Existence Of Multiple Solutions Research Articles

Existence of solutions for critical Neumann problem with superlinear perturbation in the half‐space

AbstractIn this paper, we consider the existence and multiplicity of solutions for the critical Neumann problem where , , , , , is the outward unit normal vector at the boundary , is the usual critical exponent for the Sobolev embedding and is the critical exponent for the Sobolev trace embedding . By establishing an improved Pohozaev identity, we show that problem () has no nontrivial solution if . Applying the mountain pass theorem without the condition and the delicate estimates for the mountain pass level, we obtain the existence of a positive solution for all and the different values of the parameters and . Particularly, for , , , we prove that problem () has a positive solution if and only if . Moreover, the existence of multiple solutions for () is also obtained by dual variational principle for all and suitable .

Addressing the multiplicity of optimal solutions to the Clonal Deconvolution and Evolution Problem

The Clonal Deconvolution and Evolution Problem consists on unraveling the clonal structure and phylogeny of a tumor using estimated mutation frequency values obtained from multiple biopsies containing mixtures of tumor clones. In this article, we tackle the problem from an optimization perspective and we explore the number of optimal solutions for a given instance. Even in ideal scenarios without noise, we demonstrate that the Clonal Deconvolution and Evolution Problem is highly under-determined, leading to multiple solutions. Through a comprehensive analysis, we examine the factors contributing to the multiplicity of solutions. We find that as the number of samples increases, the number of optimal solutions decreases. Additionally, we explore how this phenomenon operates across various tumor topology scenarios. To address the issue of the existence of multiple solutions, we present sufficient conditions under which the problem can have a unique solution, and we propose a linear programming-based algorithm that leverages mutation orderings to generate instances with a single solution for a given topology. This algorithm encounters numerical challenges when applied to large instance sizes so, to overcome this, we propose a heuristic adaptation that enables the algorithm’s use for instances of any size.

N-point functions in conformal quantum mechanics: a momentum space odyssey

In this paper, we study the implications of conformal invariance in momentum space for correlation functions in quantum mechanics. We find that three point functions of arbitrary operators can be written in terms of the 2F1 hypergeometric function. We then show that generic four-point functions can be expressed in terms of Appell’s generalized hypergeometric function F2 with one undetermined parameter that plays the role of the conformal cross ratio in momentum space. We also construct momentum space conformal partial waves, which we compare with the Appell F2 representation. We test our expressions against free theory and DFF model correlators, finding an exact agreement. We then analyze five, six, and all higher point functions. We find, quite remarkably, that n-point functions can be expressed in terms of the Lauricella generalized hypergeometric function, EA, with n − 3 undetermined parameters, which is in one-to-one correspondence with the number of conformal cross ratios. This analysis provides the first instance of a closed form for generic momentum space conformal correlators in contrast to the situation in higher dimensions. Further, we show that the existence of multiple solutions to the momentum space Ward identities can be attributed to the Fourier transforms of the various possible time orderings. Finally, we extend our analysis to theories with N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1, 2 supersymmetry, where we find that the constraints due to the superconformal ward identities are identical to identities involving hypergeometric functions.

Minimum Principles for Sturm–Liouville Inequalities and Applications

A minimum principle for a Sturm–Liouville (S-L) inequality is obtained, which shows that the minimum value of a nonconstant solution of a S-L inequality never occurs in the interior of the domain (a closed interval) of the solution. The minimum principle is then applied to prove that any nonconstant solutions of S-L inequalities subject to separated inequality boundary conditions (IBCs) must be strictly positive in the interiors of their domains and are increasing or decreasing for some of these IBCs. These positivity results are used to prove the uniqueness of the solutions of linear S-L equations with separated BCs. All of these results hold for the corresponding second-order differential inequalities (or equations), which are special cases of S-L inequalities (or equations). These results are applied to two models arising from the source distribution of the human head and chemical reactor theory. The first model is governed by a nonlinear S-L equation, while the second one is governed by a nonlinear second-order differential equation. For the first model, the explicit solutions are not available, and there are no results on the existence of solutions of the first model. Our results show that all the nonconstant solutions are increasing and are strictly positive solutions. For the second model, many results on the uniqueness of the solutions and the existence of multiple solutions have been obtained before. Our results are applied to prove that all the nonconstant solutions are decreasing and strictly positive.

