My partial differential equation of interest is as follows:
$y+k^2 \nabla^2 y = p(\mathbf{x})$
where $y$ and $p$ are functions of $\mathbf{x}$, which is a vector in $\mathbb{R}^D$, and the Laplacian operator $\nabla^2$ is defined as $\sum_{d=1}^{D} \frac{\partial^2}{\partial x_{d}^2}$. $k$ is a real, positive constant.
$p(\mathbf{x})$ may be an arbitrary function with good properties. (continuous, differentiable, ...) $p(\mathbf{x})$ is always positive and $p(\mathbf{x}) \to 0$ as $\mathbf{x} \to \infty$ and $\mathbf{x} \to -\infty$.
Boundary conditions are $y(\infty)=y(-\infty)=y'(\infty)=y'(-\infty)=0$.
A straightforward approach would be applying Fourier transform, but it would not yield a valid solution because of the positive sign of $k^2 \nabla^2 y$.
Does it have any solution? If so, how can it be derived?