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Take $B_1$ the unit ball in Euclidean $N$ dimensional space and suppose $3 \le N \le 10$ and take $ 1<p< \frac{N+2}{N-2}$. By some abstract theory there is a infinite sequence of smooth radial sign changing solutions, say $u_m$, of $$ -\Delta u = |u|^{p-1} u $$ in $B_1$ with $ u=0$ on $ \partial B_1$. Moreover one has $ \int_{B_1} | \nabla u_m|^2 dx \rightarrow \infty$. For simplicity assume $p=3$ and $N=3$. Consider the linearized operator $$L_m(\phi)= -\Delta \phi(x) - 3 (u_m(r))^2 \phi(x)$$ and consider the eigenvalues $ \{ \mu_{k,m} \}_{k \ge 1}$ of this linear operator. I know the number of negative eigenvalues goes to infinity as $ m \rightarrow \infty$.

Question. Is there a standard approach to get an estimate on $$ \inf \{| \mu_{k,m}|: k \ge 1 \}$$? Any comments would be greatly appreciated.

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  • $\begingroup$ What about Weyl's Law? Since $\mu_{,k,m}\to+\infty$ with $k$, you are asking, for large $m$, about the asymptotic distribution of the eigenvalues, since the minimum for a fixed, large, $m$ will be attained for some large $k$. If you are asking for an inf on $k$ and $m$, it looks like it should be zero, for that reason, since eigenvalues become close together ($N=3$, so $2/N<1$). $\endgroup$
    – username
    Commented Jul 3, 2022 at 21:15
  • $\begingroup$ Thanks for the comment. Ya I don't know anything about Weyl's Law. It sounds like you are saying that $T_m:= \inf \{ | \mu_{k,m}|: k \ge 1 \} \rightarrow 0$ as $ m \rightarrow 0$; which is what I was mainly interested in. Any hope of trying to find a bound on how quick it approaches zero? $\endgroup$
    – Math604
    Commented Jul 4, 2022 at 8:10
  • $\begingroup$ Weyl Law is proved by comparison estimation see these notes or others, it's the first link in google. If you have an estimate on $u_m$ you can follow the same steps and derive the corresponding estimate for your problem? $\endgroup$
    – username
    Commented Jul 4, 2022 at 20:16

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