Experimental characterization of bending modes and investigation of multi-stable solutions of PLA-based additively manufactured beams

This study examines the experimental characterization of three identically 3D printed polylactic acid (PLA)-based cantilever beams subjected to various vibrational excitations with investigating the possible existence of multiple solutions and compares the results of the beams to one another to reveal uncertainties and possible issues of reproducibility of the additively-manufactured systems. Free, random, and harmonic excitations are performed to evaluate the responses of identically manufactured PLA-based beams with focusing on the first three modes of vibrations. The results show that nonlinear damping and nonlinear softening are present for the first three modes of vibration for the systems under consideration. It is proved that significant uncertainties may be found in estimating the linear and nonlinear characteristics of “identically” manufactured beams. It is also demonstrated that the reproducibility of these systems is not feasible from systems under vibratory excitations. In addition, the experiments indicate that the same additively manufactured beam may encounter multi-stable solutions which questions the reliability of manufacturing 3D printed systems subjected to dynamical excitations.

Multiplicity of solutions for a class of nonhomogeneous quasilinear elliptic system with locally symmetric condition in ℝN

This paper is concerned with a class of nonhomogeneous quasilinear elliptic system driven by the locally symmetric potential and the small continuous perturbations in the whole‐space . By a variant of Clark's theorem without the global symmetric condition and Moser's iteration technique, we obtain the existence of multiple solutions when the nonlinear term satisfies some growth conditions only in a circle with center 0, and the perturbation term is any continuous function with a small parameter and no any growth hypothesis.

A hybrid Krasnosel’skiĭ-Schauder fixed point theorem for systems

We provide new results regarding the localization of the solutions of nonlinear operator systems. We make use of a combination of Krasnosel’skiĭ cone compression–expansion type methodologies and Schauder-type ones. In particular we establish a localization of the solution of the system within the product of a conical shell and of a closed convex set. By iterating this procedure we prove the existence of multiple solutions. We illustrate our theoretical results by applying them to the solvability of systems of Hammerstein integral equations. In the case of two specific boundary value problems and with given nonlinearities, we are also able to obtain a numerical solution, consistent with our theoretical results.

Form-finding of thermal-adaptive pin-bar assemblies based on eigenvalue modification

Lattice structures with tunable expansion properties have been investigated in multidisciplinary fields to control the temperature effects of structures or materials. The expected thermal adaptivity can be achieved by optimizing the structural geometry. A novel method for the form-finding of thermal-adaptive pin-bar assemblies is developed in this paper by considering the control of structural temperature effects as the minimization of the potential energy of the system. Based on the stationarity condition of the potential energy with respect to the nodal coordinates, the compatibility relationship between the thermal elongations of members and the target nodal displacements is proven to be the sufficient and necessary condition for structural thermal adaptivity. The solvability of the compatibility equation is determined by the rank equality between the compatibility matrix and its augmented form, which can be measured by the number of nonzero eigenvalues of its Gramian matrix. The analytical relationship between the eigenvalues of the Gramian matrix and the nodal coordinates is established using the matrix perturbation theory. A numerical strategy based on Newton’s method is proposed in which the eigenvalues are gradually modified by adjusting the nodal coordinates until the rank equality is satisfied. To address the existence of multiple solutions with structural thermal adaptivity, structural symmetry and periodicity constraints are introduced to narrow the solution space. The thermal-adaptive configurations of three illustrative pin-bar assemblies are analyzed using the proposed form-finding method, and the expected thermal deformations are verified for the obtained configurations using the finite element software ABAQUS. Comparing the results obtained by the proposed method with those obtained by nonlinear programming and the genetic algorithm validates the advantages of the proposed method in terms of computational time, optimality of the obtained configuration and applicability to complex structural geometries.

Infinitely many solutions for a Schrödinger–Kirchhoff equations with concave and convex nonlinearities

This paper is concerned with the following Schrödinger–Kirchhoff equation − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δu + V ( x ) u = k ( x ) | u | q − 2 u − h ( x ) | u | p − 2 u , u ∈ H 1 ( R 3 ) , where a and b are positive constants, 1 < q < 2 < p < + ∞ . Under some suitable assumptions on V ( x ) , k ( x ) and h ( x ) , we obtain the existence of multiple solutions for this problem via a variant of Clark's theorem.

