According to the answers in the the following questions: How to prove the spectrum of the Laplace operator? and What is spectrum for Laplacian in $\mathbb{R}^n$ , the spectrum of the Laplace operator $\Delta :H^2(\mathbb{R}^2)\subset L^2(\mathbb{R}^2)\to L^2(\mathbb{R}^2)$ is in fact $\sigma(\Delta)=(-\infty,0].$ However, I was not able to find a discussion on the eigenvalues of $\Delta$.

The set of eigenvalues $\sigma_p(\Delta)$ (also called point spectrum) is known to be contained in $\sigma(\Delta)$ and one can have $\sigma_p(\Delta)\subsetneq \sigma(\Delta)$. Indeed, by taking the Fourier transform $\mathcal{F}:L^2(\mathbb{R}^2)\to L^2(\mathbb{R}^2)$ of the eigenvalue problem one has
$$\Delta u(x) = \lambda u(x),\;\;\forall x\in \Bbb R^2 \;\;\;\overset{\mathcal F}{\longrightarrow}\;\;\;\;-4\pi^2|\xi|^2\hat u(\xi) =\lambda \hat u(\xi), \;\;\;\forall\xi\in \Bbb R^2,$$ and this can only be satisfied by $\hat u=u=0$. ~~This means that the only eigenvalue-eigenvector pair in this setting is $(\lambda,u)=(0,0)$~~ . Also, the same argument applies when $\Delta$ is seen as $\Delta:W^{m,p}(\mathbb{R}^2)\subset L^p(\mathbb{R}^2)\to L^p(\mathbb{R}^2)$ with $p\in [1,2)$ and $\mathcal F:L^p(\mathbb{R}^2)\to L^{p^*}(\mathbb{R}^2)$ with $1/p+1/p^*=1$.

* Question 1*.

*What happens when $p>2$ and the Fourier transform becomes distribution valued, so that the above elementary argument cannot be applied directly?*

It seems if $u\in C^2(\mathbb{R}^2)$ is in fact an eigenvalue of $\Delta$, then it cannot be in $L^p(\mathbb{R}^2)$ for any $p\in [1,2]$. Depending on the answer to Q1, this might also hold for $p>2$. In any case, it seems that the $L^p$ framework is not suitable for this problem.

* Question 2*.

*On what space(s) could one define the domain of $\Delta$ to obtain non-trivial eigenvalues?*

*Edit.* The crossed out sentence should be replaced by: "The point spectrum $\sigma_p(\Delta)$ is therefore empty."