Multiplicity Of Nontrivial Solutions Research Articles

GLOBAL BIFURCATION RESULT FOR DISCRETE BOUNDARY VALUE PROBLEM INVOLVING THE MEAN CURVATURE OPERATOR

<abstract> In this paper, by applying bifurcation technique, we obtain that there are two distinct unbounded continua $ \mathcal{C}_k^+ $ and $ \mathcal{C}_k^- $ for a class of discrete Dirichlet problem involving the mean curvature operator which bifurcate from intervals of the line of trivial solutions. Under some suitable conditions on nonlinear term near at the origin, we will show the existence and multiplicity of nontrivial solutions. </abstract>

Some uniqueness results in quasilinear subhomogeneous problems

We establish uniqueness results for quasilinear elliptic problems through the criterion recently provided in Bonheure et al. (Trans Amer Math Soc 370(10):7081–7127, 2018). We apply it to generalized p-Laplacian subhomogeneous problems that may admit multiple nontrivial nonnegative solutions. Based on a generalized hidden convexity result, we show that uniqueness holds among strongly positive solutions and nonnegative global minimizers. Problems involving nonhomogeneous operators as the so-called (p, r)-Laplacian are also treated.

Estimation of Kinetic Parameters of Additives By Semi-Analytical Solution

The electrodeposition of Cu and other metals is an essential step in the interconnect fabrication, 3-D integration, and various packaging technologies. Such electrodeposition processes are enabled by a group of organic additives. Furthermore, strong agitation is often used to expedite the electrodeposition rate when the processing time becomes critical for large features, which however introduces non-uniform mass transport within a single feature and between different features. A numerical description of how organic additives behave in conjunction with external flow at various feature geometry and dimensions is a challenging task, but much needed for chemistry and process development. Among the various requirements, the additive behavior or their electrochemical impacts on Cu deposition needs to be precisely described.This talk reports an effort to obtain the numerical parameters that describe the additive behaviors during Cu deposition on rotating disc electrodes (RDE). A one-dimensional semi-analytical model is developed for Cu electrodeposition from a two-component additive system. The steady-state mass transport differential equations of each chemical species, including diffusion and convection, are solved analytically at different potentials, while the boundary conditions at the RDE surface are coupled with each other through the surface coverages of adsorbed species. The steady state of such surface coverage results from a dynamic equilibrium between three co-existing processes, adsorption, desorption and incorporation (consumption), namely, Adsorption Rate – Desorption Rate – Consumption Rate = 0 A set of solutions, including the concentration profile of each species in electrolyte as well as the surface coverage of each adsorbate, can be obtained at a fixed potential. The equations are first solved considering only the suppressor for a range of over-potential. It was found that, for a specific range of voltages and kinetic parameters, multiple non-trivial solutions of surface coverage exist. In other words, multiple steady states exist at a same applied potential. As shown in Figure 1, this translates to a S-shaped negative differential resistance (NDR) feature in cyclic voltammogram (CV), consistent with experimental observation reported in literature. The numerically calculated NDR region was found sensitive to the additive concentration and therefore extremely useful in determining the electrochemical parameters using multiple experimental datasets with different concentrations. The swarm optimization algorithm used for parameter fitting and the correlation between multiple numerical solutions with the NDR region will be discussed in details in the talk. Figure 1

Existence of Solutions for a Schrödinger–Poisson System with Critical Nonlocal Term and General Nonlinearity

We study the existence and multiplicity of nontrivial solutions for a Schrödinger–Poisson system involving critical nonlocal term and general nonlinearity. Based on the variational method and analysis technique, we obtain the existence of two nontrivial solutions for this system.

