Energy Sign-changing Solutions Research Articles

Least energy sign-changing solutions for discrete Kirchhoff-type problems

In this paper, by variational technique coupled with quantitative lemma, we not only investigate the existence of least energy solutions, but also point out that the energy of the least energy sign-changing solutions is strictly larger than twice that of the ground state solutions.

Least energy sign-changing solutions for fractional critical Kirchhoff–Schrödinger–Poisson with steep potential well

Least energy sign-changing solutions for fractional critical Kirchhoff–Schrödinger–Poisson with steep potential well

Existence and convergence of the least energy sign-changing solutions for nonlinear Kirchhoff equations on locally finite graphs

In this paper, we study the least energy sign-changing solutions to the following nonlinear Kirchhoff equation − ( a + b ∫ V | ∇ u | 2 d μ ) Δ u + c ( x ) u = f ( u ) on a locally finite graph G = ( V , E ), where a, b are positive constants. We use the constrained variational method to prove the existence of a least energy sign-changing solution u b of the above equation if c ( x ) and f satisfy certain assumptions, and to show the energy of u b is strictly larger than twice that of the least energy solutions. Moreover, if we regard b as a parameter, as b → 0 + , the solution u b converges to a least energy sign-changing solution of a local equation − a Δ u + c ( x ) u = f ( u ).

Convergence of least energy sign-changing solutions for logarithmic Schrödinger equations on locally finite graphs

In this paper, we study the following logarithmic Schrödinger equation −Δu+λa(x)u=ulogu2inVon a connected locally finite graph G=(V,E), where Δ denotes the graph Laplacian, λ>0 is a constant, and a(x)≥0 represents the potential. Using variational techniques in combination with the Nehari manifold method based on directional derivative, we can prove that, there exists a constant λ0>0 such that for all λ≥λ0, the above problem admits a least energy sign-changing solution uλ. Moreover, as λ→+∞, we prove that the solution uλ converges to a least energy sign-changing solution of the following Dirichlet problem −Δu=ulogu2inΩ,u(x)=0on∂Ω,where Ω={x∈V:a(x)=0} is the potential well.

Infinitely many sign-changing solutions for a kind of fractional Klein-Gordon-Maxwell system

In this paper, we consider the fractional subcritical Klein-Gordon-Maxwell system as follows: $$\begin{aligned} \left\{ \begin{aligned}&(-\Delta )^{s} u+V(x)u-(2\omega +\phi (x) )\phi (x)u= f(x,u), \,\,\,&\text {in } \mathbb {R}^3, \\&(\Delta )^{s} \phi (x)=(\omega +\phi (x) )u^2, \,\,\,&\text {in } \mathbb {R}^3, \end{aligned} \right. \end{aligned}$$the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many high energy sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method.

Least energy sign-changing solutions for a class of fractional $ (p, q) $-Laplacian problems with critical growth in $ \mathbb{R}^N $

<abstract><p>This paper considers the following fractional $ (p, q) $-Laplacian equation:</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-\Delta)_{p}^{s} u+(-\Delta)_{q}^{s} u+V(x)\left(|u|^{p-2} u+|u|^{q-2} u\right) = \lambda f(u)+|u|^{q^*_s-2}u \quad \text { in } \mathbb{R}^{N}, $\end{document} </tex-math></disp-formula></p> <p>where $ s \in(0, 1), \lambda > 0, 2 < p < q < \frac{N}{s} $, $ (-\Delta)_{t}^{s} $ with $ t \in\{p, q\} $ is the fractional $ t $-Laplacian operator, and potential $ V $ is a continuous function. Using constrained variational methods, a quantitative Deformation Lemma and Brouwer degree theory, we prove that the above problem has a least energy sign-changing solution $ u_{\lambda} $ under suitable conditions on $ f $, $ V $ and $ \lambda $. Moreover, we show that the energy of $ u_{\lambda} $ is strictly larger than two times the ground state energy.</p></abstract>

Nodal solutions for an asymptotically linear Kirchhoff-type problem in ℝ N

ABSTRACT In this paper, we are concerned with nodal solutions for a class of Kirchhoff-type problems where , a, b>0, f satisfies some asymptotically linear growth conditions. First of all, when N = 3, b>0 and , b>0 sufficiently small, we attain infinitely many nodal solutions by an equivalent transformation. Based on this, via a new and direct approach, the least energy sign-changing radial solutions can be obtained for the above problem. What's more, we also establish non-existence results of nodal solutions for and b large enough.

