Recent publications

In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: $$\begin{aligned} &a(u,v-u)+b(v-u,\lambda )+j(v)-j(u)+J(u,v)-J(u,u)\geq (f,v-u)_{X}, \\ &b(u,\mu -\lambda )-\psi (\mu )+\psi (\lambda )\leq 0, \end{aligned}$$ a ( u , v − u ) + b ( v − u , λ ) + j ( v ) − j ( u ) + J ( u , v ) − J ( u , u ) ≥ ( f , v − u ) X , b ( u , μ − λ ) − ψ ( μ ) + ψ ( λ ) ≤ 0 , ( $v\in K\subseteq X$ v ∈ K ⊆ X , $\mu \in \Lambda \subset Y$ μ ∈ Λ ⊂ Y ), where $(X,(\cdot,\cdot )_{X})$ ( X , ( ⋅ , ⋅ ) X ) and $(Y,(\cdot,\cdot )_{Y})$ ( Y , ( ⋅ , ⋅ ) Y ) are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting $\psi (\cdot )=\epsilon \bar{\psi}(\cdot )$ ψ ( ⋅ ) = ϵ ψ ¯ ( ⋅ ) $(\epsilon >0)$ ( ϵ > 0 ) . Then, we deliver a convergence result for $\epsilon \to 0$ ϵ → 0 , the case $\psi \equiv 0$ ψ ≡ 0 appearing like a limit case.
The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones.

The classical Ambrosetti–Prodi problem considers perturbations of the linear Dirichlet Laplace operator by a nonlinear reaction whose derivative jumps over the principal eigenvalue of the operator. In this paper, we develop a related analysis for parametric problems driven by the nonlinear Robin (p,q)-Laplace operator (sum of a p-Laplacian and a q-Laplacian). Under hypotheses that cover both the (p−1)-linear and the (p−1)-superlinear case, we prove an optimal existence, multiplicity, and non-existence result, which is global in the parameter λ>0.

Let D≥2 be an integer. For each open and bounded set Ω⊂RD and each integer k≥1 we denote by Rk(Ω) the largest number r>0 for which there exists k disjoint open balls in Ω of radius r. Next, for each open, bounded, convex set Ω⊂RD with smooth boundary and each real number p∈(1,∞) we denote by {λk(p;Ω)}k≥1 the sequence of eigenvalues of the p-Laplace operator subject to the homogeneous Dirichlet boundary conditions, given by the Ljusternik-Schnirelman theory. For each integer k≥1 we show that there exists Mk∈[(ke)−1,1] such that for any open, bounded, convex set Ω⊂RD with smooth boundary for which Rk(Ω) is less than or equal to Mk, the k-th eigenvalue of the p-Laplacian on Ω, λk(p;Ω), is an increasing function of p on (1,∞). Moreover, there exists Nk≥Mk such that for any real number s∈(Nk,∞)∖{1} there exists an open, bounded, convex set Ω⊂RD with smooth boundary which has Rk(Ω) equal to s such that λk(p;Ω) is not a monotone function of p∈(1,∞).

This is the fourth data paper publishing lightcurve survey work of 52 Near Earth Asteroids (NEAs) using 10 telescopes available to the EURONEAR network between 2017 and 2020. Forty six targets were not observed before our runs (88% of the sample) but some of these were targeted during the same oppositions mainly by Brian Warner. We propose new periods for 20 targets (38% of the sample), confirming published data for 20 targets, while our results for 8 targets do not match published data. We secured periods for 15 targets (29% of the sample), candidate periods for 23 objects (44%), tentative periods for 11 asteroids (21%), and have derived basic information about 3 targets (6% of the sample). We calculated the lower limit of the ellipsoid shape ratios a/b for 46 NEAs (including 13 PHAs). We confirmed or suggested 4 binary objects, recommending two of them for follow-up during future dedicated campaigns.

In this paper, we study a class of cross constrained variational problem for the inhomogeneous nonlinear Schrödinger equation in RN. By constructing cross-invariant manifolds, we derive a sharp condition for blow-up phenomenon and global well-posedness of solutions.

