Consider a function $u \in C^\infty(\mathbb{R}^n)$ such that

$$ \begin{cases} u(x) > 0 & \forall x \in \mathbb{R}^n \\ \Delta u(x) \geq u(x) > 0 &\forall x \in \mathbb{R}^n \end{cases} $$

Does there exist a solution to this PDE that is bounded i.e

Does there exist a solution to the above PDE, $ v(x) $, such that

$$ v(x) < M \ \forall x\in \mathbb{R}^n $$

for some $M > 0$?

Consider a function $u \in C^\infty(\mathbb{R}^n)$ such that

$$ \begin{cases} u(x) > 0 & \forall x \in \mathbb{R}^n \\ \Delta u(x) \geq u(x) > 0 &\forall x \in \mathbb{R}^n \end{cases} $$

Does there exist a solution to this PDE that is bounded i.e

Does there exist a solution to the above PDE, $ v(x) $, such that

$$ \vert v(x) \vert < M \ \forall x\in \mathbb{R}^n $$

for some $M > 0$?

Consider a function $u \in C^\infty(\mathbb{R}^n)$ such that

$$ \begin{cases} u(x) > 0 & \forall x \in \mathbb{R}^n \\ \Delta u(x) \geq u(x) > 0 &\forall x \in \mathbb{R}^n \end{cases} $$

Does there exist a solution to this PDE that is bounded i.e

Does there exist a solution to the above PDE, $ v(x) $, such that

$$ v(x) < M \ \forall x\in \mathbb{R}^n $$

for some $M > 0$.

Consider a function $u \in C^\infty(\mathbb{R}^n)$ such that

$$ \begin{cases} u(x) > 0 & \forall x \in \mathbb{R}^n \\ \Delta u(x) \geq u(x) > 0 &\forall x \in \mathbb{R}^n \end{cases} $$

Does there exist a solution to this PDE that is bounded i.e

Does there exist a solution to the above PDE, $ v(x) $, such that

$$ v(x) < M \ \forall x\in \mathbb{R}^n $$

for some $M > 0$?