Hénon-type equations and concentration on spheres (Q2806006)

Let \(B_N(0, 1) \subset \mathbb{R}^N (N\geq3)\) be the unit ball centered at the origin and \(S_r^{k-1}:= \{y\in \mathbb{R}^k : |y| = r\}\) and \(S^{k-1} := S_1^{k-1}\). The authors consider the following type of elliptic problem: NEWLINE\[NEWLINE -\Delta u = h(x)|u|^{p-2}u, \;x\in B_N(0, 1), \;u|_{\partial B_N(0,1)} = 0, \text{ and } h(x) = |x|^\alpha, \alpha>0, p>2. \leqno{(P)} NEWLINE\]NEWLINE When \(N = 2m, m>1, x = (y_1, y_2), y_i \in \mathbb{R}^m (i = 1, 2)\) and \(2<p<2(m+1)/(m-1)\), the authors prove that, there exists \(\alpha_0 = \alpha_0(p,m)>4\) such that any least energy solution \(u_\alpha\) of (P) among the doubly symmetric solutions (i.e. the solution \(u\) with \(u(y_1, y_2) = u(|y_1|, |y_2|), \forall (y_1, y_2) \in B_{2m}(0, 1)\)) is positive in \(B_{2m}(0, 1)\) and not radially symmetric, and for some \(r_\alpha \in (0, 1)\), either NEWLINE\[NEWLINE \mathcal{M_\alpha} = \max\limits_{(y_1,y_2)\in B_{2m}(0,1)}u_\alpha(y_1, y_2) = u_\alpha(y_1, 0) \text{ for all } y_1 \in S_{r_\alpha}^{m-1}, \text{ if } h(x) = |(y_1, y_2)|^\alpha, \leqno\text{(i.)} NEWLINE\]NEWLINE or, NEWLINE\[NEWLINE \mathcal{M_\alpha} = \max\limits_{(y_1,y_2)\in B_{2m}(0,1)}u_\alpha(y_1, y_2) = u_\alpha(0,y_2) \text{ for all } y_2 \in S_{r_\alpha}^{m-1}, \text{ if } h(x) = |y_2|^\alpha. \leqno\text{(ii.)} NEWLINE\]NEWLINE Moreover, \(u_\alpha\) concentrates and blows up either on \(S^{m-1} \times\{0\}\) in case (i), or on \(\{0\}\times S^{m-1}\) in case (ii). That is, \(r_\alpha \rightarrow 1, \mathcal{M}_\alpha \approx \alpha^{2/(p-2)}\) and \(\alpha(1 - r_\alpha)\rightarrow \ell\) for some \(\ell>0\), as \(\alpha \rightarrow \infty\).NEWLINENEWLINEWhen \(N\geq 3\) is a general dimension, \(x = (x_1, x_2, \cdots, x_N)\), problem (P) with \(p\in (2, \frac{2N}{N-2})\) is also studied by the authors. They prove that, any least energy solution \(u_\alpha\) of (P) among the solutions axially symmetric w.r.t. \(\mathbb{R}e_N \subset \mathbb{R}^N\) is positive in \(B_N(0, 1)\), and there is \(r_\alpha \in [0, 1)\) such that NEWLINE\[NEWLINE \mathcal{M_\alpha} = \max\limits_{(x_1,\cdots,x_N) \in B_N(0,1)} u_\alpha(x_1,\cdots, x_N) = u_\alpha(0,\cdots, 0, r_\alpha) = u_\alpha(0,\cdots, 0, -r_\alpha). NEWLINE\]NEWLINE Moreover, \(u_\alpha\) concentrates and blows up simultaneously on \((0,\cdots, 0, 1)\) and \((0,\cdots, 0,-1)\), i.e., \(r_\alpha \rightarrow 1, \mathcal{M}_\alpha \approx \alpha^{2/(p-2)}\) and \(\alpha (1 - r_\alpha) \rightarrow \ell\) for some \(\ell> 0\), as \(\alpha \rightarrow \infty\).