Consider the equation
$$ -\Delta f+mf+\lambda f^p=0$$
on $\mathbb{R}^d$, where $d>2$,$m>0$, $p>1$ is integer, and $\lambda \in \mathbb{R}$. Are there any known results regarding the non-existence of non-zero solutions $f$ of this equation in $\mathcal{S}(\mathbb{R}^d)$ (the Schwartz space of rapidly decreasing smooth functions), at least for some values of $d$, $p$ and $\lambda$?



Non-existence of rapidly decaying solutions of certain elliptic semilinear equations
(To substantiate my comment above)


If $\lambda \geq 0$ and $p$ is odd, assume $f\in \mathcal{S}(\mathbb{R}^d)$ is a solution, then you can multiply the equation by $f$ and integrate by parts (everything converges appropriately) to find


$$ \int |\nabla f|^2 + m f^2 + \lambda f^{p+1} = 0 $$


Noting that $p+1$ is even and so $f^{p+1}\geq 0$, you find this implies $\|f\|\_{L^2} = 0$.


(Alternatively, you can also use the maximum principle: write the equation as $-\triangle f + Vf = 0$ where $V = m + \lambda f^{p-1}$. When $p$ is odd this function $V$ is manifestly positive, and by maximum principle any solution cannot have a positive local max nor a negative local min. Combined with the Schwartz assumption this forces $f\equiv 0$.)