Existence of Positive and Sign-Changing Solutions to a Coupled Elliptic System with Mixed Nonlinearity Growth

In the present paper, we make a rigorous study of the solitary wave solutions to a coupled Schrodinger system with quadratic and cubic nonlinearity. This kind of system of Schrodinger equations arises from optics theory. First, the existence and nonexistence of nontrivial solutions, respectively, in focusing and defocusing cases are considered. Second, we prove the existence of multiple nontrivial solutions by using the Crandall–Rabinowitz local bifurcation theorems and calculate the exact Morse index of these solutions. Third, the continuous dependence on the parameter and asymptotic behavior of positive ground state solutions in the focusing case are also established. Particularly, from the mathematical point of view, we prove the behavior of positive solution coincides with the physical phenomena of Bang et al. (Opt Lett 22(22):1680–1682, 1997; Phys Rev E (3) 58(4):5057–5069, 1998). Finally, we prove the existence of sign-changing solutions.

Critical groups and multiple solutions for Kirchhoff type equations with critical exponents

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

Boundary Value Problem for a Second-Order Difference Equation with Resonance

In this paper, we study the existence and multiplicity of nontrivial solutions of a second-order discrete boundary value problem with resonance and sublinear or superlinear nonlinearity. The main methods are based on the Morse theory and the minimax methods. In addition, some examples are given to illustrate our results.

Remarks on infinitely many solutions for a class of Schrödinger equations with sublinear nonlinearity

In this paper, we study the existence of multiple nontrivial solutions for the following semilinear Schrödinger equation: where the potential V is continuous and allowed to be sign‐changing and f(x,u) satisfies more general sublinear growth conditions than those in previous studies. The first result of this paper obtains the existence of a nontrivial solution of (1). The second establishes an existent result on infinitely many nontrivial solutions of (1) by using a variant fountain theorems.

Notes on Nontrivial Multiple Periodic Solutions for Second-Order Discrete Hamiltonian System

We obtain new existence of multiple nontrivial M-periodic solutions for a class of second-order discrete Hamiltonian system with the force F(t, x) being neither supquadratic nor subquadratic growth in x. Moreover, we exhibit four instructive examples to illustrate our results.

Existence of non-trivial solutions for fractional Schrödinger-Poisson systems with subcritical growth

In this paper, we are concerned with the following fractional Schrodinger-Poisson system:    (−∆s)u + u + λφu = µf(u) +|u|p−2|u|, x ∈R3 (−∆t)φ = u2, x ∈R3 where λ,µ are two parameters, s,t ∈ (0,1] ,2t + 4s > 3 ,1 < p ≤ 2∗ s and f : R → R is continuous function. Using some critical point theorems and truncation technique, we obtain the existence and multiplicity of non-trivial solutions with the help of the variational methods.

Nontrivial solutions for the fractional Schrödinger‐Poisson system with subcritical or critical nonlinearities

In this paper, we are concerned with a class of fractional Schrödinger‐Poisson system involving subcritical or critical nonlinearities. By using the Nehari manifold and variational methods, we obtain the existence and multiplicity of nontrivial solutions.

Multiple nontrivial solutions of a semilinear elliptic problem with asymmetric nonlinearity

In this paper we study the existence and multiplicity of solutions of the boundary–value problem(1){−Δu=−λ|u|q−2u+au+b(u+)p−1, in Ω;u=0, on ∂Ω, where Δ denotes the N-dimensional Laplacian, Ω is a bounded domain with smooth boundary, ∂Ω, in RN(N⩾3), u+ denotes the positive part of u:Ω→R, 1<q<2<p<2⁎=2N/(N−2), λ>0, a∈R and b>0. Using infinite-dimensional Morse Theory, we extend the results of Paiva and Presoto [14] and establish some conditions for the existence of at least four nontrivial solutions of (1).

Multiplicity results for biharmonic equations involving multiple Rellich-type potentials and critical exponents

In this paper, a biharmonic equation is investigated, which involves multiple Rellich-type potentials and a critical Sobolev exponent. By using variational methods and analytical techniques, the existence and multiplicity of nontrivial solutions to the equation are established.

