In this paper, we investigate the multiplicity of positive solutions for a class of Schrödinger–Poisson systems with concave and convex nonlinearities as follows: − Δu + λV (x)u + μϕu = a(x)|u|p−2u + b(x)|u|q−2uin ℝ3, − Δϕ = u2 in ℝ3, where λ,μ > 0 are two parameters, 1 < q < 2 < p < 4, V ∈ C(ℝ3) is a potential well, a ∈ L∞(ℝ3) and b ∈ Lp/(p−q)(ℝ3). Such problem cannot be studied by applying variational methods in a standard way, since the (PS) condition is still unsolved on H1(ℝ3) due to 2 < p < 4. By developing a novel constraint approach, we prove that the above problem admits at least two positive solutions.

中文翻译：

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通过一种新颖的约束方法研究涉及凹凸非线性的薛定谔-泊松系统

在本文中，我们研究了一类具有凹凸非线性的薛定谔-泊松系统的正解的多重性，如下所示： - Δ你 + λ五 (X)你 + μφ你 = 一种(X)|你|p-2你 + b(X)|你|q-2你在 ℝ3, - Δφ = 你2 在 ℝ3, 在哪里λ,μ > 0是两个参数，1 < q < 2 < p < 4,五 ∈ C(ℝ3)是一个潜在的井，一种 ∈ 大号∞(ℝ3)和b ∈ 大号p/(p-q)(ℝ3). 不能通过以标准方式应用变分方法来研究此类问题，因为（PS）条件仍未解决H1(ℝ3)由于2 < p < 4. 通过开发一种新颖的约束方法，我们证明上述问题至少有两个正解。