Existence Of Multiple Solutions Research Articles

Classification of String Solutions for the Self-Dual Einstein–Maxwell–Higgs Model

In this paper, we are concerned with an elliptic system arising from the Einstein–Maxwell–Higgs model which describes electromagnetic dynamics coupled with gravitational fields in spacetime. Reducing this system to a single equation and setting up the radial ansatz, we classify solutions into three cases: topological solutions, nontopological solutions of type I, and nontopological solutions of type II. There are two important constants: $$a>0$$ representing the gravitational constant and $$N\ge 0$$ representing the total string number. When $$0\le aN<2$$ , we give a complete classification of all possible solutions and prove the uniqueness of solutions for a given decay rate. In particular, we obtain a new class of topological solitons, with nonstandard asymptotic value $$\sigma <0$$ at infinity, when the total string number is sufficiently large such that $$1<aN<2$$ . We also prove the multiple existence of solutions for a given decay rate in the case $$aN \ge 2$$ . Our classification improves previous results which are known only for the case $$0<aN<1$$ .

Existence of multiple solutions for nonhomogeneous Schrödinger–Kirchhoff system involving the fractional [formula omitted]-Laplacian with sign-changing potential

In this paper, we prove the existence of multiple solutions for the following Schrödinger–Kirchhoff system involving the fractional p-Laplacian M∬R2N|u(x)−u(y)|p|x−y|N+psdxdy(−Δ)psu+V(x)|u|p−2u=Fu(x,u,v)+λg(x),x∈RN,M∬R2N|v(x)−v(y)|p|x−y|N+psdxdy(−Δ)psu+V(x)|v|p−2v=Fv(x,u,v)+λh(x),x∈RN,u(x)→0,v(x)→0,as|x|→+∞,where (−Δ)ps denotes the fractional p-Laplacian of order s∈(0,1), 2≤p<∞, ps<N, Fu=∂F∂u, Fv=∂F∂v, V(x) is allowed to be sign-changing, λ>0 and g,h:RN→R is a perturbation. Under some certain assumptions on f, we obtain the existence of multiple solutions for this problem via Ekeland’s variational principle and mountain pass theorem.

Multiple solutions for the non-Abelian Chern–Simons–Higgs vortex equations

In this paper we study the existence of multiple solutions for the non-Abelian Chern–Simons–Higgs (N×N)-system:Δui=λ(∑j=1N∑k=1NKkjKjieujeuk−∑j=1NKjieuj)+4π∑j=1niδpij,i=1,…,N; over a doubly periodic domain Ω, with coupling matrix K given by the Cartan matrix of SU(N+1), (see (1.2) below). Here, λ>0 is the coupling parameter, δp is the Dirac measure with pole at p and ni∈N, for i=1,…,N. When N=1,2 many results are now available for the periodic solvability of such system and provide the existence of different classes of solutions known as: topological, non-topological, mixed and blow-up type. On the contrary for N≥3, only recently in [27] the authors managed to obtain the existence of one doubly periodic solution via a minimization procedure, in the spirit of [46]. Our main contribution in this paper is to show (as in [46]) that actually the given system admits a second doubly periodic solutions of “Mountain-pass” type, provided that 3≤N≤5. Note that the existence of multiple solutions is relevant from the physical point of view. Indeed, it implies the co-existence of different non-Abelian Chern–Simons condensates sharing the same set (assigned component-wise) of vortex points, energy and fluxes. The main difficulty to overcome is to attain a “compactness” property encompassed by the so-called Palais–Smale condition for the corresponding “action” functional, whose validity remains still open for N≥6.

On Kirchhoff Problems Involving Critical Exponent and Critical Growth

In this paper, we establish the existence of multiple solutions to a class of Kirchhoff type equations involving critical exponent, concave term and critical growth. Our main tools are the Nehari manifold and mountain pass theorem.

Critical point approaches to Gradient-Type systems on the Sierpiński Gasket

We investigate the existence of multiple solutions for parametric quasi-linear systems of the gradient-type on the Sierpiński gasket. We give some new criteria to guarantee that the systems have at least three weak solutions by using a variational method and some critical points theorems due to Ricceri. We extend and improve some recent results. Finally, we give two examples to illustrate the main results.

Existence theorem for a class of semilinear totally characteristic elliptic equations involving supercritical cone sobolev exponents

In this paper, we prove the existence of bounded positive solutions for a class of semilinear degenerate elliptic equations involving supercritical cone Sobolev exponents. We also obtain the existence of multiple solutions by the Ljusternik-Schnirelman theory.

Existence and asymptotic behavior of standing wave solutions for a class of generalized quasilinear Schrödinger equations with critical Sobolev exponents

In this paper, we study the following generalized quasilinear Schrödinger equation − div ( ε 2 g 2 ( u ) ∇ u ) + ε 2 g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = K ( x ) | u | p − 2 u + | u | 22 ∗ − 2 u , x ∈ R N , where N ⩾ 3, ε &gt; 0, 4 &lt; p &lt; 22 ∗ , g ∈ C 1 ( R , R + ), V ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global minimum, and K ∈ C ( R N ) ∩ L ∞ ( R N ) has a positive global maximum. By using a change of variable, we obtain the existence and concentration behavior of ground state solutions for this problem with critical growth, and establish a phenomenon of exponential decay. Moreover, by Ljusternik–Schnirelmann theory, we also prove the existence of multiple solutions.

