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Positive ground states for nonlinearly coupled Choquard type equations with lower critical exponents
Boundary Value Problems ( IF 1.7 ) Pub Date : 2021-01-30 , DOI: 10.1186/s13661-021-01491-z
Huiling Wu

We study the coupled Choquard type system with lower critical exponents $$ \textstyle\begin{cases} -\Delta u+\lambda _{1}(x)u=\mu _{1}(I_{\alpha }* \vert u \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u+\beta (I_{\alpha }* \vert v \vert ^{ \frac{N+\alpha }{N}}) \vert u \vert ^{\frac{\alpha }{N}-1}u,\quad x\in {\mathbb{R}}^{N}, \\ -\Delta v+\lambda _{2}(x)v=\mu _{2}(I_{\alpha }* \vert v \vert ^{ \frac{N+\alpha }{N}}) \vert v \vert ^{\frac{\alpha }{N}-1}v+\beta (I_{\alpha }* \vert u \vert ^{ \frac{N+\alpha }{N}}) \vert v \vert ^{\frac{\alpha }{N}-1}v,\quad x\in {\mathbb{R}}^{N}, \\ u, v\in H^{1}({\mathbb{R}}^{N}), \end{cases} $$ where $N\ge 3$ , $\mu _{1}, \mu _{2}, \beta >0$ , and $\lambda _{1}(x)$ , $\lambda _{2}(x)$ are nonnegative functions. The existence of at least one positive ground state of this system is proved under certain assumptions on $\lambda _{1}$ , $\lambda _{2}$ .

中文翻译:

具有较低临界指数的非线性耦合Choquard型方程的正基态

我们研究具有较低临界指数$$的耦合Choquard类型系统$$ \ textstyle \ begin {cases}-\ Delta u + \ lambda _ {1}(x)u = \ mu _ {1}(I _ {\ alpha} * \ vert u \ vert ^ {\ frac {N + \ alpha} {N}})\ vert u \ vert ^ {\ frac {\ alpha} {N} -1} u + \ beta(I _ {\ alpha} * \ vert v \ vert ^ {\ frac {N + \ alpha} {N}})\ vert u \ vert ^ {\ frac {\ alpha} {N} -1} u,\ quad x \ in {\ mathbb {R}} ^ { N},\\-\ Delta v + \ lambda _ {2}(x)v = \ mu _ {2}(I _ {\ alpha} * \ vert v \ vert ^ {\ frac {N + \ alpha} {N} })\ vert v \ vert ^ {\ frac {\ alpha} {N} -1} v + \ beta(I _ {\ alpha} * \ vert u \ vert ^ {\ frac {N + \ alpha} {N}}) \ vert v \ vert ^ {\ frac {\ alpha} {N} -1} v,\ quad x \ in {\ mathbb {R}} ^ {N},\\ u,v \ in H ^ {1} ({\ mathbb {R}} ^ {N}),\ end {cases} $$,其中$ N \ ge 3 $,$ \ mu _ {1},\ mu _ {2},\ beta> 0 $,和$ \ lambda _ {1}(x)$,$ \ lambda _ {2}(x)$是非负函数。在$ \ lambda _ {1} $,$ \ lambda _ {2} $的某些假设下证明了该系统至少存在一个正基态。
更新日期:2021-02-01
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