Asymptotic Behavior of Least Energy Solutions for a Critical Elliptic System

2015 ◽  
Vol 2015 (21) ◽  
pp. 11045-11082 ◽  
Author(s):  
Zhijie Chen ◽  
Chang-Shou Lin
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic System ◽  
Least Energy Solutions ◽  
Least Energy

Least energy solutions for indefinite biharmonic problems via modified Nehari–Pankov manifold

2018 ◽  
Vol 20 (04) ◽  
pp. 1750047 ◽  
Author(s):  
Miaomiao Niu ◽  
Zhongwei Tang ◽  
Lushun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Potential Function ◽  
Biharmonic Equation ◽  
Positive Function ◽  
Energy Solution ◽  
Zero Set ◽  
Least Energy Solution ◽  
Least Energy Solutions ◽  
Least Energy ◽  
Biharmonic Problems

In this paper, by using a modified Nehari–Pankov manifold, we prove the existence and the asymptotic behavior of least energy solutions for the following indefinite biharmonic equation: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] is a parameter, [Formula: see text] is a nonnegative potential function with nonempty zero set [Formula: see text], [Formula: see text] is a positive function such that the operator [Formula: see text] is indefinite and non-degenerate for [Formula: see text] large. We show that both in subcritical and critical cases, equation [Formula: see text] admits a least energy solution which for [Formula: see text] large localized near the zero set [Formula: see text].

Solutions for the Problems Involving Fractional Laplacian and Indefinite Potentials

2017 ◽  
Vol 17 (3) ◽  
Author(s):  
Zhongwei Tang ◽  
Lushun Wang
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Laplacian ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Definition Of ◽  
Least Energy Solutions ◽  
Least Energy

AbstractIn this paper, we consider a class of Schrödinger equations involving fractional Laplacian and indefinite potentials. By modifying the definition of the Nehari–Pankov manifold, we prove the existence and asymptotic behavior of least energy solutions. As the fractional Laplacian is nonlocal, when the bottom of the potentials contains more than one isolated components, the least energy solutions may localize near all the isolated components simultaneously. This phenomenon is different from the Laplacian.

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