Morse index and critical/super-critical Dirichlet problems with a large parameter (Q715675)

Consider the Dirichlet problem \[ \begin{cases} - \Delta u + \lambda u = u^p & \text{ in } \Omega,\\ u = 0 & \text{ on } \partial \Omega, \end{cases}\tag{1} \] with \(\Omega \subset \mathbb{R}^N\), \(N \geq 3\), a bounded domain, with \(p \geq \frac{N+2}{N-2}\), in the regime \(\lambda \to + \infty\). In this paper, the author studies the asymptotic behavior of positive solutions of (1) and obtains that solutions of (1) cannot have uniformly bounded Morse index as \(\lambda \to +\infty\) as long as \(p < p_{JL}(N)\), the so-called Joseph-Lundgren exponent. The author points out that the Morse index going to infinity is a very important information. So the author gets the following Theorem 1. Let \(\Omega\) be a bounded domain of \(\mathbb{R}^N\), \(N \geq 3\) and \(u_n\) be a sequence of positive solutions to \[ \begin{cases} - \Delta u_n + \lambda_n u_n = u_n^p & \text{ in } \Omega,\\ u_n > 0 & \text{ in } \Omega,\\ u_n = 0 & \text{ on } \partial \Omega. \end{cases}\tag{4} \] Assume that \(\overline{k} = \sup_n m(u_n) < +\infty\), where \(m(u_n)\) is the Morse index. Then there exist \(k \in \mathbb{N}^*\), \(k \leq \overline{k}\), converging sequences \(P^1_n,\dots,P^k_n\) in \(\overline{\Omega}\) and sequences \(\varepsilon^1_n,\dots,\varepsilon^k_n\) of positive real numbers converging to 0, such that \[ \frac{\left| P^i_n - P^j_n \right|}{ \min ( \varepsilon^i_n, \varepsilon^j_n ) }\to +\infty \] holds and such that, up to a subsequence, the following properties hold: 1. for any \(x \in \Omega\) and any \(n\) \[ d_n^k(x)^{\frac{N}{2} -1 } u_n(x) \leq C \] for some \(C > 0\) where \(d_n^k(x) \) is given by \[ d_n^k(x) = \min_{i = 1, \dots, k} |x - P_n^i|. \] Moreover \[ \lim_{R \to +\infty} \lim_{n \to \infty} \sup_{x \in \Omega \setminus \cap_{i=1}^k B_{R\varepsilon_n^i} (P_n^i)} d_n^k(x)^{\frac{N}{2} -1 } u_n(x) = 0. \] 2. \(u_n \to 0\) strongly in \(C_{loc}^0 (\overline{\Omega} \setminus S)\) as \(n \to +\infty\), where \(S\) is given by \[ S := \left\{ \lim_{n \to \infty} P_n^i, i = 1, \dots, k \right\}. \] Using the previous accurate description of the asymptotic behavior as \(\lambda \to +\infty\) through a Morse index information, the author gets the following. In the critical case, by the description of the blow-up mechanism along \(u_n\) given in Theorem 1, the author obtains the following. {Theorem.} Let \(u_n\) be a solution of (4) with \(p = \frac{N+2}{N-2}\) and \(\lambda_n \to +\infty\). Assume that \(\overline{k} = \sup_n m(u_n) < +\infty\). Then for \(k \leq \overline{k}\) sequences \((P_n^i)\) and \((\varepsilon_n^i)\) such that \( \frac{\left| P^i_n - P^j_n \right|}{ \min ( \varepsilon^i_n, \varepsilon^j_n ) } \to + \infty\) for all \(i \not= j\) as \(n \to \infty\), \(u_n \to 0\) in \(C_{loc}^0 (\overline{\Omega} \setminus \{ P_n^i, i = 1, \dots, k \})\) as \(n \to \infty\), and there exists \(C > 1\) such that \(\forall n\) and \(\forall x \in \Omega\) \[ u_n(x) \leq C \sum_{i=1}^{k} (\varepsilon_n^i)^{\left(1 - \frac{N}{2} \right)} \left[ U_0^1 \left( \frac{x - P_n^i}{\varepsilon_n^i} \right) \right]^{1-\varepsilon}, \] where \(U_0^1 = \left( 1 + \frac{|x|^2}{N(N-2)} \right)^{1 - \frac{N}{2}}\) and \(0 < \varepsilon < \frac{1}{2}\). In particular, we have \[ \sup_{n \in \mathbb{N}} \int_{\Omega} u_n^{\frac{2N}{N-2}} < + \infty. \] In the supercritical case the following theorem is established. {Theorem.} Let \(\frac{N+2}{N-2} \leq p < p_{JL}(N)\). Let \(u_n\) be a solution of (4) with \(\lambda_n \to +\infty\) as \(n \to \infty\). Then we have \(m(u_n) \to +\infty\) as \(n \to \infty\), where \(p_{JL}(N)\) is the Joseph--Lundgren exponent \[ p_{JL}(N) = \begin{cases} + \infty & \text{ if } N \leq 10,\\ \frac{(N-2)^2 - 4N + 8\sqrt{N-1}}{(N -2)(N-10)} & \text{ if } N \geq 11. \end{cases} \]