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Existence and multiplicity for an elliptic problem with critical growth in the gradient and sign-changing coefficients
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

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

其中\(c _ {\ lambda} \)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\(c _ {\ lambda } \)\(c _ {+} \ gneqq 0 \)\(c _ {-} \ ge 0 \)\(c _ {+} c _ {-} \ equiv 0 \)\(\ lambda \ in \ mathbb { R} \)。这里\(c _ {\ lambda} \)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。

更新日期:2020-05-13
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