Sign-changing Solutions Research Articles

Multiple Sign-Changing Solutions to Non-local Supercritical Exponent Problem in Symmetric Domains

Multiple Sign-Changing Solutions to Non-local Supercritical Exponent Problem in Symmetric Domains

Existence of Nontrivial Solutions for Boundary Value Problems of Fourth-Order Differential Equations

This article investigates the solvability problem of fourth-order differential equations with two-point boundary conditions; specifically, conclusions regarding sign-changing solutions are obtained. The methods used in this article are fixed-point theorems on lattices. Firstly, under some sublinear conditions, the existence of three nontrivial solutions is demonstrated, including a sign-changing solution, a negative solution and a positive solution. Secondly, under some unilaterally asymptotically linear and superlinear conditions, the existence of at least one sign-changing solution is proved. Finally, this article provides several specific examples to illustrate the obtained conclusions.

Existence of ground state sign-changing solutions for a class of quasilinear scalar field equations originating from nonlinear optics

This paper is mainly concerned with the existence of ground state sign-changing solutions for a class of second order quasilinear elliptic equations in bounded domains, which derived from nonlinear optics models. Combining a non-Nehari manifold method due to Tang and Cheng [J. Differ. Equations 261, 2384–2402 (2016)] and a quantitative deformation lemma with Miranda theorem, we obtain that the problem has at least one ground state solution and one ground state sign-changing solution with two precise nodal domains and we present the concrete lower bounds of ground state solution and ground state energy. We also obtain that any weak solution of the problem has C1,σ-regularity for some σ ∈ (0, 1). With the help of Maximum Principle, we reach the conclusion that the energy of the ground state sign-changing solutions is strictly larger than twice that of the ground state solutions.

Cauchy problem for the Lane-Emden heat flow with sign-changing initial data

This article concerns the blow-up phenomenon of sign-changing solutions to the Lane-Emden heat flow. We construct sign-changing weak sub-solutions and localization of the positive and negative parts of the sign-changing solutions. We also extend the results to a nonlinear and finite diffusion equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/66/abstr.html

Nodal solutions for a weighted (N,p)-Kirchhoff-type problem with double exponential growth non-linearity

This paper investigates the existence of sign-changing solutions for a logarithmic weighted Kirchhoff [Formula: see text]-Laplacian problem in the unit ball [Formula: see text] of [Formula: see text] where [Formula: see text]. The non-linearity of the equation is assumed to have double exponential growth, which is motivated by Trudinger–Moser-type inequalities. To prove the existence of these solutions, we employ techniques such as constrained minimization within the Nehari set, a quantitative deformation lemma and results from degree theory.

Sign-changing solutions for a class of fractional Choquard equation with the Sobolev critical exponent in [formula omitted]

In this paper, we study the following fractional Choquard equation with critical exponent(−Δ)σu+u=(Iα⁎|u|p)|u|p−2u+|u|2σ⁎−2uinR3, where σ∈(34,1), α∈(2σ,3) and α+2σ3−2σ<p<2α,σ⁎:=3+α3−2σ, 2α,σ⁎ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, 2σ⁎:=63−2σ is the fractional Sobolev critical exponent and the operator (−Δ)σ stands for the fractional Laplacian of order σ. Based on the above assumptions, we establish the existence of positive ground state solutions, and if p∈(α+2σ3−2σ,2α3−2σ), we also have the corresponding regular property. Subsequently, by introducing some other additional hypotheses on σ, α, p and with the help of quantitative deformation lemma, we employ constrained minimization arguments on the sign-changing Nehari manifold to obtain the existence of ground state sign-changing solutions.

The asymptotic behavior of constant sign and nodal solutions of [formula omitted]-Laplacian problems as [formula omitted] goes to 1

In this paper we study the asymptotic behavior of solutions to the (p,q)-equation −Δpu−Δqu=f(x,u)inΩ,u=0on∂Ω,as p→1+, where N≥2, 1<p<q<1∗≔N/(N−1) and f is a Carathéodory function that grows superlinearly and subcritically. Based on a Nehari manifold treatment, we are able to prove that the (1,q)-Laplace problem given by −div∇u|∇u|−Δqu=f(x,u)inΩ,u=0on∂Ω,has at least two constant sign solutions and one sign-changing solution, whereby the sign-changing solution has least energy among all sign-changing solutions. Furthermore, the solutions belong to the usual Sobolev space W01,q(Ω) which is in contrast with the case of 1-Laplacian problems, where the solutions just belong to the space BV(Ω) of all functions of bounded variation. As far as we know this is the first work dealing with (1,q)-Laplace problems even in the direction of constant sign solutions.

