scholarly journals The existence and nonexistence results of ground state nodal solutions for a Kirchhoff type problem

2017 ◽  
Vol 16 (2) ◽  
pp. 611-627 ◽  
Author(s):  
Chun-Lei Tang ◽  
Xiao-Jing Zhong
Keyword(s):  
Ground State ◽  
Type Problem ◽  
Nodal Solutions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Existence And Nonexistence

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 Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity

2018 ◽  
Vol 9 (1) ◽  
pp. 148-167 ◽  
Author(s):  
Sitong Chen ◽  
Binlin Zhang ◽  
Xianhua Tang
Keyword(s):  
Ground State ◽  
Existence Result ◽  
Riesz Potential ◽  
Analytical Techniques ◽  
Existence Results ◽  
Type Problem ◽  
Ground State Solutions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Kirchhoff Type Problems

Abstract This paper is concerned with the following Kirchhoff-type problem with convolution nonlinearity: -\bigg{(}a+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,\mathrm{d}x\bigg{)}% \Delta u+V(x)u=(I_{\alpha}*F(u))f(u),\quad x\in{\mathbb{R}}^{3},\,u\in H^{1}(% \mathbb{R}^{3}), where {a,b>0} , {I_{\alpha}\colon\mathbb{R}^{3}\rightarrow\mathbb{R}} , with {\alpha\in(0,3)} , is the Riesz potential, {V\in\mathcal{C}(\mathbb{R}^{3},[0,\infty))} , {f\in\mathcal{C}(\mathbb{R},\mathbb{R})} and {F(t)\kern-1.0pt=\kern-1.0pt\int_{0}^{t}f(s)\,\mathrm{d}s} . By using variational and some new analytical techniques, we prove that the above problem admits ground state solutions under mild assumptions on V and f. Moreover, we give a non-existence result. In particular, our results extend and improve the existing ones, and fill a gap in the case where {f(u)=|u|^{q-2}u} , with {q\in(1+\alpha/3,2]} .

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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