scholarly journals The Existence of the Sign-Changing Solutions for the Kirchhoff-Schrödinger-Poisson System in Bounded Domains

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Cun-bin An ◽  
Jiangyan Yao ◽  
Wei Han
Keyword(s):  
Existence Result ◽  
Degree Theory ◽  
Bounded Domains ◽  
Poisson System ◽  
Convergence Properties ◽  
Doubling Property ◽  
Deformation Lemma ◽  
Quantitative Deformation Lemma ◽  
Sign Changing Solutions ◽  
Least Energy

In this paper, we study a class of the Kirchhoff-Schrödinger-Poisson system. By using the quantitative deformation lemma and degree theory, the existence result of the least energy sign-changing solution u0 is obtained. Meanwhile, the energy doubling property is proved, that is, we prove that the energy of any sign-changing solution is strictly larger than twice that of the least energy. Moreover, we also get the convergence properties of u0 as the parameters b↘0 and λ↘0.

scholarly journals The Existence Result for a Class of p-Kirchhoff-Type Problem with a Multilinear Growth Nonlinearity

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Jiangyan Yao ◽  
Wei Han
Keyword(s):  
Linear Growth ◽  
Nehari Manifold ◽  
Doubling Property ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Kirchhoff Type Problem ◽  
Quantitative Deformation Lemma ◽  
Sign Changing Solutions ◽  
Least Energy ◽  
Kirchhoff Type Problems

In this paper, we firstly discuss the existence of the least energy sign-changing solutions for a class of p-Kirchhoff-type problems with a (2p-1)-linear growth nonlinearity. The quantitative deformation lemma and Non-Nehari manifold method are used in the paper to prove the main results. Remarkably, we use a new method to verify that Mb≠∅. The main results of our paper are the existence of the least energy sign-changing solution and its corresponding energy doubling property. Moreover, we also give the convergence property of the least energy sign-changing solution as the parameter b↘0.

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

2021 ◽  
pp. 1-14
Author(s):  
Yonghui Tong ◽  
Hui Guo ◽  
Giovany Figueiredo
Keyword(s):  
Ground State ◽  
State Energy ◽  
Monotonicity Condition ◽  
Infinitely Many Solutions ◽  
Bounded Domains ◽  
Nontrivial Solutions ◽  
Deformation Lemma ◽  
Quantitative Deformation Lemma ◽  
Sign Changing Solutions ◽  
New Inequalities

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.

scholarly journals The Existence of Least Energy Sign-Changing Solution for Kirchhoff-Type Problem with Potential Vanishing at Infinity

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ting Xiao ◽  
Canlin Gan ◽  
Qiongfen Zhang
Keyword(s):  
Variational Method ◽  
Type Equation ◽  
Previous Literature ◽  
Type Problem ◽  
Nodal Domains ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Kirchhoff Type Problem ◽  
Quantitative Deformation Lemma ◽  
Least Energy

In this paper, we study the Kirchhoff-type equation: − a + b ∫ ℝ 3     ∇ u 2 d x Δ u + V x u = Q x f u , in   ℝ 3 , where a , b > 0 , f ∈ C 1 ℝ 3 , ℝ , and V , Q ∈ C 1 ℝ 3 , ℝ + . V x and Q x are vanishing at infinity. With the aid of the quantitative deformation lemma and constraint variational method, we prove the existence of a sign-changing solution u to the above equation. Moreover, we obtain that the sign-changing solution u has exactly two nodal domains. Our results can be seen as an improvement of the previous literature.

scholarly journals Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Sihua Liang ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
State Energy ◽  
Convergence Property ◽  
Degree Theory ◽  
Critical Sobolev Exponent ◽  
Topological Degree Theory ◽  
Nodal Solutions ◽  
Sobolev Exponent ◽  
Sign Changing Solutions ◽  
Least Energy ◽  
Kirchhoff Problem

AbstractIn this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^p\right) (-\Delta )^s_pu = \lambda |u|^{q-2}u\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\quad \text {in } \Omega , \\ u=0 &{}\quad \text {in } {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$ a + b [ u ] s , p p ( - Δ ) p s u = λ | u | q - 2 u ln | u | 2 + | u | p s ∗ - 2 u in Ω , u = 0 in R N \ Ω , where $$N >sp$$ N > s p with $$s \in (0, 1)$$ s ∈ ( 0 , 1 ) , $$p>1$$ p > 1 , and $$\begin{aligned}{}[u]_{s,p}^p =\iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \end{aligned}$$ [ u ] s , p p = ∬ R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y , $$p_s^*=Np/(N-ps)$$ p s ∗ = N p / ( N - p s ) is the fractional critical Sobolev exponent, $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N $$(N\ge 3)$$ ( N ≥ 3 ) is a bounded domain with Lipschitz boundary and $$\lambda $$ λ is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution $$u_b$$ u b . Moreover, for any $$\lambda > 0$$ λ > 0 , we show that the energy of $$u_b$$ u b is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as $$b \rightarrow 0$$ b → 0 .

scholarly journals On a class of Kirchhoff equations involving an anisotropic operator and potential

2020 ◽  
Author(s):  
Mohammed Massar
Keyword(s):  
Variational Methods ◽  
Kirchhoff Equations ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Fractional Equations ◽  
Quantitative Deformation Lemma ◽  
Least Energy

AbstractIn this work, we are concerned with a class of fractional equations of Kirchhoff type with potential. Using variational methods and a variant of quantitative deformation lemma, we prove the existence of a least energy sign-changing solution. Moreover, the existence of infinitely many solution is established.

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

2021 ◽  
Vol 62 (9) ◽  
pp. 091508
Author(s):  
Yuxi Meng ◽  
Xingrui Zhang ◽  
Xiaoming He
Keyword(s):  
Poisson System ◽  
Sign Changing Solutions ◽  
Least Energy
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