Ground State Sign-changing Solutions Research Articles

Ground state sign-changing solution for a logarithmic Kirchhoff-type equation in R 3

We investigate the following logarithmic Kirchhoff-type equation: ( a + b ∫ R 3 | ∇ u | 2 + V ( x ) u 2 d x ) [ − Δ u + V ( x ) u ] = | u | p − 2 u ln ⁡ | u | , x ∈ R 3 , where a , b > 0 are constants, 4 < p < 2 ∗ = 6 . Under some appropriate hypotheses on the potential function V , we prove the existence of a positive ground state solution, a ground state sign-changing solution and a sequence of solutions by using the constraint variational methods, topological degree theory, quantitative deformation lemma and symmetric mountain pass theorem. Our results complete those of Gao et al. [Appl. Math. Lett. 139(2023), 108539] with the case of 4 < p < 6 .

Existence and asymptotic behavior for ground state sign-changing solutions of fractional Schrödinger-Poisson system with steep potential well

<abstract><p>In this paper, we investigate the existence of ground state sign-changing solutions for the following fractional Schrödinger-Poisson system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \begin{cases} (-\Delta)^s u+V_{\lambda} (x)u+\mu\phi u = f(u), & \; \mathrm{in}\; \; \mathbb{R}^3, \\ (-\Delta)^t \phi = u^2, & \; \mathrm{in}\; \; \mathbb{R}^3, \end{cases} \nonumber \end{equation} $\end{document} </tex-math></disp-formula></p> <p>where $ \mu > 0, s\in(\frac{3}{4}, 1), t\in(0, 1) $ and $ V_{\lambda}(x) $ = $ \lambda V(x)+1 $ with $ \lambda > 0 $. Under suitable conditions on $ f $ and $ V $, by using the constraint variational method and quantitative deformation lemma, if $ \lambda > 0 $ is large enough, we prove that the above problem has one least energy sign-changing solution. Moreover, for any $ \mu > 0 $, the least energy of the sign-changing solution is strictly more than twice of the energy of the ground state solution. In addition, we discuss the asymptotic behavior of ground state sign-changing solutions as $ \lambda\rightarrow \infty $ and $ \mu\rightarrow0 $.</p></abstract>

Ground State Sign-Changing Solutions for a Schrödinger–Poisson System with Steep Potential Well and Critical Growth

Ground State Sign-Changing Solutions for a Schrödinger–Poisson System with Steep Potential Well and Critical Growth

Ground State Sign-Changing Solution for a Large Class of $$ p(x) \& q(x)$$-Quasilinear Elliptic Equations with Nonsmooth Nonlinearity

Ground State Sign-Changing Solution for a Large Class of $$ p(x) \& q(x)$$-Quasilinear Elliptic Equations with Nonsmooth Nonlinearity

The Existence and Concentration of Ground State Sign-Changing Solutions for Kirchhoff-Type Equations with a Steep Potential Well

The Existence and Concentration of Ground State Sign-Changing Solutions for Kirchhoff-Type Equations with a Steep Potential Well

Ground state sign-changing solutions for the Chern-Simons-Schrödinger equation with zero mass potential

Ground state sign-changing solutions for the Chern-Simons-Schrödinger equation with zero mass potential

Ground State Sign-Changing Solutions for Critical Schrödinger–Poisson System with Steep Potential Well

Ground State Sign-Changing Solutions for Critical Schrödinger–Poisson System with Steep Potential Well

Nonexistence of ground state sign-changing solutions for autonomous Schrödinger–Poisson system with critical growth

This paper is concerned with the following Schrödinger–Poisson system where 3<p<5. With the help of an odd Nehari manifold and ‘energy doubling’ property, we prove the nonexistence of ground state sign-changing solutions on . In this sense, our result explains why the existing literature can only consider the existence of the ground state sign-changing solutions in the radial Sobolev space .

Ground state sign-changing solutions for Schrödinger–Kirchhoff-type problem with critical growth

In this paper, we study the following Schrödinger–Kirchhoff-type problem: −a+b∫R3|∇u|2dx△u+u=|u|4u+f(u), x∈R3, where a &amp;gt; 0 and b &amp;gt; 0 are small enough. Under suitable assumptions on f, we obtain the existence of ground state sign-changing solution ub by the constraint variational method with Miranda’s theorem. Moreover, we prove that its energy is strictly larger than twice that of the ground state solution.