For any bounded and convex set Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^{N}$$\end{document} (N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2$$\end{document}), with smooth boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}, and any real number p>1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1,$$\end{document} we denote by up\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_{p}$$\end{document} the p-torsion function on Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}, that is the solution of the torsional creep problemΔpu=-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{p}u=-1$$\end{document} in Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}, u=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u=0$$\end{document} on ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}, where Δpu:=div(∇up-2∇u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{p}u:=div( \left| \nabla u\right| ^{p-2}\nabla u) $$\end{document} is the p-Laplace operator. Our aim is to investigate the monotonicity with respect to p for the p-torsional rigidity on Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document}, defined as TpΩ:=∫Ωupdx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_{p}\left( \Omega \right) :=\int _{\Omega }u_{p}dx$$\end{document}. More precisely, we show that there exist two constants D1∈12,e-1N+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_1\in \left[ \frac{1}{2},e^{\frac{-1}{N+1}}\right] $$\end{document} and D2∈1,N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_2\in \left[ 1,N\right] $$\end{document} such that for each bounded and convex set Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^{N}$$\end{document} with |∂Ω||Ω|≤D1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{|\partial \Omega |}{|\Omega |}\le D_1$$\end{document} the function p→Tp(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\rightarrow T_p(\Omega )$$\end{document} is decreasing on 1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( 1,\infty \right) $$\end{document}, while for each bounded and convex set Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^{N}$$\end{document}, with |∂Ω||Ω|≥D2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{|\partial \Omega |}{|\Omega |}\ge D_2$$\end{document}, the function p→Tp(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\rightarrow T_p(\Omega )$$\end{document} is increasing on 1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( 1,\infty \right) $$\end{document}. Moreover, for each real number s∈(D1,D2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (D_1,D_2)$$\end{document} there exists a bounded and convex set Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset {\mathbb {R}}^{N}$$\end{document}, with |∂Ω||Ω|=s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{|\partial \Omega |}{|\Omega |}=s$$\end{document}, such that the function p→Tp(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\rightarrow T_p(\Omega )$$\end{document} is not monotone on (1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(1,\infty )$$\end{document}.

This paper is concerned with the following singularly perturbed fractional double-phase problem with unbalanced growth and competing potentials ϵps(-Δ)psu+ϵqs(-Δ)qsu+V(x)(|u|p-2u+|u|q-2u)=W(x)g(u),inRN,u∈Ws,p(RN)∩Ws,q(RN),u>0,inRN,where s∈(0,1), 2≤p0 is a small parameter, V is the absorption potential, W is the reaction potential and g is the reaction term with subcritical growth. Assume that the potentials V, W, and the nonlinearity g satisfy some natural conditions, applying topological and variational methods, we establish the existence and concentration phenomena of positive solutions for ϵ>0 sufficiently small as well as the multiplicity result depended on the topology of the set where V attains its global minimum and W attains its global maximum. Finally, we also obtain the nonexistence result of ground state solutions under suitable conditions.

In this article, we consider a system consisting of two elastic strings with attached tip masses coupled through an elastic spring. Our aim is to analyze its exact boundary controllability properties and to characterize the spaces of controllable initial data depending on the number of controls acting on the boundaries of the strings. We show that singularities in waves are “smoothed by three orders” as they cross a point mass. Consequently, when only one control acts on the extremity of the first string, the space of controlled initial data is asymmetric, the components corresponding to the second string having to be more regular than those corresponding to the first one. Roughly speaking, if the initial data for the string which is directly controlled can be in L2×H-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2\times H^{-1}$$\end{document}, they should be at least in H3×H2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^3\times H^2$$\end{document} for the second string, located on the other part of the masses.

