Nontrivial Solutions Research Articles

Nontrivial solutions for a general p(x)-Laplacian Robin problem

We establish the existence of multiple nontrivial solutions for a class of $p(x)$-Laplacian Robin problem. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined withadequate variational methods and a variant of the Mountain Pass lemma.

Rotating SU(2) gluon matter and deconfinement at finite temperature

In this work a non-abelian gauge theory is reformulated in a rotating frame. With this new formalism, the influence of the background rotation on the color deconfinement transition for a pure SU(2) gluon system has been studied. The KvBLL caloron, which is a color neutral and topologically nontrivial solution of Yang-Mills equation at finite temperature, is reexamined under rotation and adopted as the confined vacuum of such a system. With rotation-modified solutions of the caloron's constituent solitons, i.e. dyons, the non-perturbative part of effective potential of the rotating system has been obtained. Combining with the perturbative potential by Gaussian fluctuations, the critical temperature of confinement-deconfinement phase transition is investigated. It is found that neither the rotational semi-classical potential nor Gaussian fluctuations can confine color charges more tightly when the rotation becomes faster. While only a stronger coupling constant is able to make the critical temperature increase with angular velocity, as what indicated in lattice simulations. And it is found a non-monotonic dependence of the critical temperature on the angular velocity is established because of the competition between these two contrary contributions.

Singular Hamiltonian elliptic systems involving double exponential growth in dimension two

In this research, we are interested to investigate the existence of nontrivial weak solutions to the following Hamiltonian elliptic system −div(ω(x)∇u)=g(v)|x|a,x∈B1(0),−div(ω(x)∇v)=f(u)|x|b,x∈B1(0),with Dirichlet boundary conditions, where a,b∈[0,2), the weight ω(x) is of logarithmic type and the nonlinearities f and g possess double exponential growth. To establish the existence of solutions, our approach involves utilizing the linking theorem and a finite-dimensional approximation.

Soliton dynamics of the KdV–mKdV equation using three distinct exact methods in nonlinear phenomena

Abstract The KdV–mKdV equation is investigated in this study. This equation is a useful tool to model many nonlinear phenomena in the fields of fluid dynamics, quantum mechanics, and soliton wave theory. The exact soliton solutions of the KdV–mKdV equation are extracted using three distinct exact methods, namely, the generalized projective Riccati equation method, the modified auxiliary equation method, and the generalized unified method. Many novel soliton solutions, including kink, periodic, bright, dark, and singular dark–bright soliton solutions, are obtained. Rational functions, exponential functions, trigonometric functions, and hyperbolic functions are contained in the acquired nontrivial exact solutions. The graphical simulation of some obtained solutions is depicted using 3D plots, 2D contour plots, density plots, and 2D line plots. For the first time, the KdV–mKdV equation is investigated using the proposed three exact methods, and many novel solutions, such as dark, bright, and dark–bright singular soliton solutions, are determined, which have never been reported in the literature.

On Liouville-type theorems for 𝑘-Hessian equations with gradient terms

In this paper, we investigate several Liouville-type theorems related to k k -Hessian equations with non-linear gradient terms. More specifically, we study non-negative solutions to S k [ D 2 u ] ≥ h ( u , | D u | ) S_k[D^2u]\ge h(u,|Du|) in R n \mathbb {R}^n . The results depend on some qualified growth conditions of h h at infinity. A Liouville-type result to subsolutions of a prototype equation S k [ D 2 u ] = f ( u ) + g ( u ) ϖ ( | D u | ) S_k[D^2u]=f(u)+g(u)\varpi (|Du|) is investigated. A necessary and sufficient condition for the existence of a non-trivial non-negative entire solution to S k [ D 2 u ] = f ( u ) + g ( u ) | D u | q S_k[D^2u]=f(u)+g(u)|Du|^q for 0 ≤ q > k + 1 0\le q>k+1 is also given.

