Existence Of Multiple Solutions Research Articles

Multiple solutions with constant sign of a Dirichlet problem for a class of elliptic systems with variable exponent growth

We investigate the followingDirichlet problem with variable exponents: \begin{document}$\left\{ \begin{align} &-{{\vartriangle }_{p(x)}}u=\lambda \alpha (x)|u{{\text{ }\!\!|\!\!\text{ }}^{\alpha (x)-2}}u|v{{\text{ }\!\!|\!\!\text{ }}^{\beta (x)}}+{{F}_{u}}(x,u,v),\text{ in }\Omega , &-{{\vartriangle }_{q(x)}}v=\lambda \beta (x)\text{ }\!\!|\!\!\text{ }u{{|}^{\alpha \left( x \right)}}|v{{|}^{\beta (x)\text{-2}}}v+{{F}_{v}}(x,u,v),\text{ in} \Omega , u'>We present here, in the system setting, a new set of growth conditions under which we manage to use a novel method to verify the Cerami compactness condition. By localization argument, decomposition technique and variational methods, we are able to show the existence of multiple solutions with constant sign for the problem without the well-knownAmbrosetti-Rabinowitz type growth condition. More precisely, we manage to show that the problem admitsfour, six and infinitely many solutions respectively.

Homogeneous–Heterogeneous Reactions in Boundary-Layer Flow of a Nanofluid Near the Forward Stagnation Point of a Cylinder

A mathematical model describing the homogeneous–heterogeneous reactions in the vicinity of the forward stagnation point of a cylinder immerged in a nanofluid is established. We assume that the homogeneous reaction is given by isothermal cubic autocatalator kinetics, while the heterogeneous reaction is chosen as first-order kinetics. The existence of multiple solutions through hysteresis bifurcations is discussed in detail for the various diffusion coefficients of reactant and autocatalyst.

Critical Nonlocal Systems with Concave-Convex Powers

Abstract By using the fibering method jointly with Nehari manifold techniques, we obtain the existence of multiple solutions to a fractional p-Laplacian system involving critical concave-convex nonlinearities, provided that a suitable smallness condition on the parameters involved is assumed. The result is obtained although there is no general classification for the optimizers of the critical fractional Sobolev embedding.

Multiple solutions of the $$L_{p}$$ L p -Minkowski problem

In this paper, we study the existence of multiple solutions to the \(L_{p}\)-Minkowski problem. We prove if \(p 0\), there exists a smooth positive function f on \(\mathbb S^n\) such that the \(L_{p}\)-Minkowski problem admits at least N different smooth solutions. We also construct nonsmooth, positive function f for which the \(L_{p}\)-Minkowski problem has infinitely many \(C^{1,1}\) solutions.

Multiple solutions for impulsive problems with non-autonomous perturbations

In this article, we study the existence of multiple solutions for nonlinear impulsive problems with small non-autonomous perturbations. We show the existence of at least three distinct classical solutions by using variational methods and a three critical points theorem.

Revealing the Impact of Multiple Solutions in DCOPF on the Risk Assessment of Line Cascading Failure in OPA Model

Cascading failure is one of the most damaging events to power systems. To mitigate the risk, it is essential to first evaluate the risk and then take corresponding protective measures. However, the issue of multiple solutions in DCOPF, which is a tool used in the existing OPA risk assessment models and a linear programming (LP) problem, has not been considered. In this letter, we show that the risk may be seriously under-estimated if we do not consider the existence of multiple solutions. The risk is evaluated by generating two dispatch plans based on a bilevel mixed integer linear programming (MILP) model for DCOPF. The solution to the bilevel MILP model is obtained by solving one LP and one MILP. Simulation results based on the IEEE 118-bus and 300-bus systems demonstrate the impact of multiple solutions in DCOPF and effectiveness of the proposed approach.