Least energy sign-changing solutions for Schrödinger-Poisson systems with potential well

Abstract In this article, we investigate the existence of least energy sign-changing solutions for the following Schrödinger-Poisson system − Δ u + V ( x ) u + K ( x ) ϕ u = f ( u ) , x ∈ R 3 , − Δ ϕ = K ( x ) u 2 , x ∈ R 3 , \left\{\begin{array}{ll}-\Delta u+V\left(x)u+K\left(x)\phi u=f\left(u),\hspace{1.0em}& x\in {{\mathbb{R}}}^{3},\\ -\Delta \phi =K\left(x){u}^{2},\hspace{1.0em}& x\in {{\mathbb{R}}}^{3},\\ \hspace{1.0em}\end{array}\right. where the functions V ( x ) , K ( x ) V\left(x),K\left(x) have finite limits as ∣ x ∣ → ∞ | x| \to \infty satisfying some mild assumptions. By combining variational methods with the global compactness lemma, we obtain a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.

Least energy sign-changing solutions for Kirchhoff-type problems with potential well

In this paper, we investigate the existence of least energy sign-changing solutions for the Kirchhoff-type problem −a+b∫R3|∇u|2dxΔu+V(x)u=f(u),x∈R3, where a, b > 0 are parameters, V∈C(R3,R), and f∈C(R,R). Under weaker assumptions on V and f, by using variational methods with the aid of a new version of global compactness lemma, we prove that this problem has a least energy sign-changing solution with exactly two nodal domains, and its energy is strictly larger than twice that of least energy solutions.

Least energy sign-changing solutions for a nonlocal anisotropic Kirchhoff type equation

Abstract In this paper, we investigate the existence of sign-changing solutions for the following class of fractional Kirchhoff type equations with potential ( 1 + b [ u ] α 2 ) ( ( - Δ x ) α u - Δ y u ) + V ( x , y ) u = f ( x , y , u ) , ( x , y ) ∈ ℝ N = ℝ n × ℝ m , \left( {1 + b\left[ u \right]_\alpha ^2} \right)\left( {{{\left( { - {\Delta _x}} \right)}^\alpha }u - {\Delta _y}u} \right) + V\left( {x,y} \right)u = f\left( {x,y,u} \right),\left( {x,y} \right) \in {\mathbb{R}^N} = {\mathbb{R}^n} \times {\mathbb{R}^m}, where [ u ] α = ( ∫ ℝ N ( | ( - Δ x ) α 2 u | 2 + | ∇ y u | 2 ) d x d y ) 1 2 {\left[ u \right]_\alpha } = {\left( {\int {_{{\mathbb{R}^N}}\left( {{{\left| {{{\left( { - {\Delta _x}} \right)}^{{\alpha \over 2}}}u} \right|}^2} + {{\left| {{\nabla _y}u} \right|}^2}} \right)dxdy} } \right)^{{1 \over 2}}} . Based on variational approach and a variant of the quantitative strain lemma, for each b > 0, we show the existence of a least energy nodal solution ub . In addition, a convergence property of ub as b ↘ 0 is established.

Least energy sign-changing solutions of Kirchhoff equation on bounded domains

<abstract><p>We deal with sign-changing solutions for the Kirchhoff equation</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} -(a+ b\int _{\Omega}|\nabla u|^{2}dx)\Delta u = \lambda u+\mu|u|^{2}u, \; \ x\in\Omega, \\ u = 0, \; \ x\in \partial\Omega, \end{cases} $\end{document} </tex-math></disp-formula></p> <p>where $ a, b > 0 $ and $ \lambda, \mu\in\mathbb{R} $ being parameters, $ \Omega\subset \mathbb{R}^{3} $ is a bounded domain with smooth boundary $ \partial\Omega $. Combining Nehari manifold method with the quantitative deformation lemma, we prove that there exists $ \mu^{\ast} > 0 $ such that above problem has at least a least energy sign-changing (or nodal) solution if $ \lambda < a\lambda_{1} $ and $ \mu > \mu^{\ast} $, where $ \lambda_{1} > 0 $ is the first eigenvalue of $ (-\Delta u, H^{1}_{0}(\Omega)) $. It is noticed that the nonlinearity $ \lambda u+\mu|u|^{2}u $ fails to satisfy super-linear near zero and super-three-linear near infinity, respectively.</p></abstract>

