Ground State Solutions for a Class of Periodic Kirchhoff-Type Equation in $${\mathbb {R}}^3$$ Involving Critical Sobolev Exponent

2019 ◽  
Author(s):  
Sofiane Khoutir
Keyword(s):  
Ground State ◽  
Type Equation ◽  
Critical Sobolev Exponent ◽  
Ground State Solutions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Sobolev Exponent

Ground State Solutions for the Nonlinear Schrödinger–Bopp–Podolsky System with Critical Sobolev Exponent

2020 ◽  
Vol 20 (3) ◽  
pp. 511-538 ◽  
Author(s):  
Lin Li ◽  
Patrizia Pucci ◽  
Xianhua Tang
Keyword(s):  
Ground State ◽  
Critical Sobolev Exponent ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
State Solution ◽  
Ground State Solutions ◽  
Nonlinear Schrödinger ◽  
Sobolev Exponent

AbstractIn this paper, we study the existence of ground state solutions for the nonlinear Schrödinger–Bopp–Podolsky system with critical Sobolev exponent\left\{\begin{aligned} \displaystyle{}{-}\Delta u+V(x)u+q^{2}\phi u&% \displaystyle=\mu|u|^{p-1}u+|u|^{4}u&&\displaystyle\phantom{}\mbox{in }\mathbb% {R}^{3},\\ \displaystyle{-}\Delta\phi+a^{2}\Delta^{2}\phi&\displaystyle=4\pi u^{2}&&% \displaystyle\phantom{}\mbox{in }\mathbb{R}^{3},\end{aligned}\right.where {\mu>0} is a parameter and {2<p<5}. Under certain assumptions on V, we prove the existence of a nontrivial ground state solution, using the method of the Pohozaev–Nehari manifold, the arguments of Brézis–Nirenberg, the monotonicity trick and a global compactness lemma.

The existence of a ground-state solution for a class of Kirchhoff-type equations in ℝN

2016 ◽  
Vol 146 (2) ◽  
pp. 371-391 ◽  
Author(s):  
Jiu Liu ◽  
Jia-Feng Liao ◽  
Chun-Lei Tang
Keyword(s):  
Ground State ◽  
Type Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation ◽  
Image Position

In this paper, we study the following Kirchhoff-type equation: where a, b are positive constants and N = 1, 2, 3. Under appropriate assumptions on V, K and g, we obtain a ground-state solution by using the approach developed by Szulkin and Weth in 2010.

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