Concentration behavior of solutions for fractional Schrödinger equations involving critical exponent

2020 ◽  
Vol 61 (7) ◽  
pp. 071513
Author(s):  
Quanqing Li ◽  
Kaimin Teng ◽  
Jian Zhang ◽  
Wenbo Wang
Keyword(s):  
Critical Exponent ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations ◽  
Concentration Behavior

Existence of normalized solutions for nonlinear fractional Schrödinger equations with trapping potentials

2018 ◽  
Vol 149 (03) ◽  
pp. 617-653 ◽  
Author(s):  
Miao Du ◽  
Lixin Tian ◽  
Jun Wang ◽  
Fubao Zhang
Keyword(s):  
Fractional Laplacian ◽  
Energy Estimates ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Schrödinger Equations ◽  
Nonlinear Fractional Schrödinger Equations ◽  
Embedding Constant ◽  
Normalized Solutions ◽  
Concentration Behavior ◽  
Nonlocal Nature

AbstractIn this paper, we study the existence, nonexistence and mass concentration of L2-normalized solutions for nonlinear fractional Schrödinger equations. Comparingwith the Schrödinger equation, we encounter some new challenges due to the nonlocal nature of the fractional Laplacian. We first prove that the optimal embedding constant for the fractional Gagliardo–Nirenberg–Sobolev inequality can be expressed by exact form, which improves the results of [17, 18]. By doing this, we then establish the existence and nonexistence of L2-normalized solutions for this equation. Finally, under a certain type of trapping potentials, by using some delicate energy estimates we present a detailed analysis of the concentration behavior of L2-normalized solutions in the mass critical case.

scholarly journals Sign-Changing Solutions for a Class of Zero Mass Nonlocal Schrödinger Equations

2019 ◽  
Vol 19 (1) ◽  
pp. 113-132 ◽  
Author(s):  
Vincenzo Ambrosio ◽  
Giovany M. Figueiredo ◽  
Teresa Isernia ◽  
Giovanni Molica Bisci
Keyword(s):  
Careful Analysis ◽  
Continuous Functions ◽  
Nehari Manifold ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Deformation Lemma ◽  
Fractional Schrödinger Equations ◽  
Fractional Spaces ◽  
The Nehari Manifold ◽  
Sign Changing Solutions

Abstract We consider the following class of fractional Schrödinger equations: (-\Delta)^{\alpha}u+V(x)u=K(x)f(u)\quad\text{in }\mathbb{R}^{N}, where {\alpha\in(0,1)} , {N>2\alpha} , {(-\Delta)^{\alpha}} is the fractional Laplacian, V and K are positive continuous functions which vanish at infinity, and f is a continuous function. By using a minimization argument and a quantitative deformation lemma, we obtain the existence of a sign-changing solution. Furthermore, when f is odd, we prove that the above problem admits infinitely many nontrivial solutions. Our result extends to the fractional framework some well-known theorems proved for elliptic equations in the classical setting. With respect to these cases studied in the literature, the nonlocal one considered here presents some additional difficulties, such as the lack of decompositions involving positive and negative parts, and the non-differentiability of the Nehari Manifold, so that a careful analysis of the fractional spaces involved is necessary.

Sign in / Sign up

Export Citation Format

Share Document