Instabilities and chaos in low-Prandtl number rayleigh-Bénard convection

We investigate instabilities and chaos in Rayleigh-Bénard convection (RBC) of low Prandtl-number (Pr) fluids with stress-free boundary conditions. Three dimensional direct numerical simulations (DNS) of the governing equations of RBC are done for investigation. DNS shows the existence of multiple solutions near the onset of convection. To understand the origin of these solutions and their bifurcations, we construct a low dimensional model from the DNS data. The low dimensional model unfolds a rich bifurcation structure in RBC of low Prandtl-number fluids. The bifurcation structure involves various bifurcations including pitchfork, Neimark-Sacker, Homoclinic gluing, Hopf bifrucations, and attractor merging crisis near the onset of convection. The results of the model are consistent with that of the DNS.

Multiple solutions of nonlinear equations involving the square root of the Laplacian

In this paper, we examine the existence of multiple solutions of parametric fractional equations involving the square root of the Laplacian in a smooth bounded domain () and with Dirichlet zero-boundary conditions, i.e.The existence of at least three -bounded weak solutions is established for certain values of the parameter requiring that the nonlinear term f is continuous and with a suitable growth. Our approach is based on variational arguments and a variant of Caffarelli–Silvestre’s extension method.

Iterative scheme for a coupled system of fractional‐order differential equations with three‐point boundary conditions

We investigate sufficient conditions for existence of multiple solutions to a coupled system of fractional‐order differential equations with three‐point boundary conditions. By coupling the method of upper and lower solutions together with the method of monotone iterative technique, we develop conditions for iterative solutions. Based on these conditions, we study maximal and minimal solutions to the problem under consideration. We also study error estimates and provide an example. Copyright © 2016 John Wiley & Sons, Ltd.

Multiple solutions for semi-linear corner degenerate elliptic equations with singular potential term

The present paper is concern with the Dirichlet problem for semi-linear corner degenerate elliptic equations with singular potential term. We first give the preliminary of the framework and then discuss the weighted corner type Hardy inequality. By using the variational method, we prove the existence of multiple solutions for the Dirichlet boundary-value problem.

Existence of Multiple Solutions for ap-Laplacian System in ℝNwith Sign-changing Weight Functions

AbstractIn this paper, we consider the quasi-linear elliptic problemwhereand the weight H(x); h1(x); h2(x) are continuous functions that change sign in ℝN. We will prove that the problem has at least two positive solutions by using the Nehari manifold and the fibering maps associated with the Euler functional for this problem.

Form drag effect on the onset of non-linear convection and Hopf bifurcation in binary fluid saturating a tall porous cavity

This paper reports a numerical study of natural convection in at all porous enclosure filled with a binary fluid. The Darcy–Dupuis model, which includes effects of the form drag force, is adopted to describe the flow in the porous medium. The two vertical walls of the cavity are subject to constant gradients of temperature while the two horizontal ones are kept adiabatic and impermeable. Concentration gradients are assumed to be induced either by the imposition of constant gradients of solute on the vertical walls of the system (a=0; double diffusive convection) or by the Soret effect (a=1). Governing parameters of the problem under study are the thermal Rayleigh number RT, form drag parameter G, buoyancy ratio φ, Lewis number. Le, normalized porosity ε, and aspect ratio of the cavity A. The case of equal and opposing thermal and solutal buoyancy forces, φ=-1, is considered. For this situation, an equilibrium solution corresponding to the rest state is possible and the resulting onset of motion can be either supercritical or subcritical. A semi-analytical solution, valid for an infinite layer (A≫1) assuming parallel flow, is derived. Based on the linear stability theory, the onset of motion from the rest state is predicted for both double diffusive and Soret convection. The onset of Hopf bifurcation, characterizing the transition from a convective steady state to oscillatory state, is also studied. The influence of the governing parameters on the onset of motion and the resulting fluid flow, temperature and concentration fields is discussed in detail. The existence of supercritical, subcritical and oscillatory convective modes is demonstrated. A good agreement is found between the predictions of the parallel flow approximation and the numerical results obtained by solving the full governing equations. The existence of multiple solutions and traveling waves for a given set of the governing parameter is demonstrated and leads to the existence of a bistability phenomenon. Overall, the form drag behaves as a stabilizing effect and is seen to affect considerably the onset of subcritical convection and Hopf bifurcation.

