We consider the following critical nonlocal Schrödinger problem with general nonlinearities − ε 2a+ε b∫ R3|∇ u|2Δ u+V(x)u=f(u)+u5,x∈ R3,u∈ H1(R3), $$\begin{array}{} \displaystyle \left\{\begin{array}{} &-\left(\varepsilon^{2}a+\varepsilon b\displaystyle\int\limits_{\mathbb{R}^{3}}|\nabla u|^{2}\right){\it\Delta} u+V(x)u=f(u)+u^{5}, &x \in \mathbb{R}^{3},\\ &u\in H^{1}(\mathbb{R}^{3}), \end{array}\right. \end{array}$$( SK ε ) and study the existence of semiclassical ground state solutions of Nehari-Pohožaev type to ( SK ε ), where f ( u ) may behave like | u | q –2 u for q ∈ (2, 4] which is seldom studied. With some decay assumption on V , we establish an existence result which improves some exiting works which only handle q ∈ (4, 6). With some monotonicity condition on V , we also get a ground state solution v̄ ε and analysis its concentrating behaviour around global minimum x ε of V as ε → 0. Our results extend some related works.

中文翻译：

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一类临界薛定谔问题的基态解

我们考虑以下具有一般非线性的临界非局部薛定谔问题 − ε 2a+ε b∫ R3|∇ u|2Δ u+V(x)u=f(u)+u5,x∈ R3,u∈ H1(R3), $$\begin{array}{} \displaystyle \left\{\begin{array}{} &-\left(\varepsilon^{2}a+\varepsilon b\displaystyle\int\limits_{\mathbb{R}^ {3}}|\nabla u|^{2}\right){\it\Delta} u+V(x)u=f(u)+u^{5}, &x \in \mathbb{R}^ {3},\\ &u\in H^{1}(\mathbb{R}^{3}), \end{array}\right。\end{array}$$( SK ε ) 并研究 Nehari-Pohožaev 类型到 ( SK ε ) 的半经典基态解的存在性，其中 f ( u ) 可能表现为 | 你| q –2 u for q ∈ (2, 4] 很少被研究。通过对 V 的一些衰减假设，我们建立了一个存在结果，该结果改进了一些仅处理 q ∈ (4, 6) 的现有工作。在一些单调性条件下,