Ground state solutions for a modified fractional Schrödinger equation with critical exponent

2020 ◽  
Vol 43 (6) ◽  
pp. 2924-2944
Author(s):  
Xian Wu ◽  
Wei Zhang ◽  
Xingwei Zhou
Keyword(s):  
Schrödinger Equation ◽  
Critical Exponent ◽  
Ground State ◽  
Schrodinger Equation ◽  
Fractional Schrödinger Equation ◽  
Ground State Solutions ◽  
Fractional Schrodinger Equation

scholarly journals Ground state solutions for fractional Schrödinger equation with variable potential and Berestycki–Lions type nonlinearity

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Jing Chen ◽  
Zu Gao
Keyword(s):  
Schrödinger Equation ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlocal Problem ◽  
Minimization Method ◽  
Fractional Schrödinger Equation ◽  
Ground State Solutions ◽  
The One ◽  
Least Energy Solutions ◽  
Fractional Schrodinger Equation

Abstract We consider the following nonlinear fractional Schrödinger equation: $$ (-\triangle )^{s} u+V(x)u=g(u) \quad \text{in } \mathbb{R} ^{N}, $$ ( − △ ) s u + V ( x ) u = g ( u ) in  R N , where $s\in (0, 1)$ s ∈ ( 0 , 1 ) , $N>2s$ N > 2 s , $V(x)$ V ( x ) is differentiable, and $g\in C ^{1}(\mathbb{R} , \mathbb{R} )$ g ∈ C 1 ( R , R ) . By exploiting the minimization method with a constraint over Pohoz̆aev manifold, we obtain the existence of ground state solutions. With the help of Pohoz̆aev identity we also process the existence of the least energy solutions for the above equation. Our results improve the existing study on this nonlocal problem with Berestycki–Lions type nonlinearity to the one that does not need the oddness assumption.

scholarly journals Ground State Solution of Pohožaev Type for Quasilinear Schrödinger Equation Involving Critical Exponent in Orlicz Space

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 779 ◽  
Author(s):  
Jianqing Chen ◽  
Qian Zhang
Keyword(s):  
Schrödinger Equation ◽  
Critical Exponent ◽  
Orlicz Space ◽  
Ground State ◽  
Schrodinger Equation ◽  
Nonlinear Term ◽  
Ground State Solution ◽  
Existence Results ◽  
Quasilinear Schrödinger Equation ◽  
Ground State Solutions

We study the following quasilinear Schrödinger equation involving critical exponent - Δ u + V ( x ) u - Δ ( u 2 ) u = A ( x ) | u | p - 1 u + λ B ( x ) u 3 N + 2 N - 2 , u ( x ) > 0 for x ∈ R N and u ( x ) → 0 as | x | → ∞ . By using a monotonicity trick and global compactness lemma, we prove the existence of positive ground state solutions of Pohožaev type. The nonlinear term | u | p - 1 u for the well-studied case p ∈ [ 3 , 3 N + 2 N - 2 ) , and the less-studied case p ∈ [ 2 , 3 ) , and for the latter case few existence results are available in the literature. Our results generalize partial previous works.

Solutions for Fractional Schrödinger Equation Involving Critical Exponent via Local Pohozaev Identities

2020 ◽  
Vol 20 (1) ◽  
pp. 185-211 ◽  
Author(s):  
Yuxia Guo ◽  
Ting Liu ◽  
Jianjun Nie
Keyword(s):  
Schrödinger Equation ◽  
Critical Exponent ◽  
Schrodinger Equation ◽  
Critical Sobolev Exponent ◽  
Laplacian Operator ◽  
Extension Problem ◽  
Infinitely Many Solutions ◽  
Fractional Schrödinger Equation ◽  
Finite Dimensional Reduction ◽  
Fractional Schrodinger Equation

AbstractWe consider the following fractional Schrödinger equation involving critical exponent:\left\{\begin{aligned} &\displaystyle(-\Delta)^{s}u+V(y)u=u^{2^{*}_{s}-1}&&% \displaystyle\text{in }\mathbb{R}^{N},\\ &\displaystyle u>0,&&\displaystyle y\in\mathbb{R}^{N},\end{aligned}\right.where {N\geq 3} and {2^{*}_{s}=\frac{2N}{N-2s}} is the critical Sobolev exponent. Under some suitable assumptions of the potential function {V(y)}, by using a finite-dimensional reduction method, combined with various local Pohazaev identities, we prove the existence of infinitely many solutions. Due to the nonlocality of the fractional Laplacian operator, we need to study the corresponding harmonic extension problem.

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