Multiplicity and concentration of nontrivial solutions for a class of fractional Kirchhoff equations with steep potential well

2021 ◽  
Author(s):  
Liuyang Shao ◽  
Haibo Chen
Keyword(s):  
Potential Well ◽  
Nontrivial Solutions ◽  
Kirchhoff Equations ◽  
Steep Potential Well

Existence and Concentration of Solutions for Choquard Equations with Steep Potential Well and Doubly Critical Exponents

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Yong-Yong Li ◽  
Gui-Dong Li ◽  
Chun-Lei Tang
Keyword(s):  
Critical Exponents ◽  
Ground State ◽  
Mountain Pass Theorem ◽  
Riesz Potential ◽  
Nehari Manifold ◽  
Linking Theorem ◽  
Potential Well ◽  
Nontrivial Solutions ◽  
Choquard Equation ◽  
Steep Potential Well

AbstractIn this paper, we investigate the non-autonomous Choquard equation-\Delta u+\lambda V(x)u=(I_{\alpha}\ast F(u))F^{\prime}(u)\quad\text{in}\ \mathbb{R}^{N},where N\geq 4, \lambda>0, V\in C(\mathbb{R}^{N},\mathbb{R}) is bounded from below and has a potential well, I_{\alpha} is the Riesz potential of order \alpha\in(0,N) and F(u)=\frac{1}{2_{\alpha}^{*}}\lvert u\rvert^{2_{\alpha}^{*}}+\frac{1}{2_{*}^{\alpha}}\lvert u\rvert^{2_{*}^{\alpha}}, in which 2_{\alpha}^{*}=\frac{N+\alpha}{N-2} and 2_{*}^{\alpha}=\frac{N+\alpha}{N} are upper and lower critical exponents due to the Hardy–Littlewood–Sobolev inequality, respectively. Based on the variational methods, by combining the mountain pass theorem and Nehari manifold, we obtain the existence and concentration of positive ground state solutions for 𝜆 large enough if 𝑉 is nonnegative in \mathbb{R}^{N}; further, by the linking theorem, we prove the existence of nontrivial solutions for 𝜆 large enough if 𝑉 changes sign in \mathbb{R}^{N}.

scholarly journals Lévy Flights in a Steep Potential Well

2004 ◽  
Vol 115 (5/6) ◽  
pp. 1505-1535 ◽  
Author(s):  
Aleksei V. Chechkin ◽  
Vsevolod Yu. Gonchar ◽  
Joseph Klafter ◽  
Ralf Metzler ◽  
Leonid V. Tanatarov
Keyword(s):  
Potential Well ◽  
Lévy Flights ◽  
Levy Flights ◽  
Steep Potential Well
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