In this paper, we investigate the non-autonomous Choquard equation -Δ⁢u+λ⁢V⁢(x)⁢u=(Iα∗F⁢(u))⁢F′⁢(u) in⁢RN,-\Delta u+\lambda V(x)u=(I_{\alpha}\ast F(u))F^{\prime}(u)\quad\text{in}\ \mathbb{R}^{N}, where N≥4N\geq 4, λ>0\lambda>0, V∈C⁢(RN,R)V\in C(\mathbb{R}^{N},\mathbb{R}) is bounded from below and has a potential well, IαI_{\alpha} is the Riesz potential of order α∈(0,N)\alpha\in(0,N) and F⁢(u)=12α*⁢|u|2α*+12*α⁢|u|2*αF(u)=\frac{1}{2_{\alpha}^{*}}\lvert u\rvert^{2_{\alpha}^{*}}+\frac{1}{2_{*}^{\alpha}}\lvert u\rvert^{2_{*}^{\alpha}}, in which 2α*=N+αN-22_{\alpha}^{*}=\frac{N+\alpha}{N-2} and 2*α=N+αN2_{*}^{\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 RN\mathbb{R}^{N}; further, by the linking theorem, we prove the existence of nontrivial solutions for 𝜆 large enough if 𝑉 changes sign in RN\mathbb{R}^{N}.

中文翻译：

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具有陡势势阱和双临界指数的二次方程解的存在性和集中性

本文研究investigateRN，-\ Delta中的非自治Choquard方程-Δ⁢u+λ⁢V⁢（x）⁢u=（Iα*F⁢（u））⁢F′⁢（u） u + \ lambda V（x）u =（I _ {\ alpha} \ ast F（u））F ^ {\ prime}（u）\ quad \ text {in} \ \ mathbb {R} ^ {N}，其中N≥4N\ geq 4，λ> 0 \ lambda> 0，C（\ mathbb {R} ^ {N}，\ mathbb {R}）中的V∈C⁢（RN，R）V \ in有一个势阱，IαI_ {\ alpha}是阶数α∈（0，N）\ alpha \ in（0，N）和F⁢（u）=12α*⁢| u |2α* + 12 *的Riesz势α⁢| u | 2 *αF（u）= \ frac {1} {2 _ {\ alpha} ^ {*}} \ lvert u \ rvert ^ {2 _ {\ alpha} ^ {*}} + \ frac {1 } {2 _ {*} ^ {\ alpha}} \ lvert u \ rvert ^ {2 _ {*} ^ {\ alpha}}，其中2α* = N +αN-22_ {\ alpha} ^ {*} = \由于Hardy–Littlewood，frac {N + \ alpha} {N-2}和2 *α= N +αN2_ {*} ^ {\ alpha} = \ frac {N + \ alpha} {N}分别是上下临界指数–索伯列夫不等式。根据变分方法，将山路定理和Nehari流形结合起来，如果\在RN \ mathbb {R} ^ {N}中为非负值，我们将获得足够大的ground的正基态解的存在和集中。此外，通过链接定理，我们证明了如果RN \ mathbb {R} ^ {N}中的𝑉改变符号，则𝜆的非平凡解的存在就足够大。