Flexible head-following motion planning for scalable and bendable continuum robots

Continuum robots, which are characterized by high length-to-diameter ratios and flexible structures, show great potential for various applications in confined and irregular environments. Due to the combination of motion modes, the existence of multiple solutions, and the presence of complex obstacle constraints, motion planning for these robots is highly challenging. To tackle the challenges of online and flexible operation for continuum robots, we propose a flexible head-following motion planning method that is suitable for scalable and bendable continuum robots. Firstly, we establish a piecewise constant curvature (PCC) kinematic model for scalable and bendable continuum robots. The article proposes an adaptive auxiliary points model and a method for updating key nodes in head-following motion to enhance the precise tracking capability for paths with different curvatures. Additionally, the article integrates the strategy for adjusting the posture of local joints of the robot into the head-following motion planning method, which is beneficial for achieving safe obstacle avoidance in local areas. The article concludes by presenting the results of multiple sets of motion simulation experiments and prototype experiments. The study demonstrates that the algorithm presented in this paper effectively navigates and adjusts posture to avoid obstacles, meeting the real-time demands of online operations. The average time for a single-step solution is 4.41×10−5 s, and the average tracking accuracy for circular paths is 7.8928 mm.

The existence of multiple solutions for a class of upper critical Choquard equation in a bounded domain

Abstract In this article, we consider the following Choquard equation with upper critical exponent: − Δ u = μ f ( x ) ∣ u ∣ p − 2 u + g ( x ) ( I α * ( g ∣ u ∣ 2 α * ) ) ∣ u ∣ 2 α * − 2 u , x ∈ Ω , -\Delta u=\mu f\left(x){| u| }^{p-2}u+g\left(x)({I}_{\alpha }* \left(g{| u| }^{{2}_{\alpha }^{* }})){| u| }^{{2}_{\alpha }^{* }-2}u,\hspace{1.0em}x\in \Omega , where μ &gt; 0 \mu \gt 0 is a parameter, N &gt; 4 N\gt 4 , 0 &lt; α &lt; N 0\lt \alpha \lt N , I α {I}_{\alpha } is the Riesz potential, N N − 2 &lt; p &lt; 2 \frac{N}{N-2}\lt p\lt 2 , Ω ⊂ R N \Omega \subset {{\mathbb{R}}}^{N} is a bounded domain with smooth boundary, and f f and g g are continuous functions. For μ \mu small enough, using variational methods, we establish the relationship between the number of solutions and the profile of potential g g .

A multiplicity result in finite-dimensional vector spaces

Let f 1 , … , f n be n ( n ≥ 2 ) continuous real-valued functions on R such that lim | t | → + ∞ ∫ 0 t f k ( s ) d s t 2 = − ∞ for all k = 1 , … , n . This sole condition is far from ensuring the existence of multiple solutions for the classical problem { − ( x k + 1 − 2 x k + x k − 1 ) = f k ( x k ) k = 1 , … , n , x 0 = x n + 1 = 0. However, as a by-product of a much more general result, we get the following: for each ρ ∈ R and for each i = 1 , … , n , there exists ( λ , μ ) ∈ R 2 such that the problem { − ρ ( x k + 1 − 2 x k + x k − 1 ) = f k ( x k ) k = 1 , … , n , k ≠ i − ρ ( x i + 1 − 2 x i + x i − 1 ) = f i ( x i ) + λ x i + μ x 0 = x n + 1 = 0 has at least three solutions.

On multiple solutions of the Grad–Shafranov equation

The Grad–Shafranov equation (GSE) for axisymmetric MHD equilibria is a nonlinear, scalar PDE which in principle can have zero, one or more non-trivial solutions. The conditions for the existence of multiple solutions has been little explored in the literature so far. We develop a simple analytic model to calculate multiple solutions in the large aspect ratio limit. We compare the results to the recently developed deflated continuation method to find multiple solutions in a realistic geometry and right-hand side of the GSE using the finite element method. The analytic model is surprisingly accurate in calculating multiple solutions of the GSE for given boundary conditions and the two methods agree well in limiting cases. We examine the effect of plasma shaping and aspect ratio on the multiple solutions and show that shaping generally does not alter the number of solutions. We discuss implications for predictive modelling, equilibrium reconstruction, plasma stability and disruptions.

Multiplicity of solutions for a singular system with sign-changing potential

Abstract This paper focuses on a singular system with a sign-changing potential in Γ, a bounded domain with a Lipschitz boundary in ℝ d {\mathbb{R}^{d}} . By imposing appropriate conditions on the weight potential, which is allowed to change sign, we establish the existence of multiple solutions using the shape optimization approach. This study represents one of the earliest endeavors to explore and analyze the occurrence of multiple solutions in fractional singular systems involving sign-changing potentials. By explicitly addressing this particular aspect, our paper contributes significantly to the limited body of literature that exists in this specific field.

On solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory with Berestycki–Lions conditions on $ \mathbb{R}^3 $

&lt;abstract&gt;&lt;p&gt;In this paper, the existence of multiple solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory was investigated. The potential and the primitive of the nonlinearity in this kind of elliptic equations are both allowed to be sign-changing. Besides, we assumed that the nonlinearity satisfies the Berestycki–Lions type conditions. By employing Ekeland's variational principle, mountain pass theorem, Pohožaev identity, and various other techniques, two nontrivial solutions were obtained under some suitable conditions.&lt;/p&gt;&lt;/abstract&gt;

Multiple solutions of nonlinear Neumann inclusions

We prove new results on the existence of multiple solutions for elliptic inclusions with nonlinear boundary conditions of Neumann type. Our approach is topological and relies on the fixed point index for multivalued map.

Micropolar nanoparticles flow on a stretching/shrinking sheet with multiple slips

The purpose of this paper is to examine the flow of micropolar fluid containing ternary nanoparticles, with multiple slips over a stretching/shrinking sheet. Also, the influences of internal radiation and with the first order chemical reaction are examined to study the fluid properties. Mathematical modeling for the defined problem is framed and yields coupled nonlinear PDEs which are then reduced to nonlinear ODEs by choosing suitable similarity variables. The resulting ODEs are solved by analytical technique and solution is represented as an incomplete gamma function and a hypergeometric function. The regulating parameters, such as Eringen parameter, chemical reaction parameter, inverse Darcy number, Prandtl number and radiation parameter impact are examined for the profiles of velocity, microrotation, energy and concentration of the fluid through graphs. Furthermore, the skin friction drag of the defined flow is also discussed. This investigation reveals the existence of multiple solutions for the case of shrinking and for the case of stretching it shows a unique solution. Also, the tri-hybrid nanofluid flow velocity is found to increase and decrease for the first and second solutions respectively. The velocity, energy and concentration of the flow are found to decrease due to slip conditions.

Analog RF Circuit Sizing by a Cascade of Shallow Neural Networks

A deep neural network architecture for the automatic sizing of analog circuit components is proposed, with a focus on radio frequency (RF) applications in the 2 to 5 GHz region. It addresses the challenges of the typically small number of examples for network training and the existence of multiple solutions, of which impractical values for integrated circuit implementation. We address these issues by restricting the learning to one component size at a time, thanks to a cascade of dedicated shallow neural networks (SNN), where each network constrains the prediction of the next ones. Moreover, the SNNs are individually tuned by a genetic algorithm for the prediction order and accuracy. This reduction of the solution space at each step allows the use of small training sets, and the introduced constraints between SNNs handle component interdependencies. The method is successfully validated on three different types of RF microcircuits: a low-noise amplifier (LNA), a voltage-controlled oscillator (VCO), and a mixer, using 180 nm and 130 nm CMOS implementations. All the predictions were within 5 % of the true values, both at the component and performance levels, and all the responses were obtained in less than 5 s, after 4 to 47 min. training on a regular PC station. The obtained results show that the proposed method is fast and applicable to arbitrary analog circuit topologies, with no need to retrain the developed neural network for each new set of desired circuit performances.

Existence of Multiple Solutions for Elliptic Equations with Indefinite Potential

Existence of Multiple Solutions for Elliptic Equations with Indefinite Potential

Normalized solutions for Schrödinger-Poisson equation with prescribed mass: The Sobolev subcritical case and the Sobolev critical case with mixed dispersion

In this paper, we study the following Schrödinger Poisson equation{−Δu−λu−γ(|x|−1⁎|u|2)u=f(u),x∈R3;∫R3u2dx=c, where c>0, λ∈R and f∈C(R,R). When γ<0 and f satisfies some weaker L2-supercritical conditions in Sobolev subcritical case, we show the existence of normalized solutions. When f(u)=μ|u|q−2u+|u|4u with μ>0, γ>0 and 2<q<103, a more complicated situation, we obtain the existence of multiple solutions. Our main tools are some technically simpler than the Ghoussoub minimax principle [18], which not only allow us to construct a bounded (PS) sequence under some weaker conditions on f than before, but help to yield richer results. In particular, our results generalize and improve some ones in [21] and some other related literatures.