Non existence of solution of a certain PDE problem In my introductory course of PDE appears the following remark without proof.
**It must be observed that if $\mu$ is the first positive eigenvalue (so, there is a postive eigenfunction associated to $\mu$) of minus laplacian operator with zero Dirichlet conditions in a regular and bounded domain $D$, then the following problem has not solution 
I have tried to use Fourier transform to obtain this result but I would know if there exists any easier way to prove it.
 A: Let me rewrite a comment of @mattos in a form which is more accessible for beginners. 
Suppose, by contradiction, that your boundary value problem has a solution, say, $f$. 
And let us denote by $\varphi$ a positive eigenfunction associated with $\mu$.
Then, multiplying the equation by $\varphi$ and integrating over $D$, we get
$$
-\int_D \varphi \nabla^2 f \, dx = \mu \int_D f \varphi \,dx + \int_D \varphi \, dx.
$$
Applying the Divergence Theorem and recalling that $\varphi,f=0$ on $\partial D$, we see that
$$
-\int_D \varphi \nabla^2 f \, dx = \int_D (\nabla f, \nabla \varphi) \, dx.
$$
On the other hand, since $\varphi$ is the eigenfunction associated with $\mu$, we have, by definition,
$$
-\nabla^2 \varphi = \mu \varphi ~\text{in}~ D, \quad \varphi=0 ~\text{on}~ D.
$$
Hence, multiplying this equation by $f$ and integrating over $D$, we get
$$
-\int_D f\nabla^2 \varphi \, dx \equiv \int_D (\nabla f, \nabla \varphi) \, dx
=
\mu \int_D f \varphi \,dx.
$$
Combining the formulas above, we conclude that
$$
\int_D \varphi \, dx = 0.
$$
But this is impossible since $\varphi$ is strictly positive by assumption (and hence we should have $\int_D \varphi \, dx >0$). Thus, we've got a contradiction, and hence your boundary value problem cannot have a solution.
