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 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 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 Existence of solutions for a class of Kirchhoff-type equations with indefinite potential

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jian Zhou ◽  
Yunshun Wu
Keyword(s):  
Morse Theory ◽  
Existence Of Solutions ◽  
Type Problem ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Indefinite Potential

AbstractIn this paper, we consider the existence of solutions of the following Kirchhoff-type problem: $$\begin{aligned} \textstyle\begin{cases} - (a+b\int _{\mathbb{R}^{3}} \vert \nabla u \vert ^{2}\,dx )\Delta u+ V(x)u=f(x,u) , & \text{in }\mathbb{R}^{3}, \\ u\in H^{1}(\mathbb{R}^{3}),\end{cases}\displaystyle \end{aligned}$$ { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + V ( x ) u = f ( x , u ) , in  R 3 , u ∈ H 1 ( R 3 ) , where $a,b>0$ a , b > 0 are constants, and the potential $V(x)$ V ( x ) is indefinite in sign. Under some suitable assumptions on f, the existence of solutions is obtained by Morse theory.

scholarly journals On a p x -Biharmonic Kirchhoff Problem with Navier Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Hafid Lebrimchi ◽  
Mohamed Talbi ◽  
Mohammed Massar ◽  
Najib Tsouli
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Nontrivial Solution ◽  
Existence Of Solutions ◽  
Type Problem ◽  
Navier Boundary Conditions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Kirchhoff Problem

In this article, we study the existence of solutions for nonlocal p x -biharmonic Kirchhoff-type problem with Navier boundary conditions. By different variational methods, we determine intervals of parameters for which this problem admits at least one nontrivial solution.

Sign in / Sign up

Export Citation Format

Share Document