Abstract In this paper we prove an existence result of multiple positive solutions for the following quasilinear problem: { - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u = | u | p - 2 ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u-\Delta(u^{2})u&\displaystyle=|u|% ^{p-2}u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a smooth and bounded domain in ℝ N , N ≥ 3 {\mathbb{R}^{N},N\geq 3} . More specifically we prove that, for p near the critical exponent 22 * = 4 ⁢ N / ( N - 2 ) {22^{*}=4N/(N-2)} , the number of positive solutions is estimated below by topological invariants of the domain Ω: the Ljusternick–Schnirelmann category and the Poincaré polynomial. With respect to the case involving semilinear equations, many difficulties appear here and the classical procedure does not apply immediately. We obtain also en passant some new results concerning the critical case.

中文翻译：

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具有几乎临界非线性的拟线性薛定谔方程的多重正解

摘要 本文证明了以下拟线性问题的多个正解的存在性结果：{ - Δ ⁢ u - Δ ⁢ ( u 2 ) ⁢ u = | 你| p - 2 ⁢ u in ⁢ Ω , u = 0 on ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle-\Delta u-\Delta(u^{2})u&\displaystyle=|u| % ^{p-2}u&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end {对齐}\右。其中 Ω 是 ℝ N 中的光滑有界域，N ≥ 3 {\mathbb{R}^{N},N\geq 3} 。更具体地说，我们证明，对于接近临界指数的 p 22 * = 4 ⁢ N / ( N - 2 ) {22^{*}=4N/(N-2)} ，正解的数量通过拓扑估计如下域 Ω 的不变量：Ljusternick-Schnirelmann 范畴和 Poincaré 多项式。对于涉及半线性方程的情况，这里出现了许多困难，经典程序并不能立即适用。我们还获得了一些关于关键案例的新结果。