Existence and concentration of ground state sign-changing solutions for Kirchhoff type equations with steep potential well and nonlinearity

2018 ◽  
Vol 49 (2) ◽  
pp. 1
Author(s):  
Bitao Cheng ◽  
Jianhua Chen ◽  
Xianhua Tang
Keyword(s):  
Ground State ◽  
Potential Well ◽  
Kirchhoff Type ◽  
Steep Potential Well ◽  
Sign Changing Solutions

Ground State Solutions for Kirchhoff Type Quasilinear Equations

2019 ◽  
Vol 19 (2) ◽  
pp. 353-373
Author(s):  
Xiangqing Liu ◽  
Junfang Zhao
Keyword(s):  
Ground State ◽  
Quasilinear Equations ◽  
Ground State Solutions ◽  
Kirchhoff Type ◽  
Sign Changing Solutions ◽  
Nehari Method

AbstractIn this paper, we are concerned with quasilinear equations of Kirchhoff type, and prove the existence of ground state signed solutions and sign-changing solutions by using the Nehari method.

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}.

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