scholarly journals On ground state solutions for the Schrödinger–Poisson equations with critical growth

2014 ◽  
Vol 412 (1) ◽  
pp. 435-448 ◽  
Author(s):  
Zhisu Liu ◽  
Shangjiang Guo
Keyword(s):  
Ground State ◽  
Critical Growth ◽  
Ground State Solutions ◽  
Poisson Equations

Ground State Solutions for Quasilinear Schrödinger Equations with Critical Growth and Lower Power Subcritical Perturbation

2019 ◽  
Vol 19 (1) ◽  
pp. 219-237 ◽  
Author(s):  
Yinbin Deng ◽  
Wentao Huang ◽  
Shen Zhang
Keyword(s):  
Ground State ◽  
Nonlinear Function ◽  
Critical Growth ◽  
Quasilinear Schrödinger Equation ◽  
Monotone Method ◽  
Lower Power ◽  
Ground State Solutions ◽  
Change Of Variables ◽  
The Given ◽  
Edinburgh Sect

Abstract We study the following generalized quasilinear Schrödinger equation: -(g^{2}(u)\nabla u)+g(u)g^{\prime}(u)|\nabla u|^{2}+V(x)u=h(u),\quad x\in% \mathbb{R}^{N}, where {N\geq 3} , {g\colon\mathbb{R}\rightarrow\mathbb{R}^{+}} is an even differentiable function such that {g^{\prime}(t)\geq 0} for all {t\geq 0} , {h\in C^{1}(\mathbb{R},\mathbb{R})} is a nonlinear function including critical growth and lower power subcritical perturbation, and the potential {V(x)\colon\mathbb{R}^{N}\rightarrow\mathbb{R}} is positive. Since the subcritical perturbation does not satisfy the (AR) condition, the standard variational method cannot be used directly. Combining the change of variables and the monotone method developed by Jeanjean in [L. Jeanjean, On the existence of bounded Palais–Smale sequences and application to a Landesman–Lazer-type problem set on {\mathbf{R}}^{N} , Proc. Roy. Soc. Edinburgh Sect. A 129 1999, 4, 787–809], we obtain the existence of positive ground state solutions for the given problem.

scholarly journals On ground state solutions for a Kirchhoff type equation with critical growth

2016 ◽  
Vol 72 (3) ◽  
pp. 729-740 ◽  
Author(s):  
Chun-Yu Lei ◽  
Hong-Min Suo ◽  
Chang-Mu Chu ◽  
Liu-Tao Guo
Keyword(s):  
Ground State ◽  
Type Equation ◽  
Critical Growth ◽  
Ground State Solutions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Equation
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