# finding Morse index for the following functional

not sure if this meets the standards here in this forum. I was dealing with the following functional $$I(u)=\frac{1}{p}\int_{\Omega}|\nabla u|^{p}dx-\frac{1}{q}\lambda\int_{\Omega}|u|^qdx$$ for $$p \geq 2$$ and $$p-1, $$p^*=\frac{Np}{N-p}$$, $$p, $$\Omega$$ is a bounded domain in $$\mathbb{R}^N$$. I was interested to check the Morse index of $$I$$. Towards this, I computed the double derivative which is as follows. $$\begin{eqnarray}I''(u)(w,w)&=\int_{\Omega}\{|\nabla u|^{p-2}|\nabla w|^2dx+2(p-2)|\nabla u|^{p-4}(\nabla u.\nabla w)^2\}dx\nonumber\\ &-\lambda \int_{\Omega}(q-1)|u|^{q-2}|w|^2dx.\end{eqnarray}$$ I wasn't able to figure out what will be the largest dimension of a subspace of $$W_0^{1,p}(\Omega)$$ such that $$I"(u)$$ is nonnegative?. Is it at least possible to do so?.

• same question on MSE math.stackexchange.com/questions/3483440/… Dec 21, 2019 at 8:32
• You may consult the paper ''Morse index and uniqueness for positive solutions of radial $p$-Laplace equations" by Aftalion and Pacella in Trans of AMS (2004).
– GabS
Jan 7, 2020 at 16:54

I guess you mean "the largest dimension of a subspace of $$W_0^{1,p}(\Omega)$$ such that $$I''(u)$$ is negative"? Otherwise, the largest dimension equals infinity, due to the spectral theory of the linearized $$p$$-Laplacian, see, e.g., Theorem 1.1 in [Castorina, Esposito, Sciunzi; 2011].
Moreover, the choice of $$W_0^{1,p}(\Omega)$$ doesn't seem to be optimal for $$p \neq 2$$, see again [Castorina, Esposito, Sciunzi; 2011].
Finally, in general, the Morse index depends on $$u$$. E.g., if $$u$$ is a local minimizer of $$I$$ (i.e., in the case $$p-1), then the Morse index equals $$0$$. On the other hand, if $$p=2$$, $$q>p$$, and $$\Omega$$ is a ball, then the Morse index of the positive radial solution is $$1$$.