Existence and multiplicity of solutions for Kirchhoff type equations involving fractional p-Laplacian without compact condition

The purpose of this paper is mainly to investigate the following fractional Kirchhoff equation in $${{\mathbb {R}}}^N$$$$\begin{aligned} \left( a+b\iint _{{{\mathbb {R}}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\mathrm{d}x\mathrm{d}y\right) ^{p-1} (-\Delta )^s_p u+\lambda V(x)|u|^{p-2}u=f(x,u), \end{aligned}$$where $$0 0$$ are constants, $$\lambda $$ is a parameter, V is sign-changing potential function satisfying some assumptions which may not guarantee the compactness of the corresponding Sobolev embedding. Under some suitable conditions, we prove the existence and multiplicity of nontrivial solutions by applying some new tricks for the above equation.

The stationary quantum Zakharov system perturbed by a local nonlinearity

We study a class of stationary quantum Zakharov systems with a perturbation as follows Δ2u−Δu+λV(x)u=uϕ+μf(x)|u|p−2uin R3,−Δϕ+ϕ=u2,in R3,where λ>0, μ∈R, p>1, and both f(x) and V(x) are nonnegative functions. By using the Nehari manifold, we prove the nonexistence, existence and multiplicity of nontrivial solutions, depending on the parameters λ,μ and p.

Multiplicity of periodic bouncing solutions for generalized impact Hamiltonian systems

Applying the Generalized Nonsmooth Saddle Point Theorem, we obtain multiple nontrivial periodic bouncing solutions for systems ddot{x}=f(t,x) with new conditions. In particular, we generalize the collision axis from x=0 to the axis x=a, where a is an arbitrary constant.

Existence and multiplicity of non-trivial solutions for the fractional Schr\xf6dinger\u2013Poisson system with superlinear terms

In this paper, we study the following fractional Schrödinger–Poisson system with superlinear terms \t\t\t{(−Δ)su+V(x)u+K(x)ϕu=f(x,u),x∈R3,(−Δ)tϕ=K(x)u2,x∈R3,\\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}$$ \\textstyle\\begin{cases} (-\\Delta )^{s}u+V(x)u+K(x)\\phi u=f(x,u), & x \\in \\mathbb{R}^{3}, \\\\ (-\\Delta )^{t}\\phi =K(x)u^{2}, & x \\in \\mathbb{R}^{3}, \\end{cases} $$\\end{document} where s,tin (0,1), 4s+2t>3. Under certain assumptions of external potential V(x), nonnegative density charge K(x) and superlinear term f(x,u), using the symmetric mountain pass theorem, we obtain the existence and multiplicity of non-trivial solutions.

Existence of multiple non-trivial solutions for a nonlocal problem

In this paper, we are concerned with the following general nonlocal problem\begin{equation*}\begin{cases}-\mathcal{L}_K u=\lambda_1u+f(x,u)& \text{in} \Omega,\\u=0& \text{in} \mathbb{R}^N\backslash\Omega,\end{cases}\end{equation*}where $\lambda_1$ denotes the first eigenvalue of the nonlocal integro-differential operator $-\mathcal{L}_K$, $\Omega\subset\mathbb{R}^N$ is open, bounded domain with smooth boundary. Under several structural assumptions on $f$, we verify that the problem possesses at least two non-trivial solutions and locate the region in different parts of the Hilbert space by variational method. As a particular case, we derive an existence theorem for the following equation driven by the fractional Laplacian\begin{equation*}\begin{cases}(-\Delta)^su=\lambda_1u+f(x,u)& \text{in} \Omega,\\u=0& \text{in} \mathbb{R}^N\backslash\Omega.\end{cases}\end{equation*}

Multiple Nontrivial Solutions for a Class of Biharmonic Elliptic Equations with Sobolev Critical Exponent

In this paper, we study the existence and multiplicity of nontrivial solutions for a class of biharmonic elliptic equation with Sobolev critical exponent in a bounded domain. By using the idea of the previous paper, we generalize the results and prove the existence and multiplicity of nontrivial solutions of the biharmonic elliptic equations.

Multiplicity results for a class of fractional differential equations with impulse

In this paper, we apply Morse theory, local linking arguments and the Clark theorem to a study of the multiplicity of nontrivial solutions for a class of impulsive fractional differential equations with Dirichlet boundary conditions.