According to the answers in the the following questions: [How to prove the spectrum of the Laplace operator?][1] and [What is spectrum for Laplacian in $\mathbb{R}^n$ ][2], 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
$$\Delta u(x) = \lambda u(x),\;\;\forall x\in \Bbb R^2 \;\;\;\overset{\mathcal F}{\longrightarrow}\;\;\;\;-4\pi^2|\xi|^2\hat u(\xi) =\lambda \hat u(\xi), \;\;\;(\text{a.e})\;\xi\in \Bbb R^2,$$ and this can only be satisfied by $\hat u=u=0$. <s>This means that the only eigenvalue-eigenvector pair in this setting is $(\lambda,u)=(0,0)$</s> . 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?*

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

  [1]: https://math.stackexchange.com/questions/790401/how-to-prove-the-spectrum-of-the-laplace-operator?noredirect=1&lq=1
  [2]: https://math.stackexchange.com/questions/766479/what-is-spectrum-for-laplacian-in-mathbbrn%5D