Nodal solutions for Q-Laplacian problem with exponential nonlinearities on the Heisenberg group

This paper deals with the existence of nodal solutions (sign-changing) to the following elliptic equation involving Q-Laplacian:{−ΔQu=λf(ξ,u)inΩ,u=0on∂Ω, where ΔQ(⋅):=divHn(|∇Hn(⋅)|HnQ−2∇Hn(⋅)) is the Q-Laplacian operator, Ω is an open, smooth bounded domain in the Heisenberg group Hn, Q=2n+2 is the homogeneous dimension of Hn, λ is a positive parameter, and nonlinear term f(ξ,u) behaves like exp⁡(β|s|QQ−1) as s→∞. Under suitable conditions on f, together with the constraint variational method and the quantitative deformation lemma, we obtain the existence, energy estimates and the convergence property of the least energy sign-changing solutions for the above problem in subcritical and critical exponential growth cases.

Least energy sign-changing solutions of fractional Kirchhoff–Schrödinger–Poisson system with critical and logarithmic nonlinearity

In the present paper, we deal with the following fractional Kirchhoff–Schrödinger–Poisson system with logarithmic and critical nonlinearity: where and Ω is a bounded domain in with Lipschitz boundary. Combining constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has a least energy sign-changing solution . Moreover, we show that the energy of is strictly larger than two times the ground state energy. Finally, we regard b as a parameter and show the convergence property of as .

Least energy sign-changing solutions for a class of fractional Kirchhoff–Poisson system

Using the constraint variational method and a quantitative deformation lemma, we establish the existence of the least energy sign-changing solutions for the fractional Kirchhoff–Poisson system, a+b∫R3|(−Δ)s2u|2dx(−Δ)su+V(x)u+ϕ(x)u=f(x,u),(−Δ)tϕ=u2,x∈R3, where a > 0 is a constant, b∈R+ is a parameter, s, t ∈ (0, 1) and 4s + 2t > 3, (−Δ)s stands for the fractional Laplacian, V is a continuous, positive function, and f is nonlinear function satisfying suitable growth assumptions. Moreover, for any b > 0, we prove that the energy of the least energy sign-changing solution is strictly larger than twice the ground state energy. Furthermore, we show a convergence property of the least energy sign-changing solutions as the parameter b goes to zero. Our results complement an in-depth study of Wang, Radulescu, and Zhang [J. Math. Phys. 60, 011506 (2019)] in the sense that we are concerned with the nodal characteristics of the ground states.

Sign-changing solutions for a kind of Klein–Gordon–Maxwell system

In this paper, the subcritical Klein–Gordon–Maxwell system −Δu + V(x)u − (2ω + ϕ)ϕu = f(u) coupled with Δϕ=(ω+ϕ)u2,x∈R3 is studied. Under general assumptions on V, f, by using the abstract critical theorems, which are derived from the minimax method and the method of invariant sets of descending flow, we obtain the existence of sign-changing solutions for this system, in particular, a sequence of high energy sign-changing solutions is obtained if f is odd.

