Does this common PDE have a name? What is the name of this common 2D PDE:

$$\nabla^2 u - k^2 u=0$$

where $k\in\mathbb{R}$.

And where can I find a general solution and the way to derive it? THanks!

Edit:

Some people have suggested that this is the Helmoltz equation. My concern is the following: the Helmoltz equation does admit a solution with products of trigonometric functions and corresponding bessel J functions:

$$f(r,\phi)= A_n\sin(n\phi)J_n(\lambda r)$$

The one I wrote, with a minus instead of a plus, does not admit such solutions. Infact admits exponential solutions:

$$f(r,\phi)= \exp\left[\frac{\lambda}{\sqrt{2}} (r\cos\phi+r\sin\phi)\right]$$

because of its structure.
If you define $\nabla^2$ as $\nabla^2 u = -\sum u_{x_ix_i}$, this is the Helmholtz equation.  If you then give it Dirichlet boundary conditions, it is the equation of Dirichlet eigenvalues, and it will have solutions.  If, however, you are defining $\nabla^2 u = \sum u_{x_ix_i}$, the Dirichlet problem for your equation (i.e. zero boundary condition) will have no nontrivial solutions by the spectral theorem, so it isn't very interesting.
It is called Helmotz equation. When derived from the wave equation, it appears with $+k^2\,u$ instead of $-k^2\,u$.