The paper considers a simple and well-known method for reducing the differentiability order of an ordinary differential equation, defining the first derivative as a function that will become the new variable. Practically, we attach to the initial equation a supplementary one, very similar to the flow equation from the dynamical systems. This is why we name it as the “attached flow equation”. Despite its apparent simplicity, the approach asks for a closer investigation because the reduced equation in the flow variable could be difficult to integrate. To overcome this difficulty, the paper considers a class of second-order differential equations, proposing a decomposition of the free term in two parts and formulating rules, based on a specific balancing procedure, on how to choose the flow. These are the main novelties of the approach that will be illustrated by solving important equations from the theory of solitons as those arising in the Chafee–Infante, Fisher, or Benjamin–Bona–Mahony models.

This paper presents a dynamic analysis of the ParReEx multibody mechanism, which has been designed for human wrist joint rehabilitation. The starting point of the research is a virtual prototype of the ParReEx multibody mechanism. This model is used to simulate the dynamics of the multibody mechanism using ADAMS in three simulation scenarios: (a) rigid kinematic elements without friction in joints, (b) rigid kinematic elements with friction in joints, and (c) kinematic elements as deformable solids with friction in joints. In all three cases, the robot is used by a virtual patient in the form of a mannequin. Results such as the connecting forces in the kinematic joints and the torques necessary to operate the ParReEx robot modules are obtained by dynamic simulation in MSC.ADAMS. The torques obtained by numerical simulation are compared with those obtained experimentally. Finite element structural optimization (FEA) of the flexion/extension multibody mechanism module is performed. The results demonstrate the operational safety of the ParReEx multibody mechanism, which is structurally capable of supporting the external loads to which it is subjected.

This research aims to develop a conceptual model to establish the influence of digital core investment and digital innovation on digital resilience at the enterprise level. The data were collected through a questionnaire-based survey of managers and IT specialists of companies. The analysis was performed using structural equation modeling with SPSS Statistics and Amos software. Based on the literature review, the study identifies the main factors that can ensure digital resilience and assesses their impact on Romania’s private and public companies. The research results confirm the hypotheses presented in the article, emphasizing that digital resilience is the result of the collaboration of several factors with different effects, determined by using Industry 4.0 technologies. Thus, digital core and digital innovation investments help improve digital resilience. Moreover, digital core investments have a positive impact on the digital resilience of enterprises, mediated by digital innovation investments. The study’s novelty consists in the realization of a model of interconnected analysis of several variables specific to digital and innovative technologies to ensure the resilience framework at the company level. The research offers valuable results which can be used by companies in Romania or other European Union countries to ensure their digital resilience.

Black chokeberries are a valuable source of anthocyanins and other phenolic compounds, but they are underutilized due to their unpalatable astringent taste. The aim of this study was to determine the potential of using black chokeberry juice as a health-promoting ingredient in apple juice with a view to develop a new functional food product and to increase the dietary consumption of bioactive compounds. Mixed juices were prepared from apple (A) juice and black chokeberry (BC) juice at 95:5 (ABC5), 90:10 (ABC10), 85:15 (ABC15), and 80:20 (ABC20) volumetric ratios. Comparative studies on the effect of heat treatment (90 °C, 10 min) and storage (four months, 20 °C) on the physicochemical and antioxidant properties of apple, black chokeberry, and mixed juices were carried out. The soluble solids content, titratable acidity, total phenolic, total anthocyanin and ascorbic acid content, and antioxidant activity increased while the total soluble solids/titratable acidity ratio decreased with increasing addition levels of BC juice. Mixing A juice with BC juice at 95:5 and 90:10 volumetric ratios improved the color and enhanced the palatability and general acceptability of the juice. The percentage losses of anthocyanins and polyphenols registered after heat treatment and storage increased with increasing addition levels of BC juice.

