scholarly journals Existence of positive solutions for a semilinear Schrödinger equation in R N $\mathbb{R}^{N}$

2015 ◽  
Vol 2015 (1) ◽  
Author(s):  
Houqing Fang ◽  
Jun Wang
Keyword(s):  
Schrödinger Equation ◽  
Positive Solutions ◽  
Schrodinger Equation ◽  
Existence Of Positive Solutions ◽  
Semilinear Schrödinger Equation

Positive solutions for a quasilinear Schrödinger equation involving Hardy potential and critical exponent

2014 ◽  
Vol 16 (06) ◽  
pp. 1450034 ◽  
Author(s):  
Xiaoyu Zeng ◽  
Yimin Zhang ◽  
Huan-Song Zhou
Keyword(s):  
Schrödinger Equation ◽  
Critical Exponent ◽  
Positive Solutions ◽  
Schrodinger Equation ◽  
Quasilinear Equation ◽  
Hardy Potential ◽  
Quasilinear Schrödinger Equation ◽  
Semilinear Equation ◽  
Existence Of Positive Solutions ◽  
Nehari Method

We are concerned with positive solutions of a quasilinear Schrödinger equation with Hardy potential and critical exponent. Different from the semilinear equation, the Hardy term in our equation is not only singular, but also nonlinear. It seems unlikely to get solutions for our equation in H1(ℝN) ∩ L∞(ℝN) by using Nehari method as Liu–Liu–Wang [Ground states for quasilinear Schrödinger equations with critical growth, Calc. Var. Partial Differential Equations 46 (2013) 641–669]. In this paper, by transforming the quasilinear equation to a semilinear equation, we established the existence of positive solutions for the quasilinear Schrödinger equation in H1(ℝN) under suitable conditions.

scholarly journals Positive solutions for a class of generalized quasilinear Schrödinger equation involving concave and convex nonlinearities in Orilicz space

2021 ◽  
pp. 1-26
Author(s):  
Yan Meng ◽  
Xianjiu Huang ◽  
Jianhua Chen
Keyword(s):  
Schrödinger Equation ◽  
Positive Solutions ◽  
Schrodinger Equation ◽  
Mountain Pass Theorem ◽  
Mountain Pass ◽  
Quasilinear Schrödinger Equation ◽  
Existence Of Positive Solutions ◽  
Change Of Variable ◽  
Concave And Convex Nonlinearities

In this paper, we study the following generalized quasilinear Schrödinger equation − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = λ f ( x , u ) + h ( x , u ) , x ∈ R N , where λ > 0 , N ≥ 3 , g ∈ C 1 ( R , R + ) . By using a change of variable, we obtain the existence of positive solutions for this problem with concave and convex nonlinearities via the Mountain Pass Theorem. Our results generalize some existing results.

scholarly journals Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian

2012 ◽  
Vol 142 (6) ◽  
pp. 1237-1262 ◽  
Author(s):  
Patricio Felmer ◽  
Alexander Quaas ◽  
Jinggang Tan
Keyword(s):  
Schrödinger Equation ◽  
Positive Solutions ◽  
Nonlinear Schrödinger Equation ◽  
Schrodinger Equation ◽  
Fractional Laplacian ◽  
Nonlinear Schrodinger Equation ◽  
Nonlinear Schrödinger ◽  
Symmetry Properties ◽  
Existence Of Positive Solutions ◽  
Image Position

We study the existence of positive solutions for the nonlinear Schrödinger equation with the fractional LaplacianFurthermore, we analyse the regularity, decay and symmetry properties of these solutions.

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