Ground state solutions for Kirchhoff-type problems with convolution nonlinearity and Berestycki–Lions type conditions

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Die Hu ◽  
Xianhua Tang ◽  
Shuai Yuan ◽  
Qi Zhang
Keyword(s):  
Ground State ◽  
Ground State Solutions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problems

scholarly journals Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity

2018 ◽  
Vol 9 (1) ◽  
pp. 148-167 ◽  
Author(s):  
Sitong Chen ◽  
Binlin Zhang ◽  
Xianhua Tang
Keyword(s):  
Ground State ◽  
Existence Result ◽  
Riesz Potential ◽  
Analytical Techniques ◽  
Existence Results ◽  
Type Problem ◽  
Ground State Solutions ◽  
Kirchhoff Type ◽  
Kirchhoff Type Problem ◽  
Kirchhoff Type Problems

Abstract This paper is concerned with the following Kirchhoff-type problem with convolution nonlinearity: -\bigg{(}a+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,\mathrm{d}x\bigg{)}% \Delta u+V(x)u=(I_{\alpha}*F(u))f(u),\quad x\in{\mathbb{R}}^{3},\,u\in H^{1}(% \mathbb{R}^{3}), where {a,b>0} , {I_{\alpha}\colon\mathbb{R}^{3}\rightarrow\mathbb{R}} , with {\alpha\in(0,3)} , is the Riesz potential, {V\in\mathcal{C}(\mathbb{R}^{3},[0,\infty))} , {f\in\mathcal{C}(\mathbb{R},\mathbb{R})} and {F(t)\kern-1.0pt=\kern-1.0pt\int_{0}^{t}f(s)\,\mathrm{d}s} . By using variational and some new analytical techniques, we prove that the above problem admits ground state solutions under mild assumptions on V and f. Moreover, we give a non-existence result. In particular, our results extend and improve the existing ones, and fill a gap in the case where {f(u)=|u|^{q-2}u} , with {q\in(1+\alpha/3,2]} .

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