Equaivalent norm on Sobolev space

Question

Let $1<q<p^*=\frac{Np}{N-p}$ if $1<p<N$ and $1<q<\infty$ if $p\geq N$. Let $\Omega\subset\mathbb{R}^N$ be a bounded smooth domain and $u\in W^{1,p}(\Omega)$ be such that $\int_{\Omega}|u|^{q-2}u\,dx=0$. Then can we define an equivalent norm on $W^{1,p}(\Omega)$ by the gradient $L^p$ norm as $$ \|u\|_{W^{1,p}(\Omega)}:=\left(\int_{\Omega}|\nabla u|^p\,dx\right)^\frac{1}{p}. $$ I can see if we assume that $\int_{\Omega}u\,dx=0$, then by the Sobolev inequality, we have $$ \int_{\Omega}|u|^p\,dx\leq C\int_{\Omega}|\nabla u|^p\,dx. $$ Hence, we can get the above equivalent norm. But under the hypothesis $\int_{\Omega}|u|^{q-2}u\,dx=0$, it is unclear how to proceed. Can somebody please help? Thanks.

Answer 1

This seems to be way too difficult for a homework problem as this could be simply part of someone's PhD thesis, if one is interested in a quantitative analysis of what is happening underhood. Because this is about the second eigenvalue of the $p$-Laplacian (you used letter $q$ here) with a Neumann boundary condition. The $p$ and $q$ are also seemingly switching places... Anyway this is the eigenvalue problem:

$$ \begin{aligned} -\operatorname{div}\left(|\nabla u|^{q-2} \nabla u\right) = \lambda|u|^{q-2} u & \quad \text{ in } \Omega, \\ \frac{\partial u}{\partial n} = 0 & \quad \text{ on } \partial \Omega. \end{aligned} $$

The condition $\int_{\Omega}|u|^{q-2}u\,dx=0$ is actually the compatibility condition for the homogeneous Neumann boundary value such that the eigenfunction corresponding to the smallest eigenvalue is trivial. Then the goal is to estimate the second eigenvalue, which is the Rayleigh quotient:

$$ \text{Minimize }\quad \frac{\int_{\Omega}|\nabla v|^{p} d x}{\int_{\Omega}|v|^{p} d x} \quad \text{ over } \; W^{1,q}(\Omega)/ \left\{v: \int_{\Omega}|v|^{q-2}v\,dx=0\right\}. $$

This problem is well-studied for Dirichlet boundary condition, i.e., the space above is changed to $W^{1, q}_0(\Omega)$, e.g., see Reference [1] Section 5.4, also Remark 5.5,

However, I am not sure about the Neumann BC.

[1]: Lê, An. "Eigenvalue problems for the $p$-Laplacian." Nonlinear Analysis: Theory, Methods & Applications 64, no. 5 (2006): 1057-1099.

Answer 2

Here is the sketch of the proof by contradiction: Assume the claim does not hold. Then for each $n$ there is $u_n$ such that $$ \|u\|_{L^p} > n \|\nabla u_n\|_{L^p} $$ with $\int |u_n|^{q-2}u_n=0$. Wlog $\|u\|_{L^p}=1$. This implies $\|\nabla u_n\|_{L^p}\to0$.

Then $(u_n)$ is bounded in $W^{1,p}$, which implies that we can extract a subsequence with $u_n \rightharpoonup u$ in $W^{1,p}$, $u_n \to u$ in $L^q(\Omega)$ (by compact embedding), $u_n(x)\to u(x)$ a.e., and there is $g\in L^q$, $g\ge0$ such that $|u_n(x)|\le g(x)$ a.e.. The latter claim is proven in every proof that $L^p$ is complete.

Since $\nabla u_n\to0$, it follows $u=const$. Using dominated convergence, we can pass to the limit in $0=\int_\Omega |u_n|^{q-2}u_n \to \int_\Omega |u|^{q-2}u$. Since $u$ is a constant function, this implies $u=0$. This is in contradiction to $1=\|u_n\|_{L^p} \to \|u\|_{L^p}$.