Let \(\Omega \subset \mathbb {R}^{N}\), \(N \ge 2\), be a smooth bounded domain. We consider the boundary value problem

where \(c_{\lambda }\) and h belong to \(L^q(\Omega )\) for some \(q > N/2\), \(\mu \) belongs to \(\mathbb {R}{\setminus } \{0\}\) and we write \(c_{\lambda }\) under the form \(c_{\lambda }:= \lambda c_{+} - c_{-}\) with \(c_{+} \gneqq 0\), \(c_{-} \ge 0\), \(c_{+} c_{-} \equiv 0\) and \(\lambda \in \mathbb {R}\). Here \(c_{\lambda }\) and h are both allowed to change sign. As a first main result we give a necessary and sufficient condition which guarantees the existence (and uniqueness) of solution to (\(P_{\lambda }\)) when \(\lambda \le 0\). Then, assuming that \((P_0)\) has a solution, we prove existence and multiplicity results for \(\lambda > 0\). Our proofs rely on a suitable change of variable of type \(v = F(u)\) and the combination of variational methods with lower and upper solution techniques.

令\（\ Omega \ subset \ mathbb {R} ^ {N} \），\（N \ ge 2 \）为平滑有界域。我们考虑边值问题

其中\（c _ {\ lambda} \）和h属于\（L ^ q（\ Omega）\）对于某些\（q> N / 2 \），\（\ mu \）属于\（\ mathbb { R} {\ setminus} \ {0 \} \），然后以\（c _ {\ lambda}：= \ lambda c _ {+}-c _ {-} \）的形式写\（c _ {\ lambda } \）与\（c _ {+} \ gneqq 0 \），\（c _ {-} \ ge 0 \），\（c _ {+} c _ {-} \ equiv 0 \）和\（\ lambda \ in \ mathbb { R} \）。这里\（c _ {\ lambda} \）和h都允许更改符号。作为第一个主要结果，我们给出一个必要的充分条件，当（\ lambda \ le 0 \）时，它保证（\（P _ {\ lambda} \））的解的存在（和唯一性）。然后，假设\（（P_0）\）有一个解，我们证明\（\ lambda> 0 \）的存在性和多重性结果。我们的证明依赖于\（v = F（u）\）类型的变量的适当更改以及变分方法与上下求解技术的结合。