Some remarks about the Morse index for convex Hamiltonian systems

We investigate the (linearized) Morse index of solutions to Hamiltonian systems, with a focus on convex Hamiltonian functions and sign-changing radial solutions. For strongly coupled systems, we describe the profile of the radial solutions and give an estimate of their Morse index.

A compact embedding result for nonlocal Sobolev spaces and multiplicity of sign-changing solutions for nonlocal Schrödinger equations

A compact embedding result for nonlocal Sobolev spaces and multiplicity of sign-changing solutions for nonlocal Schrödinger equations

Optimal pinwheel partitions for the Yamabe equation

Abstract We establish the existence of an optimal partition for the Yamabe equation in R N made up of mutually linearly isometric sets, each of them invariant under the action of a group of linear isometries. To do this, we establish the existence of a solution to a weakly coupled competitive Yamabe system, whose components are invariant under the action of the group, and each of them is obtained from the previous one by composing it with a linear isometry. We show that, as the coupling parameter goes to − ∞ , the components of the solutions segregate and give rise to an optimal partition that has the properties mentioned above. Finally, taking advantage of the symmetries considered, we establish the existence of infinitely many sign-changing solutions for the Yamabe equation in R N that are different from those previously found by Ding, and del Pino, Musso, Pacard and Pistoia.

Sign-changing solution for an elliptic equation with critical growth at the boundary

Sign-changing solution for an elliptic equation with critical growth at the boundary

The sign-changing solutions for a class of Kirchhoff-type problems with critical Sobolev exponents in bounded domains

The sign-changing solutions for a class of Kirchhoff-type problems with critical Sobolev exponents in bounded domains

Sign-changing solutions for critical Choquard equation on bounded domain

We consider the existence of sign-changing solutions for the following nonlinear Choquard equation{−Δu=(Iα⁎|u|p)|u|p−2u+λu,x∈Ω,u∈H01(Ω), where N≥3,α∈(0,N),λ∈R, p∈(1,N+αN−2] and Ω⊂RN is a smooth bounded domain. Iα:RN→R is the Riesz potential. In critical case p=N+αN−2, if Ω is symmetric about the axis x1, we firstly develop the limit profiles for the symmetric Palais-Smale sequence by the concentration compactness principle. Then we conclude that the problem admits an odd solution with exactly two nodal domains for λ∈(0,λ1),N≥4andα∈(2,N). In contrast, the local Brezis-Nirenberg type problem −Δu=|u|4N−2u+λu,u∈H01(Ω) can not permit such type of odd symmetry solution.

Infinitely many sign-changing solutions for planar Schrödinger–Poisson system

In this paper, we consider the following planar Schrödinger–Poisson system { − Δ u + V ( x ) u + ϕu = | u | q − 2 u , in R 2 , Δϕ = u 2 , in R 2 , where the potential V : R 2 → R is continuous and coercive. By using the method of perturbation and invariant sets of descending flow, we obtain the existence of infinitely many sign-changing solutions for this system.

Sign-changing Solutions for Fourth Order Elliptic Equation with Concave-convex Nonlinearities

In this paper, we study the following fourth order elliptic equation: Δ²u - Δu + V(x)u = κ(x)|u|q-2u + |u|p-2u in RN, where Δ² := Δ(Δ) is the biharmonic operator, 4 &gt; N, 1 &lt; q &lt; 2 &lt; p &lt; 2* := 2N / (N - 4). Assuming that V(x) satisfies a class of coercive conditions and the nonnegative weighted function κ(x) belongs to Lp / (p-q)(RN), we obtain the existence of one sign-changing solution with the help of constraint variational method and quantitative deformation lemma. The novelty of this paper is that when the nonlinearity is the combination of concave and convex functions, we are able to obtain the existence of sign-changing solutions. Some recent results are improved and generalized significantly.

