# A supercritical scalar field equation with a forcing term

## TL;DR

This paper classifies the solvability of a supercritical scalar field elliptic equation with a forcing term in Euclidean space, depending on the parameters and measure conditions, extending known results up to the Joseph-Lundgren exponent.

## Contribution

It provides a complete classification of solutions for the scalar field equation with forcing term for subcritical exponents, including the supercritical range up to the Joseph-Lundgren exponent.

## Key findings

- Solved the elliptic problem for p < p_{JL} under measure conditions.
- Established existence and non-existence criteria based on measure integrability.
- Extended classification results to supercritical exponents.

## Abstract

This paper is concerned with the elliptic problem for a scalar field equation with a forcing term \begin{equation} \tag{P}-\Delta u+u=u^p+ \kappa \mu \quad \mbox{in} \quad{\bf R}^N, \quad u>0 \quad \mbox{in} \quad {\bf R}^N, \quad u(x)\to 0\quad \mbox{as} \quad |x| \to \infty, \end{equation} where $N\ge 2$, $p>1$, $\kappa>0$ and $\mu$ is a Radon measure in ${\bf R}^N$ with a compact support. Under a suitable integrability condition on $\mu$, we give a complete classification of the solvability of problem~(P) with $1<p<p_{JL}$. Here $p_{JL}$ is the Joseph-Lundgren exponent defined by $$ p_{JL} :=\infty\quad\mbox{if}\quad N\le 10, \qquad p_{JL}:=\frac{(N-2)^2-4N+8\sqrt{N-1}}{(N-2)(N-10)}\quad \text{if} \quad N\ge 11. $$