Multiple solutions for the nonhomogeneous discrete nonlinear Schrödinger equation

In the present paper, we consider the discrete nonlinear Schrödinger equation −Δun+vnun−ωun=γgn(un)+hn,n∈Z,where vn, hn are real valued sequences, ω∈R, γ=±1, and gn is a function sequence. We study this non-periodic discrete nonlinear Schrödinger equation with infinitely growing potential. Under some suitable conditions, the existence of multiple solutions is proved by using the Ekeland’s variational principle and the Mountain Pass Theorem in critical point theory. It is the first time to investigate the nonhomogeneous discrete nonlinear Schrödinger equation.

Multiple solutions for a fractional $p$-Kirchhoff problem with subcritical and critical Hardy-Sobolev exponent

In this paper, by using the variational method and the theory of genus, we obtain the existence of multiple solutions to a fractional $p$-Kirchhoff problem with subcritical and critical Hardy-Sobolev exponent.

Multiple Solutions of Sublinear Quasilinear Schrödinger Equations with Small Perturbations

AbstractIn this paper, we consider the existence of multiple solutions for the quasilinear Schrödinger equation $$\left\{ {\matrix{ {-\Delta u-\Delta (\vert u \vert ^\alpha )\vert u \vert ^{\alpha -2}u = g(x,u) + \theta h(x,u),\;\;x\in \Omega } \hfill \cr {u = 0,\;\;x\in \partial \Omega ,} \hfill \cr } } \right.$$ where Ω is a bounded smooth domain in ℝN (N ≥ 1), α ≥ 2 and θ is a parameter. Under the assumption that g(x, u) is sublinear near the origin with respect to u, we study the effect of the perturbation term h(x, u), which may break the symmetry of the associated energy functional. With the aid of critical point theory and the truncation method, we show that this system possesses multiple small negative energy solutions.

Existence of multiple solutions for a fractional p-Laplacian system with concave-convex term

In this article, we study the existence of multiple solutions for the following system driven by a nonlocal integro-differential operator with zero Dirichlet boundary conditions (0.1){(-Δ)psu=a(x)|u|q-2u+2αα+βc(x)|u|α-2u|v|β,in Ω,(-Δ)psu=b(x)|v|q-2v+2βα+βc(x)|u|α|v|β-2v,in Ω,u=v=0,inRn\\Ω,where O is a smooth bounded domain in Rn, n > ps with s ɛ (0, 1) fixed, a(x), b(x), c(x) ≥ 0 and a(x), b(x), c(x) ɛ L∞(0), 1 <q < p and α, β > 1 satisfy p < α + β < p*, p* =npn-ps. By Nehari manifold and fibering maps with proper conditions, we obtain the multiplicity of solutions to problem (0.1).

Dual nature solution of water functionalized copper nanoparticles along a permeable shrinking cylinder: FDM approach

Present framework is established to deal the characteristics of axisymmetric mixed convection flow with heat transfer of water based copper (Cu-water) nanofluid along a porous shrinking cylinder with slip effects. The physical problem is modeled in the form of set of partial differential equations (PDEs) including conservation of mass, momentum and energy equations along with defined boundary conditions. Such PDEs are adapted to ordinary differential equations (ODEs) by utilizing similarity transformation technique. System of equations is emerged with the expressions of nanoparticle and based fluid. Numerical solution of governing ODEs is sought by using Finite Difference Method (FDM) against the range of several pertinent physical parameters. Base fluid (water) is analyzed in the presence of nanoparticles. The model encounters buoyancy opposing and assisting flow regions. Present study reveals that the solution in the assisting flow region is unique whereas multiple solutions exist in the opposing flow region. The results also indicate the existence of multiple solutions for certain amount of mass suction parameter. Moreover, increase in slip parameters and nanoparticle volume fraction enhances the range of suction parameter where the similarity solutions exist. It is further found that due to increase of nanoparticle volume fraction the skin friction increases whereas heat transfer rate decreases.

Analytical prediction of multiple solutions for MHD Jeffery–Hamel flow and heat transfer utilizing KKL nanofluid model

In this framework, the novel analytical approach is presented to predict the dual solutions of Jeffery–Hamel (JH) transport model utilizing KKL (Koo–Kleinstreuer–Li) Al2O3 model with magnetic field, Ohmic heating and viscous dissipation. The predictor homotopy analysis method (PHAM) is applied to realize the existence of multiple solutions (bifurcation) for stretching/shrinking parameter and channel angle. It is observed that the dual solutions exist only for convergent channel. The eigenvalue problem is constructed to perform stability analysis which shows the physically stability of the upper branch. A numerical validation with Runge–Kutta–Fehlberg (RKF) shooting method using MATLAB is also carried out for verification. The Reynolds number is responsible to increase the velocity of fluid for both branches of the solution. For the increasing values of Ec and M, the Nusselt number decreases and increases respectively.

