Calculus of Variations and Partial Differential Equations ( IF 2.1 ) Pub Date : 2020-05-13 , DOI: 10.1007/s00526-020-01755-z Colette De Coster , Antonio J. Fernández
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.
中文翻译:
具有梯度和符号变换系数的临界增长的椭圆问题的存在性和多重性
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。