, allowing to calculate any physical relevant quantity as a function of time. Explicit analytic formula are given in both methods for the first few time steps of the dynamics. Up to the third time step the results are identical. Some arguments are presented why beyond the third time step the results differ for certain values of the model parameters. Furthermore, fixed-point equations are derived in the stationary limit. Numerical simulations confirm our theoretical findings.Comment: 26 pages in Latex, 8 eps figure

Modulation of interferon-[alpha] secretion by activated platelets in systemic lupus erythematosus.

Type I interferons play a key role in systemic lupus erythematosus (SLE) pathogenesis as an &#x22;IFN signature&#x22; is found in the majority of patients with active SLE. Immune complexes are internalized by plasmacytoid dendritic cells (DC) via Fc-[gamma] ReceptorIIA, reach the endosomal compartment and activate IFN-[alpha] secretion through TLR7/9-dependent pathways. Naturally occurring differences in expression of the TLR7/9 gene as well as factors that modulate TLR7/9 expression, including CD154 could therefore contribute to SLE pathogenesis. Although its origin is not elucidated CD154 is hyperexpressed in SLE patients, and is important for the differentiation of autoantibody-secreting cells. We hypothesized that platelets which are an abundant source of CD154, and which can mediate proinflammatory effects could be an actor involved in SLE pathogenesis. Platelets from SLE patients are activated _in vivo_ by circulating immune complexes which are abundant in SLE sera, via a CD32-dependent mechanism. Activated platelets formed aggregates with antigen-presenting cells in SLE patients and enhanced interferon-[alpha] secretion induced by immune-complexes stimulated plasmacytoid DCs. Finally, _in vivo_ depletion of platelets and megakaryocytes in NZBxNZW(F1) lupus prone mice improved all parameters assessing disease activity, whereas transfusion of activated platelets worsened the disease course. Altogether, these data identify platelets as a mediator of SLE pathogenesis and a new therapeutical target

Superlinear elliptic equations with unbalanced growth and nonlinear boundary condition

In this paper we first prove the existence of a new equivalent norm in the Musielak-Orlicz Sobolev spaces in a very general setting and we present a new result on the boundedness of the solutions of a wide class of nonlinear Neumann problems, both of independent interest. Moreover, we study a variable exponent double phase problem with a nonlinear boundary condition and prove the existence of multiple solutions under very general assumptions on the nonlinearities. To be more precise, we get constant sign solutions (nonpositive and nonnegative) via a mountain-pass approach and a sign-changing solution by using an appropriate subset of the corresponding Nehari manifold along with the Brouwer degree and the Quantitative Deformation Lemma