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 Nodal solutions for double phase Kirchhoff problems with vanishing potentials

2020 ◽  
pp. 1-26
Author(s):  
Teresa Isernia ◽  
Dušan D. Repovš
Keyword(s):  
Continuous Function ◽  
Continuous Functions ◽  
Type Problem ◽  
Nodal Solutions ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Double Phase ◽  
Kirchhoff Type Problem ◽  
Quantitative Deformation Lemma ◽  
Vanishing Potentials

We consider the following ( p , q )-Laplacian Kirchhoff type problem − ( a + b ∫ R 3 | ∇ u | p d x ) Δ p u − ( c + d ∫ R 3 | ∇ u | q d x ) Δ q u + V ( x ) ( | u | p − 2 u + | u | q − 2 u ) = K ( x ) f ( u ) in  R 3 , where a , b , c , d > 0 are constants, 3 2 < p < q < 3, V : R 3 → R and K : R 3 → R are positive continuous functions allowed for vanishing behavior at infinity, and f is a continuous function with quasicritical growth. Using a minimization argument and a quantitative deformation lemma we establish the existence of nodal solutions.

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.

Semi-classical solutions for Kirchhoff type problem with a critical frequency

2020 ◽  
pp. 1-38 ◽  
Author(s):  
Qilin Xie ◽  
Xu Zhang
Keyword(s):  
Variational Method ◽  
Critical Frequency ◽  
Classical Solutions ◽  
Positive Parameter ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Classical State ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Image Position

Abstract In the present paper, we consider the following Kirchhoff type problem $$ -\Big(\varepsilon^2+\varepsilon b \int_{\mathbb R^3} | \nabla v|^2\Big) \Delta v+V(x)v=|v|^{p-2}v \quad {\rm in}\ \mathbb{R}^3, $$ where b > 0, p ∈ (4, 6), the potential $V\in C(\mathbb R^3,\mathbb R)$ and ɛ is a positive parameter. The existence and multiplicity of semi-classical state solutions are obtained by variational method for this problem with several classes of critical frequency potentials, i.e., $\inf _{\mathbb R^N} V=0$ . As to Kirchhoff type problem, little has been done for the critical frequency cases in the literature, especially the potential may vanish at infinity.

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.

Existence and multiplicity of solutions for an indefinite Kirchhoff-type equation in bounded domains

2016 ◽  
Vol 146 (2) ◽  
pp. 435-448 ◽  
Author(s):  
Juntao Sun ◽  
Tsung-fang Wu
Keyword(s):  
Variational Principle ◽  
Mountain Pass Theorem ◽  
Type Equation ◽  
Bounded Domains ◽  
Type Problem ◽  
Smooth Bounded Domain ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Kirchhoff Type Problem ◽  
Image Position

We study the indefinite Kirchhoff-type problem where Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

scholarly journals On 𝓅 ( x ) -Kirchhoff-type equation involving 𝓅 ( x ) -biharmonic operator via genus theor

2020 ◽  
Vol 72 (6) ◽  
pp. 842-851
Author(s):  
S. Taarabti ◽  
Z. El Allali ◽  
K. Ben Haddouch
Keyword(s):  
Weak Solutions ◽  
Variational Approach ◽  
Type Equation ◽  
Biharmonic Operator ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Existence And Multiplicity ◽  
Genus Theory ◽  
Kirchhoff Type Equation ◽  
Kirchhoff Type Problem

UDC 517.9 The paper deals with the existence and multiplicity of nontrivial weak solutions for the 𝓅 ( x ) -Kirchhoff-type problem, u = Δ u = 0 o n ∂ Ω . By using variational approach and Krasnoselskii’s genus theory, we prove the existence and multiplicity of solutions for the 𝓅 ( x ) -Kirchhoff-type equation.

scholarly journals A Kirchhoff-type problem involving concave-convex nonlinearities

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yuan Gao ◽  
Lishan Liu ◽  
Shixia Luan ◽  
Yonghong Wu
Keyword(s):  
Energy Level ◽  
Variational Methods ◽  
Ground State ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Least Energy

AbstractA Kirchhoff-type problem with concave-convex nonlinearities is studied. By constrained variational methods on a Nehari manifold, we prove that this problem has a sign-changing solution with least energy. Moreover, we show that the energy level of this sign-changing solution is strictly larger than the double energy level of the ground state solution.

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