Weighted Choquard Equation Perturbed with Weighted Nonlocal Term

We investigate the following problem -div(v(x)|∇u|m-2∇u)+V(x)|u|m-2u=|x|-θ∗|u|b|x|α|u|b-2|x|αu+λ|x|-γ∗|u|c|x|β|u|c-2|x|βuinRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\\mathrm{div}(v(x)|\ abla u|^{m-2}\ abla u)+V(x)|u|^{m-2}u= \\left( |x|^{-\ heta }*\\frac{|u|^{b}}{|x|^{\\alpha }}\\right) \\frac{|u|^{b-2}}{|x|^{\\alpha }}u+\\lambda \\left( |x|^{-\\gamma }*\\frac{|u|^{c}}{|x|^{\\beta }}\\right) \\frac{|u|^{c-2}}{|x|^{\\beta }}u \\quad \ ext { in }{\\mathbb {R}}^{N}, \\end{aligned}$$\\end{document}where b,c,α,β>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$b, c, \\alpha , \\beta >0$$\\end{document}, θ,γ∈(0,N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ heta ,\\gamma \\in (0,N)$$\\end{document}, N≥3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 3$$\\end{document}, 2≤m<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2\\le m< \\infty$$\\end{document} and λ∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\in {\\mathbb {R}}$$\\end{document}. Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.

Least energy sign-changing solutions for a class of Schrödinger–Poisson system on bounded domains

This paper is concerned with the Schrödinger–Poisson system −Δu + ϕu = λu + μ|u|2u and −Δϕ = u2 setting on a bounded domain Ω⊂R3 with smooth boundary and λ,μ∈R being parameters. By using variational techniques in combination with the nodal Nehari manifold method, we show the existence of μ̄&amp;gt;0 such that for all (λ,μ)∈(−∞,λ1)×(μ̄,+∞), the above system has one least energy sign-changing solution, where λ1 &amp;gt; 0 is the first eigenvalue of −Δ,H01(Ω). The results of this paper are complementary to those in Alves and Souto [Z. Angew. Math. Phys. 65, 1153–1166 (2014)].

Sign-changing solutions to a gauged nonlinear Schrödinger equation with critical exponential growth

We study the existence and asymptotic behavior of least energy sign-changing solutions to a gauged nonlinear Schrödinger equation with critical exponential growth where are constants and Under some suitable assumptions on , we apply the constraint minimization argument to establish a least energy sign-changing solution with precisely two nodal domains. Moreover, we show that the energy of is strictly larger than two times of the ground state energy and analyze the asymptotic behavior of as . Our results generalize the existing ones, see Li G. et al. (Sign-changing solutions to a gauged nonlinear Schrödinger equation. J Math Anal Appl. 2017;455:1559–1578) and Liu Z. et al. (Existence and multiplicity of sign-changing standing waves for a gauged nonlinear Schrödinger equation in . Nonlinearity. 2019;32:3082–3111) for example, to the gauged nonlinear Schrödinger equation with critical exponential growth.

POSITIVE AND SIGN-CHANGING SOLUTIONS FOR THE FRACTIONAL KIRCHHOFF EQUATION WITH CRITICAL GROWTH

<abstract> We are interested in the existence of positive and sign-changing solutions for a fractional Kirchhoff equation. Under some mild conditions on the potentials $ V $ and $ h $, using variational methods, we prove the existence of positive ground state solutions and least energy sign-changing solutions. </abstract>

Least energy sign-changing solutions for Schrödinger-Poisson system with critical growth

<p style='text-indent:20px;'>In this paper, we investigate the existence and asymptotic behavior of least energy sign-changing solutions for the following Schrödinger-Poisson system <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+V(x)u+\lambda\phi(x)u = |u|^4u+ f(u),\ \ \ &amp;\ x \in \mathbb{R}^{3},\\ -\Delta \phi = u^2, \ \ \ &amp;\ x \in \mathbb{R}^{3}, \end{cases} \end{equation*} $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula> is a parameter. Under some suitable conditions on <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ V $\end{document}</tex-math></inline-formula>, we get a least energy sign-changing solution <inline-formula><tex-math id="M4">\begin{document}$ u_\lambda $\end{document}</tex-math></inline-formula> via variational method and its energy is strictly larger than twice that of least energy solutions. Moreover, the asymptotic behavior of <inline-formula><tex-math id="M5">\begin{document}$ u_\lambda $\end{document}</tex-math></inline-formula> as <inline-formula><tex-math id="M6">\begin{document}$ \lambda\rightarrow 0^+ $\end{document}</tex-math></inline-formula> is also analyzed.