Nontrivial solutions for indefinite Schrödinger–Poisson systems and Kirchhoff equations

In this paper we consider 4-superlinear Schrödinger–Poisson systems. The potential V here is indefinite so that the Schrödinger operator −Δ+V possesses a finite-dimensional negative space. By Morse theory, we obtain the existence of nontrivial solutions for this problem. Similar result for Kirchhoff equations on R3 is also presented.

The Gauge Equation in Statistical Manifolds: An Approach through Spectral Sequences

The gauge equation is a generalization of the conjugacy relation for the Koszul connection to bundle morphisms that are not isomorphisms. The existence of nontrivial solution to this equation, especially when duality is imposed upon related connections, provides important information about the geometry of the manifolds under consideration. In this article, we use the gauge equation to introduce spectral sequences that are further specialized to Hessian structures.

Oscillatory properties of selfadjoint boundary problems of the fourth order

A series of results and methods is presented, which make it possible to trace the relationship between the number of inner zeros of nontrivial solutions of fourth order selfadjoint boundary problems with separated boundary conditions and the negative inertia index.

Elliptic p-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by−Δp1u1=−|u1|p1−2u1in Ω,−Δp2u2=−|u2|p2−2u2in Ω,|∇u1|p1−2∇u1⋅ν=g1(x,u1,u2)on ∂Ω,|∇u2|p2−2∇u2⋅ν=g2(x,u1,u2)on ∂Ω, where ν(x) is the outer unit normal of Ω at x∈∂Ω, Δpi denotes the pi-Laplacian and gi:∂Ω×R×R→R are Carathéodory functions that satisfy general growth and structure conditions for i=1,2. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of gi near zero related to the first eigenvalue of the pi-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, (g1,g2)=∇g with a smooth function (s1,s2)↦g(x,s1,s2). By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the pi-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution.

On a class of fractional p( x, y)−Kirchhoff type problems with indefinite weight

This paper is concerned with a class of fractional \(p(x,y)-\)Kirchhoff type problems with Dirichlet boundary data along with indefinite weight of the following form \begin{equation*} \left\lbrace\begin{array}{ll} M\left(\int_{Q}\frac{1}{p(x,y)}\frac{|u(x)-u(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dx\,dy\right)\\ (-\triangle_{p(x)})^s+|u(x)|^{q(x)-2}u(x) & \\ =\lambda V(x)|u(x)|^{r(x)-2}u(x)& \text{in }\Omega,\\ u=0, & \text{in }\mathbb{R}^N\Omega. \end{array}\right. \end{equation*} By means of direct variational approach and Ekeland’s variational principle, we investigate the existence of nontrivial weak solutions for the above problem in case of the competition between the growth rates of functions \(p\) and \(r\) involved in above problem, this fact is essential in describing the set of eigenvalues of this problem.

Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system

We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative components increases with the number of equations. Our proofs use a combination of four key elements: a convenient variational approach, the asymptotic behavior of solutions (concentration), the Lusternik–Schnirelman theory, and new estimates on the category of suitable configuration spaces.

Existence and stability of steady-state bifurcations in a prey–predator system with a nonmonotonic functional response

A cross-diffusion prey–predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied. By considering the diffusive rate of the prey as a bifurcation parameter, we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species. First a priori estimate for positive steady states is obtained. Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue. The structure of the global steady-state bifurcation is discussed. We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory. To illustrate our theoretical results some numerical simulations are given. Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.

On the Pohozaev identity for the fractional p$p$‐Laplacian operator in RN$\mathbb {R}^N$

AbstractIn this paper, we show the existence of a nontrivial weak solution for a nonlinear problem involving the fractional ‐Laplacian operator and a Berestycki–Lions type nonlinearity. This solution satisfies a Pohozaev identity. Moreover, we prove that any sufficiently smooth solution fulfills the Pohozaev identity.

New kink-type solution of the equation for artificial axon

In the paper a (1 + 1)-dimension equation of motion for the artificial axon is considered. The artificial axon is a dynamical structure like a neuron. They are widely used in biophysics, for example, in studying the physiological processes. A topological non-trivial solution of one-kink type for this equation is constructed in an analytical form. The modified direct Hirota method for solving the nonlinear partial derivatives equations is applied. The special cases are considered for different voltages on the contacts of axon.