Ground state sign-changing solutions for critical Choquard equations with steep well potential

In this paper, we study sign-changing solution of the Choquard type equation − Δ u + ( λ V ( x ) + 1 ) u = ( I α ∗ | u | 2 α ∗ ) | u | 2 α ∗ − 2 u + μ | u | p − 2 u in R N , where N ≥ 3 , α ∈ ( ( N − 4 ) + , N ) , I α is a Riesz potential, p ∈ [ 2 α ∗ , 2 N N − 2 ) , 2 α ∗ := N + α N − 2 is the upper critical exponent in terms of the Hardy–Littlewood–Sobolev inequality, μ &gt; 0 , λ &gt; 0 , V ∈ C ( R N , R ) is nonnegative and has a potential well. By combining the variational methods and sign-changing Nehari manifold, we prove the existence and some properties of ground state sign-changing solution for λ , μ large enough. Further, we verify the asymptotic behaviour of ground state sign-changing solutions as λ → + ∞ and μ → + ∞ , respectively.

Ground state sign-changing solutions for a class of quasilinear Schrödinger equations

Abstract In this paper, we consider the following quasilinear Schrödinger equation: − Δ u + V ( x ) u + κ 2 Δ ( u 2 ) u = K ( x ) f ( u ) , x ∈ R N , -\Delta u+V\left(x)u+\frac{\kappa }{2}\Delta \left({u}^{2})u=K\left(x)f\left(u),\hspace{1.0em}x\in {{\mathbb{R}}}^{N}, where N ≥ 3 N\ge 3 , κ &gt; 0 \kappa \gt 0 , f ∈ C ( R , R ) f\in {\mathcal{C}}\left({\mathbb{R}},{\mathbb{R}}) , V ( x ) V\left(x) and K ( x ) K\left(x) are positive continuous potentials. Under given conditions, by changing variables and truncation argument, the energy of ground state solutions of the Nehari type is achieved. We also prove the existence of ground state sign-changing solutions for the aforementioned equation. Our results are the generalization work of M. B. Yang, C. A. Santos, and J. Z. Zhou, Least action nodal solution for a quasilinear defocusing Schrödinger equation with supercritical nonlinearity, Commun. Contemp. Math. 21 (2019), no. 5, 1850026, https://doi.org/10.1142/S0219199718500268.

Sign-changing solutions for a fractional Schrödinger–Poisson system

In this paper, we deal with the existence and multiplicity of sign-changing solutions for fractional Schrödinger–Poisson system: (℘) where , and f is a continuous function. Based on perturbation approach and the method of invariant sets of descending flow, we obtain the existence and multiplicity of sign-changing solutions of system (). In addition, by applying the constrained variational method incorporated with Brouwer degree theory, we prove that system () possesses at least one ground state sign-changing solution. Furthermore, we show that the least energy of sign-changing solutions exceed twice than the least energy, and when f is odd, system () admits infinitely many nontrivial solutions.

Existence and asymptotic behavior of ground state sign-changing solutions for a nonlinear Schrödinger–Poisson–Kirchhoff system in ℝ3

In this paper, we consider the existence and asymptotic behavior of ground state sign-changing solutions for the following nonlinear Schrödinger–Poisson–Kirchhoff system: −(a+b∫R3|∇u|2dx)△u+V(x)u+k(x)ϕu=λf(x)u+g(x,u) in ℝ3,−△ϕ=k(x)u2, in ℝ3. Under some mild assumptions, by using a series of constructive Nehari manifolds and a quantitative deformation lemma, we prove that the Schrödinger–Poisson–Kirchhoff system possesses at least one ground state sign-changing solution ub for all b>0. Moreover, for any sequence {bn}→0+ (n→∞), there exists a subsequence {bnk}, such that {ubnk} converges to u0, where u0 is a sign-changing solution of the equation −a△u+V(x)u=λf(x)u+g(x,u), x∈ℝ3.

Ground State Sign-Changing Solutions for a Kirchhoff Equation with Asymptotically 3-Linear Nonlinearity

Ground State Sign-Changing Solutions for a Kirchhoff Equation with Asymptotically 3-Linear Nonlinearity

Sufficient and necessary conditions for ground state sign-changing solutions to the Schrödinger–Poisson system with cubic nonlinearity on bounded domains

In this paper we are interested in the existence of ground state sign-changing solutions to the following Schrödinger–Poisson system −Δu+ϕu=λu+μ|u|2uinΩ−Δϕ=u2inΩu=ϕ=0on∂Ω,where Ω is a bounded smooth domain of R3, μ>0, λ<λ1 and λ1 is the first eigenvalue of −Δ,H01(Ω). Using construction technique, we show that the above system possesses one ground state sign-changing solution if and only if μ>0, which improves the recent results by Khoutir (2021).

