Positive and nodal solutions for an elliptic equation with critical growth (Q2796986)

The authors study weak solutions to the semilinear elliptic equation NEWLINE\[NEWLINE-\operatorname{div} (p(x) \nabla u) = \lambda |u|^{q-2} u + |u|^{2^\ast -2} u, \qquad x \in \Omega,NEWLINE\]NEWLINE endowed with homogeneous Dirichlet boundary conditions. Here, \(\Omega \subset \mathbb{R}^N\) is a bounded, smooth domain with \(N \geq 4\), \(q \in [2, 2^\ast)\) with \(2^\ast := \frac{2N}{N-2}\), \(\lambda >0\), and \(p \in H^1 (\Omega) \cap C(\bar{\Omega})\) is strictly positive and has a certain behavior near its minimum.NEWLINENEWLINEOn the one hand, the authors prove that the problem has a positive solution for any \(q \in (2, 2^\ast)\) and \(\lambda >0\). On the other hand, they show that the problem has a sign changing solution if \(q=2\) and \(\lambda\) is large enough depending on the first eigenvalue of a corresponding linear eigenvalue problem. These results extend those for the case \(q=2\) from [\textit{H. Brezis} and \textit{L. Nirenberg}, Commun. Pure Appl. Math. 36, 437--477 (1983; Zbl 0541.35029)] and [\textit{R. Hadiji} and \textit{H. Yazidi}, Chin. Ann. Math., Ser. B 28, No. 3, 327--352 (2007; Zbl 1186.35063)]. The proofs particularly rely on applications of the generalized Mountain Pass Theorem.