Modern agriculture produces a very large amount of agricultural waste that remains unused. The use as a reinforcer of these renewable resources for the realization of composite materials, and the finding of useful industrial applications, constitutes or provokes the groups of researchers in this field. The study conducted in this article falls in this direction. Composites were fabricated with the chopped wheat straw reinforcement and epoxy resin matrix or hybrid resins with 50% and 70% Dammar volume proportions. Some mechanical properties of this type of composite materials were studied based on tensile strength, SEM analysis, water absorption/loss, vibration behavior and compression strength. The strength–strain and strain–strain diagrams, the modulus of elasticity, the breaking strength and the elongation at break were obtained. Compared to the epoxy resin composition, those with 50 and 70% Dammar, respectively, have a 47 and 55% lower breaking strength and a 30 and 84% higher damping factor, respectively. Because the values of these mechanical properties were limited, and in practice superior properties are needed, sandwich composites were manufactured, with the core of previously studied compositions, to which the outer faces of linen fabric were applied. These composites were applied to the bend (in three points), obtaining the force–deformation diagrams. The obtained properties show that they can be used in construction (paneling, shells, etc.), or in the furniture industry.

In this paper, well-posedness of a general class of elliptic mixed hemivariational–variational inequalities is studied. This general class includes several classes of the previously studied elliptic mixed hemivariational–variational inequalities as special cases. Moreover, our approach of the well-posedness analysis is easily accessible, unlike those in the published papers on elliptic mixed hemivariational–variational inequalities so far. First, prior theoretical results are recalled for a class of elliptic mixed hemivariational–variational inequalities featured by the presence of a potential operator. Then the well-posedness results are extended through a Banach fixed-point argument to the same class of inequalities without the potential operator assumption. The well-posedness results are further extended to a more general class of elliptic mixed hemivariational–variational inequalities through another application of the Banach fixed-point argument. The theoretical results are illustrated in the study of a contact problem. For comparison, the contact problem is studied both as an elliptic mixed hemivariational–variational inequality and as an elliptic variational–hemivariational inequality.

In this paper, we develop new proof techniques and analytical methods to prove the existence of ground state solutions for the following planar Schrödinger-Poisson system with zero mass{−Δu+ϕu=f(u),x∈R2,Δϕ=2πu2,x∈R2, where f∈C(R,R) has the critical exponential growth at infinity and there is no monotonicity restriction on f(u)/u3. In particular, by using delicate estimates we obtain a desired upper bound for the Mountain Pass level just with the optimal asymptotic condition κ=liminf|t|→∞t2F(t)eα0t2>0 to restore the compactness in the presence of critical exponential growth, which significantly improves analogous assumptions on asymptotic behavior of t2F(t)eα0t2 or tf(t)eα0t2 at infinity in the previous works. Moreover, we use a different approach from the one of Du and Weth (2017) [21] dealing with the power nonlinearities to establish the Pohozaev type identity, which not only allows critical exponential growth nonlinearities, but also deals with the non-autonomous case containing a linear term V(x)u in the first equation, both of which are not covered in the existing literature. To our knowledge, there has not been any work in the literature on the subject, even for the simpler equation: −Δu=f(u) in R2.

In this paper, we focus on the existence of positive solutions to the following planar Schrödinger-Newton system with general subcritical growth{−Δu+u+ϕu=f(u)inR2,Δϕ=u2inR2, where f is a smooth reaction. We introduce a new variational approach, which enables us to study the above problem in the Sobolev space H1(R2). The analysis developed in this paper also allows to investigate the relationship between a Schrödinger-Newton system of Riesz-type and a Schrödinger-Newton system of logarithmic-type. Furthermore, this new approach can provide a new look at the planar Schrödinger-Newton system and may it have some potential applications in various related problems.

We study the following class of stationary Schrödinger equations of Choquard type−Δu+V(x)u=[|x|−μ⁎(Q(x)F(u))]Q(x)f(u),x∈R2, where the potential V and the weight Q decay to zero at infinity like (1+|x|γ)−1 and (1+|x|β)−1 for some (γ,β) in variously different ranges, ⁎ denotes the convolution operator with μ∈(0,2), and F is the primitive of f that fulfills a critical exponential growth in the Trudinger-Moser sense. By establishing a version of the weighted Trudinger-Moser inequality, we investigate the existence of nontrivial solutions of mountain-pass type for the given problem. Furthermore, we shall establish that the nontrivial solution is a bound state, namely a solution belonging to H1(R2), for some particular (γ,β).

