scholarly journals Ground State Solutions for a Nonlocal Equation in $$\mathbb {R}^2$$ Involving Vanishing Potentials and Exponential Critical Growth

2021 ◽  
Author(s):  
Francisco S. B. Albuquerque ◽  
Marcelo C. Ferreira ◽  
Uberlandio B. Severo
Keyword(s):  
Ground State ◽  
Critical Growth ◽  
Ground State Solutions ◽  
Nonlocal Equation ◽  
Exponential Critical Growth ◽  
Vanishing Potentials

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.

Sign in / Sign up

Export Citation Format

Share Document