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Register a new userHigh energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents
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We study the coupled Hartree system $$ left{begin{array}{ll} -Delta u+ V_1(x)u =alpha_1big(|x|^{-4}ast u^{2}big)u+beta big(|x|^{-4}ast v^{2}big)u &mbox{in} mathbb{R}^N,[1mm] -Delta v+ V_2(x)v =alpha_2big(|x|^{-4}ast v^{2}big)v +betabig(|x|^{-4}ast u^{2}big)v &mbox{in} mathbb{R}^N, end{array}right. $$ where $Ngeq 5$, $beta>max{alpha_1,alpha_2}geqmin{alpha_1,alpha_2}>0$, and $V_1,,V_2in L^{N/2}(mathbb{R}^N)cap L_{text{loc}}^{infty}(mathbb{R}^N)$ are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with $V_1=V_2=0$ we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that $|V_1|_{L^{N/2}(mathbb{R}^N)}+|V_2|_{L^{N/2}(mathbb{R}^N)}>0$ is suitably small.
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In this paper, we consider the strongly coupled nonlinear Kirchhoff-type system with vanshing potentials: [begin{cases} -left(a_1+b_1int_{mathbb{R}^3}| abla u|^2dxright)Delta u+lambda V(x)u=frac{alpha}{alpha+beta}|u|^{alpha-2}u|v|^{beta},&xinmathbb{R}^3, -left(a_2+b_2int_{mathbb{R}^3}| abla v|^2dxright)Delta v+lambda W(x)v=frac{beta}{alpha+beta}|u|^{alpha}|v|^{beta-2}v,&xinmathbb{R}^3, u,vin mathcal{D}^{1,2}(mathbb{R}^3), end{cases}] where $a_i>0$ are constants, $lambda,b_i>0$ are parameters for $i=1,2$, $alpha,beta>1$ satisfy $alpha+betale4$, the nonlinear term $F(x,u,v)=|u|^alpha|v|^beta$ is not 4-superlinear at infinity, $V(x)$, $W(x)$ are nonnegative continuous potentials. By establishing some new estimates and truncation technique, we obtain the existence of positive vector solutions for the above system when $b_1+b_2$ small and $lambda$ large. Moreover, the asymptotic behavior of these vector solutions is also explored as $textbf{b}=(b_1,b_2)to bf{0}$ and $lambdatoinfty$. In particular, our results extend some known ones in previous papers that only deals with the case where $alpha,beta>2$ with $alpha+beta<6$.
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