Ground State Sign-Changing Solution for Schrödinger-Poisson System with Critical Growth

This article is devoted to study the nonlinear Schrödinger-Poisson system with pure power nonlinearities $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+ \phi u=|u|^{p-1}u+|u|^4u, &{}x\in {\mathbb {R}}^3, \\ -\Delta \phi = u^2, &{}x\in {\mathbb {R}}^3, \end{array} \right. \end{aligned}$$ where $$4< p<5$$ . By employing constraint variational method and a variant of the classical deformation lemma, we show the existence of one ground state sign-changing solution with precisely two nodal domains, which improves and generalizes the existing results by Wang, Zhang and Guan (J. Math. Anal. Appl. 479 (2019), 2284–2301).

Existence of Ground State Sign-Changing Solutions of Fractional Kirchhoff-Type Equation with Critical Growth

In this paper, we study the following fractional Kirchhoff-type equation $$\begin{aligned}{\left\{ \begin{array}{ll} -(a+ b\int _{{\mathbb {R}}^{N}}\int _{{\mathbb {R}}^{N}}|u(x)-u(y)|^{2}K(x-y)dxdy){\mathcal {L}}_{K}u=|u|^{2_{\alpha }^{*}-2}u+\mu f(u), ~\ x\in \Omega ,\\ u=0, ~\ x\in {\mathbb {R}}^{N}\backslash \Omega , \end{array}\right. } \end{aligned}$$ where $$\Omega \subset {\mathbb {R}}^{N}$$ is a bounded domain with a smooth boundary, $$\alpha \in (0,1)$$ , $$2\alpha<N<4\alpha $$ , $$2_{\alpha }^{*}$$ is the fractional critical Sobolev exponent and $$\mu , a, b>0$$ ; $${\mathcal {L}}_{K}$$ is non-local integrodifferential operator. Under suitable conditions on f, for $$\mu $$ large enough, by using constraint variational method and the quantitative deformation lemma, we obtain a ground state sign-changing (or nodal) solution to this problem, and its energy is strictly larger than twice that of the ground state solutions.

Ground state sign-changing solutions and infinitely many solutions for fractional logarithmic Schrödinger equations in bounded domains

We consider a class of fractional logarithmic Schrödinger equation in bounded domains. First, by means of the constraint variational method, quantitative deformation lemma and some new inequalities, the positive ground state solutions and ground state sign-changing solutions are obtained. These inequalities are derived from the special properties of fractional logarithmic equations and are critical for us to obtain our main results. Moreover, we show that the energy of any sign-changing solution is strictly larger than twice the ground state energy. Finally, we obtain that the equation has infinitely many nontrivial solutions. Our result complements the existing ones to fractional Schrödinger problems when the nonlinearity is sign-changing and satisfies neither the monotonicity condition nor Ambrosetti-Rabinowitz condition.

GROUND STATE SIGN–CHANGING SOLUTIONS FOR FRACTIONAL KIRCHHOFF TYPE EQUATIONS IN &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ \mathbb{R}^{3} $&lt;/tex-math&gt;&lt;/inline-formula&gt;

<abstract> In this paper, we investigate the existence of ground state sign–changing solutions for the following fractional Kirchhoff equation <p class="disp_formula">$ \left(a+b\int_{\mathbb{R}^{3}}|\left(-\triangle\right)^{\frac{\alpha}{2}}u|^{2}\mathrm{d}x\right) (-\triangle)^{\alpha}u+V(x)u = K(x)f(u) \rm{ \ \ in }\mathbb{R}^{3}, $ where $ \alpha\in (0, 1) $, $ a, b $ are positive parameters, $ V(x), \; K(x) $ are nonnegative continuous functions and $ f $ is a continuous function with quasicritical growth. By establishing a new inequality, we prove the above system possesses a ground state sign–changing solutions $ u_{b} $ with precisely two nodal domains, and its energy is strictly larger than twice that of the ground state solutions of Nehari–type. Moreover, we obtain the convergence property of $ u_{b} $ as the parameter $ b\rightarrow0 $. Our conditions weaken the usual increasing condition on $ f(t)/|t|^{3} $. </abstract>

Ground state sign-changing solutions for fractional Laplacian equations with critical nonlinearity

<abstract> In this paper, we investigate the existence of the least energy sign-changing solutions for nonlinear elliptic equations driven by nonlocal integro-differential operators with critical nonlinearity. By using constrained minimization method and topological degree theory, we obtain a least energy sign-changing solution for them under much weaker conditions. As a particular case, we drive an existence theorem of sign-changing solutions for the fractional Laplacian equations with critical growth. </abstract>