In this paper, we are interested in considering the following singular elliptic problem with concaveconvex nonlinearities$$\left\{ {\begin{array}{*{20}{l}} { - \Delta u - \frac{\mu }{{|x{|^2}}}u = f(x)|u{|^{p - 2}}u + g(x)|u{|^{q - 2}}u,}&{in\;\;\Omega \backslash \{ 0\} ,} \\ {u = 0,}&{on\;\;2\Omega ,} \end{array}} \right.$$where Ω ⊂ ℝ^N(N ≥ 3) is a smooth bounded domain with 0 ∈ Ω, \(0 < \mu < \bar\mu = \frac{{{{(N - 2)}^2}}}{4}\), 1 < q < 2 < p < 2* and \(2* = \frac{{2N}}{{N - 2}}\) is the Sobolev critical exponent, the coefficient functions f, g may change sign on Ω. By the Nehari method, we obtain two solutions, and one of them is a ground state solution. Under some stronger conditions, we point that the two solutions are positive solutions by the strong maximum principle.

中文翻译：

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奇异椭圆问题的多重正解，涉及凹凸非线性和符号变化势

在本文中，我们有兴趣考虑以下具有凸-非线性非线性的奇异椭圆问题$$ \ left \ {{\ begin {array} {* {20} {l}} {-\ Delta u-\ frac {\ mu} {{| x {| ^ 2}}} u = f（x）| u {| ^ {p-2}} u + g（x）| u {| ^ {q-2}} u，}＆{ in \; \; \ Omega \反斜杠\ {0 \}，} \\ {u = 0，}＆{on \; \; 2 \ Omega，} \ end {array}} \ right。$$其中Ω⊂ ℝ ^ñ（ñ ≥3）是用0∈Ω，光滑的有界域\（0 <\亩<\条\亩= \压裂{{{{（N - 2）} ^ 2}}} {4} \ ），1 < q <2 < p <2 *和\（2 * = \ frac {{2N}} {{N-2}} \）是Sobolev临界指数，系数函数f，g可能会改变Ω的符号。通过Nehari方法，我们获得了两个解，其中一个是基态解。在某些更强的条件下，我们指出，按照强最大原理，这两个解都是正解。