Existence of a solution to a semilinear elliptic equation

Abstract

We consider the equation $-\Delta u =f(u)-\frac{1}{|\Omega|}\int_{\Omega} f(u)d\mathbf{x}$, where the domain $\Omega= \mathbb{T}^N$, the $N$-dimensional torus, with $N=2$ or $N=3$. And $f$ is a given smooth function of $u$ for$u(\mathbf{x}) \in G \subset \mathbb{R}$. We prove that there exists a solution $u$ to this equation which is unique if $|\frac{df}{du}(u_0)|$ is sufficiently small, where $u_0 \in G$ is a given constant. And we prove that the solution $u$ is not unique if $\frac{df}{du}(u_0) $ is a simple eigenvalue of $-\Delta$.

Keywords

• elliptic| existence| uniqueness| semilinear| bifurcation