A Liouville-type theorem for the coupled Schrödinger systems and the uniqueness of the sign-changing radial solutions

In this paper, we study the sign-changing radial solutions of the following coupled Schrödinger system{−Δuj+λjuj=μjuj3+∑i≠jβijui2ujin B1,uj∈H0,r1(B1) for j=1,⋯,N. Here, λj,μj>0 and βij=βji are constants for i,j=1,⋯,N and i≠j. B1 denotes the unit ball in the Euclidean space R3 centred at the origin. For any P1,⋯,PN∈N, we prove the uniqueness of the radial solution (u1,⋯,uj) with uj changes its sign exactly Pj times for any j=1,⋯,N in the following case: λj≥1 and |βij| are small for i,j=1,⋯,N and i≠j. New Liouville-type theorems and boundedness results are established for this purpose.

Blow-up of solutions to the wave equations with memory terms in Schwarzschild spacetime

The main goal of this work is to discuss formation of singularities of solutions to the Cauchy problems for single wave equation and coupled system of wave equations with nonlinear memory terms in Schwarzschild spacetime. For the initial value problem of nonlinear wave equations, blow-up results of sign changing solutions are derived by employing the suitable test function technique which is related to the Riemann-Liouville fractional derivative and contradiction argument. Additionally, upper bound lifespan estimates of solutions to the small initial value problem of wave equations are established by imposing certain conditions on the initial values. It is worth to mention that crucial difficulty is the setting of test function framework, which has not been applied to the Schwarzschild spacetime in the previous literatures. To the best of our knowledge, the results in Theorems 1.1-1.4 are new.

Entire Sign-Changing Solutions to the Fractional Critical Schrodinger Equation

Entire Sign-Changing Solutions to the Fractional Critical Schrodinger Equation

Sign-changing solutions for coupled Schrödinger system

In this paper we study the following nonlinear Schrödinger system: {−Δu+αu=|u|p−1u+2q+1λ|u|p−32u|v|q+12,x∈R3,−Δv+βv=|v|q−1v+2p+1λ|u|p+12|v|q−32v,x∈R3,u(x)→0,v(x)→0,as |x|→∞,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ extstyle\\begin{cases} -\\Delta u+\\alpha u = \\vert u \\vert ^{p-1}u+\\frac{2}{q+1} \\lambda \\vert u \\vert ^{ \\frac{p-3}{2}}u \\vert v \\vert ^{\\frac{q+1}{2}},\\quad x \\in \\mathbb{R}^{3}, \\\\ -\\Delta v+\\beta v = \\vert v \\vert ^{q-1}v+\\frac{2}{p+1} \\lambda \\vert u \\vert ^{ \\frac{p+1}{2}} \\vert v \\vert ^{\\frac{q-3}{2}}v ,\\quad x \\in \\mathbb{R}^{3}, \\\\ u(x)\\rightarrow 0,\\qquad v(x)\\rightarrow 0,\\quad \ ext{as } \\vert x \\vert \\rightarrow \\infty , \\end{cases} $$\\end{document} where 3≤p,q<5\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$3\\leq p, q<5$\\end{document}, α, β are positive parameters. We show that there exists λk>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda _{k}>0$\\end{document} such that the equation has at least k radially symmetric sign-changing solutions and at least k seminodal solutions for each k∈N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$k\\in \\mathbb{N}$\\end{document} and λ∈(0,λk)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda \\in (0, \\lambda _{k})$\\end{document}. Moreover, we show the existence of a least energy radially symmetric sign-changing solution for each λ∈(0,λ0)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda \\in (0, \\lambda _{0})$\\end{document} where λ0∈(0,λ1]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\lambda _{0}\\in (0, \\lambda _{1}]$\\end{document}.

Yamabe systems, optimal partitions and nodal solutions to the Yamabe equation

We give conditions for the existence of regular optimal partitions, with an arbitrary number \ell\geq 2 of components, for the Yamabe equation on a closed Riemannian manifold (M,g) . To this aim, we study a weakly coupled competitive elliptic system of \ell equations, related to the Yamabe equation. We show that this system has a least energy solution with nontrivial components if \dim M\geq 10 , (M,g) is not locally conformally flat, and satisfies an additional geometric assumption whenever \dim M=10 . Moreover, we show that the limit profiles of the components of the solution separate spatially as the competition parameter goes to -\infty , giving rise to an optimal partition. We show that this partition exhausts the whole manifold, and we prove the regularity of both the interfaces and the limit profiles, together with a free boundary condition. For \ell=2 the optimal partition obtained yields a least energy sign-changing solution to the Yamabe equation with precisely two nodal domains.