A critical exponent for Hénon type equation on the hyperbolic space

2015 ◽  
Vol 129 ◽  
pp. 343-370 ◽  
Author(s):  
Shoichi Hasegawa
Keyword(s):  
Critical Exponent ◽  
Hyperbolic Space ◽  
Type Equation

Ground State and Multiple Solutions for Kirchhoff Type Equations With Critical Exponent

2018 ◽  
Vol 61 (2) ◽  
pp. 353-369 ◽  
Author(s):  
Dongdong Qin ◽  
Yubo He ◽  
Xianhua Tang
Keyword(s):  
Variational Methods ◽  
Critical Exponent ◽  
Positive Solutions ◽  
Ground State ◽  
Multiple Solutions ◽  
Type Equation ◽  
Ground State Solution ◽  
Nehari Manifold ◽  
Kirchhoff Type ◽  
The Nehari Manifold

AbstractIn this paper, we consider the following critical Kirchhoff type equation:By using variational methods that are constrained to the Nehari manifold, we prove that the above equation has a ground state solution for the case when 3 < q < 5. The relation between the number of maxima of Q and the number of positive solutions for the problem is also investigated.

scholarly journals Groundstates for Choquard type equations with Hardy–Littlewood–Sobolev lower critical exponent

2019 ◽  
Vol 150 (3) ◽  
pp. 1377-1400 ◽  
Author(s):  
Daniele Cassani ◽  
Jean Van Schaftingen ◽  
Jianjun Zhang
Keyword(s):  
Critical Exponent ◽  
Ground State ◽  
Riesz Potential ◽  
Type Equation ◽  
Ground State Solution ◽  
State Solution ◽  
Choquard Equation ◽  
Resonant Case ◽  
Image Position ◽  
Nonlocal Nonlinear

AbstractFor the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, $$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$where $N\ges 3$, Vμ,ν :ℝN → ℝ is an external potential defined for μ, ν > 0 and x ∈ ℝN by Vμ,ν(x) = 1 − μ/(ν2 + |x|2) and $I_\alpha : {\open R}^N \to 0$ is the Riesz potential for α ∈ (0, N), we exhibit two thresholds μν, μν > 0 such that the equation admits a positive ground state solution if and only if μν < μ < μν and no ground state solution exists for μ < μν. Moreover, if μ > max{μν, N2(N − 2)/4(N + 1)}, then equation still admits a sign changing ground state solution provided $N \ges 4$ or in dimension N = 3 if in addition 3/2 < α < 3 and $\ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} $, namely in the non-resonant case.

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