Gaps of consecutive eigenvalues of Laplace operator and the existence of multiple solutions for superlinear elliptic problem

This paper is mainly related to multiple nontrivial solutions of the elliptic boundary value problem{−Δu=|u|p−2u+f(x,u),x∈Ω,u=0,x∈∂Ω, for p∈(2,2⁎). It is reasonable to guess that for dim⁡Ω≥2 above problem possesses infinitely many distinct solutions since this is proved to be true for ODE. However, so far one does not even know if there exists a fourth nontrivial solution. By using a new homological linking theorem, Morse theory, and some precise estimates we disclose the relationship among the gaps of consecutive eigenvalues of Laplace operator, growth trend of nonlinear terms and the existence of multiple solutions of superlinear elliptic boundary value problem. Moreover, as p is close to 2, we get the fourth nontrivial solution under appropriate hypotheses, where f(x,u) satisfies Ambrosetti–Rabinowitz condition.

Multiplicity of solutions for critical Kirchhoff type equations

ABSTRACTWe prove the existence of multiple solutions for critical Kirchhoff equations involving positive operators in closed manifolds.

Existence of multiple solutions of some second order impulsive differential equations

This paper uses critical point theory and variational methods to investigate the multiple solutions of a boundary value problem for second order impulsive differential equations. The conditions for the existence of multiple solutions are established.

Multiple solutions of the Kirchhoff-type problem in RN

Abstract In this paper, we concern with a class of quasilinear Kirchhoff-type problem. By using the Ekeland’s Variational Principle and Mountain Pass Theorem, the existence of multiple solutions is obtained. Besides, we also take this problem as an example to give the main frame of using critical point theory to find the weak solutions of nonlinear partial differential equation.

Multiple solutions for a nonlinear discrete fourth order p-Laplacian equation

In this paper we study the existence of multiple solutions for a nonlinear discrete fourth order $p$-Laplacian equation. Similar kind of problems appears in some models from population biology. The proof of the main results is based on the three critical points theorem due to B. Ricceri. An example is given. We obtain criteria for the existence of three solutions for the following fourth order problem ...

Multiplicity of solutions to a four-point boundary value problem of a differential system via variational approach

By constructing an adequate real functional and choosing an appropriate admissible function space, the existence of multiple solutions to a four-point boundary value problem, which may be taken as an extension of Sturm-Liouville boundary value problems, is proved via a variational approach for a second-order differential system with a p-Laplacian.

Critical growth fractional elliptic systems with exponential nonlinearity

We study the existence of positive solutions for the system of fractional elliptic equations of the type, (−Δ)12u=pp+qλf(x)∣u∣p−2u∣v∣q+h1(u,v)eu2+v2,in(−1,1),(−Δ)12v=qp+qλf(x)∣u∣p∣v∣q−2v+h2(u,v)eu2+v2,in(−1,1),u,v>0in (−1,1),u=v=0inR∖(−1,1) where 1<p+q<2, h1(u,v)=(α+2u2)∣u∣α−2u∣v∣β,h2(u,v)=(β+2v2)∣u∣α∣v∣β−2v and α+β>2. Here (−Δ)12 is the fractional Laplacian operator. We show the existence of multiple solutions for suitable range of λ by analyzing the fibering maps and the corresponding Nehari manifold. We also study the existence of positive solutions for a superlinear system with critical growth exponential nonlinearity.

On Nonhomogeneous Elliptic Equations with Critical Sobolev Exponent and Prescribed Singularities

In this paper we consider a class of nonhomogeneous elliptic equations involving multi-polar Hardy type potentials and a Sobolev critical nonlinearity in an open domain of $\mathbb{R}^{N}$, $N \geq 3$. By Ekeland's Variational Principle and the Mountain Pass Lemma, we prove the existence of multiple solutions under sufficient conditions on the data and the considered parameters.