Multiplicity and limit of solutions for logarithmic Schrödinger equations on graphs

Let Ω be a finite connected subset of a locally finite graph G = (V, E) with the vertex set V and the edge set E. We investigate the logarithmic Schrödinger equation on Ω with the nonlinear term |u|p−2u log u2. For p > 2, through two different approaches which are the Brouwer degree theory and mountain-pass theorem, we obtain the existence of ground state solutions. We also apply the Brouwer degree theory together with the constraint variational method to prove that the equation admits a sign-changing solution which implies the multiplicity of solutions to the equation. Finally, we illustrate that as p → 2, up to a subsequence, the solutions for p > 2 shall converge to a non-trivial solution of the equation with p = 2.

On Integer Solutions of the Ternary Bi-Quadratic Diophantine Equation 14(x^2+y^2)-24xy=17z^4

Abstract: This analysis is discussed on the non-trivial distinct integral solution to the ternary bi-quadratic diophantine equation 4 ) 24 17 2 2 14(x  y  xy  z , in four dissimilar patterns. Few connections between the solutions are obtained and also unique polygonal numbers are described

Infinitely many solutions for three quasilinear Laplacian systems on weighted graphs

We investigate a generalized poly-Laplacian system with a parameter on weighted finite graphs, a generalized poly-Laplacian system with a parameter and Dirichlet boundary value on weighted locally finite graphs, and a (p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$(p,q)$\\end{document}-Laplacian system with a parameter on weighted locally finite graphs. We utilize a critical points theorem built by Bonanno and Bisci [Bonanno, Bisci, and Regan, Math. Comput. Model. 52(1-2):152–160, 2010], which is an abstract critical points theorem without compactness condition, to obtain that these systems have infinitely many nontrivial solutions with unbounded norm when the parameters locate some well-determined range.

On a fractional (p,q)-Laplacian equation with critical nonlinearities

In this paper, we consider the following fractional ( p , q ) -Laplacian equations with critical Hardy-Sobolev exponents { ( − Δ ) p s 1 u + ( − Δ ) q s 2 u = λ | u | r − 2 u + μ | u | p s 1 ∗ ( α ) − 2 u | x | α in Ω , u = 0 in R N ∖Ω , where 0 < s 2 < s 1 < 1 < q < r ≤ p < N s 1 , 0 $ ]]> λ , μ > 0 are two parameters, 0 ≤ α < p s 1 and p s 1 ∗ ( α ) = p ( N − α ) N − p s 1 is the fractional Hardy-Sobolev critical exponent, Ω ⊂ R N is an open bounded domain with smooth boundary. By using variational methods, we show that the problem has a nontrivial nonnegative weak solution.

Kirchhoff‐type critical fractional Laplacian system with convolution and magnetic field

AbstractIn this paper, we consider a class of upper critical Kirchhoff‐type fractional Laplacian system with Choquard nonlinearities and magnetic fields. With the help of the limit index theory and the concentration–compactness principles for fractional Sobolev spaces, we establish the existence of infinitely many nontrivial radial solutions for the above system. A distinguished feature of this paper is that the above Kirchhoff‐type system is degenerate, that is, the Kirchhoff term is zero at zero.

Degenerate Schrödinger--Kirchhoff {(p,N)}-Laplacian problem with singular Trudinger--Moser nonlinearity in ℝ N

Abstract In this paper, we deal with the existence of nontrivial nonnegative solutions for a ( p , N ) {(p,N)} -Laplacian Schrödinger–Kirchhoff problem in ℝ N {\mathbb{R}^{N}} with singular exponential nonlinearity. The main features of the paper are the ( p , N ) {(p,N)} growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger–Moser inequality, and a completely new Brézis–Lieb-type lemma for singular exponential nonlinearity.