A novel kind of multiple steady states characteristics in the dividing wall column

Abstract In this article, one kind of multiple steady states (MSS) phenomenon was investigated for a dividing wall column (DWC). The four-section model constructed in Aspen Plus was employed to simulate two DWC cases: mixture of n-hexane, n-heptane and n-octane; system of methanol, ethanol and n-propanol. It can be seen that there is a range of vapor split ratio in which multiple solutions of reflux ratio exist for fixed DWC configuration with the same feed and product streams. The width and the curve shapes of the MSS region, and the number of solutions change with the liquid split ratio. This MSS phenomenon was further explained using the component recovery around the prefractionator and the component recycling flow inside the DWC. This MSS phenomenon is helpful for DWC design by knowing the probable existence of multiple solutions in advance.

Multiple solutions of asymmetric potential bistable energy harvesters: numerical simulation and experimental validation

In this paper, we investigate the multiple solutions of nonlinear asymmetric potential bistable energy harvesters (BEHs) under harmonic excitations. Basins of attraction under certain excitations explain the existence of multiple solutions in the asymmetric potential BEHs and indicate that the asymmetric system has a higher probability to oscillate in the deeper potential well under low and moderate excitation levels. Thus, the appearance of asymmetric potentials in BEHs has a negative influence on the output performance. Average output powers under different excitation frequencies and initial conditions illustrate that the asymmetric potential BEHs are more likely to achieve high-energy branch (HEB) with initial conditions in the shallower potential well, and the probability is influenced by the degree of asymmetry of the BEHs. Finally, experiments are carried out, and results under constant and sweep frequency excitations demonstrate that the output performance will be actually improved for the asymmetric potential BEHs if the initial oscillations are from the shallower potential well.

Existence of multiple solutions to an elliptic problem with measure data

In this paper we prove the existence of multiple nontrivial solutions of the following equation: $$\begin{aligned} -\Delta _{p}u &= \lambda |u|^{q-2}u+f(x,u)+\mu \quad \text {in}\,\,\Omega u & = 0 \quad \text {on}\,\, \partial \Omega ; \end{aligned}$$ where $$\Omega \subset \mathbb {R}^N$$ is a smooth bounded domain with $$N \ge 3$$ , $$1< q^{\prime }< q < p-1; \; \lambda ,\;$$ and f satisfies certain conditions, $$\mu >0$$ is a Radon measure, $$q^{\prime }=\frac{q}{q-1}$$ is the conjugate of q.

Existence of Multiple Solutions to a Discrete Fourth Order Periodic Boundary Value Problem via Variational Method

By using variational methods and critical point theory, we obtain criteria for the existence of at least three solutions for a generalized fourth order nonlinear difference equation together with periodic boundary conditions. Various special cases of the above problem are discussed. An example is included to illustrate the results.

Multiplicity and asymptotic behavior of solutions for Kirchhoff type equations involving the Hardy\u2013Sobolev exponent and singular nonlinearity

In this paper, we study a class of critical elliptic problems of Kirchhoff type: \t\t\t[a+b(∫R3|∇u|2−μu2|x|2dx)2−α2](−Δu−μu|x|2)=|u|2∗(α)−2u|x|α+λf(x)|u|q−2u|x|β,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\biggl[a+b \\biggl( \\int_{\\mathbb{R}^{3}}\\vert \\nabla u\\vert ^{2}-\\mu \\frac{u^{2}}{\\vert x\\vert ^{2}}\\,dx \\biggr)^{\\frac{2-\\alpha }{2}} \\biggr]\\biggl(-\\Delta u- \\mu \\frac{u}{\\vert x\\vert ^{2}}\\biggr) = \\frac{\\vert u\\vert ^{2^{*}(\\alpha )-2}u }{\\vert x\\vert ^{\\alpha }}+\\lambda \\frac{f(x)\\vert u\\vert ^{q-2}u }{\\vert x\\vert ^{\\beta }}, $$\\end{document} where a,b>0, mu in [0,1/4), alpha , beta in [0,2), and qin (1,2) are constants and 2^{*}(alpha )=6-2alpha is the Hardy–Sobolev exponent in mathbb{R}^{3}. For a suitable function f(x), we establish the existence of multiple solutions by using the Nehari manifold and fibering maps. Moreover, we regard b>0 as a parameter to obtain the convergence property of solutions for the given problem as bsearrow 0^{+} by the mountain pass theorem and Ekeland’s variational principle.

On the existence of multiple solutions to boundary value problems for semilinear elliptic degenerate operators

ABSTRACTIn this paper, we study the existence of multiple solutions for the boundary value problem where Ω is a bounded domain with smooth boundary in are Carathéodory functions and is the Grushin operator. This result is a generalization of that of Rabinowitz and of Luyen and Tri.

Existence and multiplicity of solutions for ‐Laplacian equations in

AbstractIn this paper, we study a class of sublinear or superlinear ‐Laplacian equations in . Some new criteria to guarantee that the existence of multiple solutions for the considered problem is established by using the Morse theory and minimax methods.