In this paper, we establish concentration and multiplicity properties of ground state solutions to the following perturbed double phase problem with competing potentials: -ϵpΔpu-ϵqΔqu+V(x)(|u|p-2u+|u|q-2u)=K(x)f(u),inRN,u∈W1,p(RN)∩W1,q(RN),u>0,inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\epsilon ^{p}\Delta _{p} u-\epsilon ^{q}\Delta _{q} u +V(x)(|u|^{p-2}u+|u|^{q-2}u)=K(x)f(u),&{} \quad \hbox {in}~\mathbb {R}^{N},\\ u\in W^{1,p}(\mathbb {R}^{N})\cap W^{1,q}(\mathbb {R}^{N}),\ u>0, &{} \quad \hbox {in}~\mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$\end{document}where 1<p<q<N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p<q<N$$\end{document}, Δsu=div(|∇u|s-2∇u)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{s}u=\hbox {div}(|\nabla u|^{s-2}\nabla u)$$\end{document}, with s∈{p,q}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in \{p,q\}$$\end{document}, is the s-Laplacian operator, and ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon $$\end{document} is a small positive parameter. We assume that the potentials V, K and the nonlinearity f are continuous but are not necessarily of class C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1}$$\end{document}. Under some natural hypotheses, using topological and variational tools from Nehari manifold analysis and Ljusternik–Schnirelmann category theory, we study the existence of positive ground state solutions and the relation between the number of positive solutions and the topology of the set where V attains its global minimum and K attains its global maximum. Moreover, we determine two concrete sets related to the potentials V and K as the concentration positions and we describe the concentration of ground state solutions as ϵ→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \rightarrow 0$$\end{document}. The asymptotic convergence and the exponential decay of positive solutions are also explored. Finally, we establish a sufficient condition for the non-existence of ground state solutions.

Let Ω ⊂ R N ( N ≥ 2) be a simply connected bounded domain, containing the origin, with C ² boundary denoted by ∂ Ω. Denote by Ω e x t := R N ∖ Ω ¯ $\Omega^{\mathrm{ext}}:=\mathbb{R}^{N} \backslash \bar{\Omega}$ the exterior of Ω. We consider the perturbed eigenvalue problem
− Δ p u − Δ q u = μ K ( x ) | u | p − 2 u for x ∈ Ω ext u ( x ) = 0 for x ∈ ∂ Ω u ( x ) → 0 , as | x | → ∞ , $$\left\{\begin{array}{lcl} -\Delta_{p} u-\Delta_{q} u=\mu K(x)|u|^{p-2} u & \text { for } & x \in \Omega^{\text {ext }} \\ u(x)=0 & \text { for } & x \in \partial \Omega \\ u(x) \rightarrow 0, & \text { as } & |x| \rightarrow \infty, \end{array}\right.$$
where p, q ∈ (1 ,N ), p ≠ q $p \neq q$ and K is a positive weight function defined on Ω ext having the property that K ∈ L ∞ (Ω ext ) ∩ L N/p (Ω ext ) . We show that the set of parameters μ for which the above eigenvalue problem possesses nontrivial weak solutions is exactly an unbounded open interval.

The specific equipment, installation and machinery infrastructure of an electric power system have always required specially designed data acquisition systems and devices to ensure their safe operation and monitoring. Besides maintenance, periodical upgrade must be ensured for these systems, to meet the current practical requirements. Monitoring, testing, and diagnosis altogether represent key activities in the development process of electric power elements. This work presents the detailed structure and implementation of a complex, configurable system which can assure efficient monitoring, testing, and diagnosis for various electric power infrastructures, with proven efficiency through a comprehensive set of experimental results obtained in real running conditions. The developed hardware and software implementation is a robust structure, optimized for acquiring a large variety of electrical signals, also providing easy and fast connection within the monitored environment. Its high level of configurability and very good price–performance ratio makes it an original and handy solution for electric power infrastructures.

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