Asymptotic behavior of minimal solutions of $-\Delta u=\lambda f(u)$ as $\lambda\to-\infty$

TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of solutions to a nonlinear elliptic PDE as the parameter tends to negative infinity, revealing the emergence of large solutions in the limit.

Contribution

It establishes the existence and uniqueness of solutions for all negative parameters and characterizes their asymptotic behavior, including the appearance of large solutions.

Findings

Solutions exist and are unique for all negative mbda.

As mbda  , solutions exhibit specific asymptotic behavior.

Large solutions naturally emerge in the asymptotic expansion.

Abstract

We consider the Dirichlet problem $-\Delta u=\lambda f(u)$ with $\lambda<0$ and $f$ non-negative and non-decreasing. We show existence and uniqueness of solutions $u_\lambda$ for any $\lambda$ and discuss their asymptotic behavior as $\lambda\to-\infty$. In the expansion of $u_\lambda$ large solutions naturally appear.