What is known, for $N\geq3$, about the existence of **nontrivial** real-valued solutions $u=u(x)$ of the following semilinear elliptic equation:
$$
\left\{ \enspace
\begin{aligned}
 &\Delta u = f(u) && \text{in ${\mathbb R}^N$,} \\
 &u(x) \to 0 && \text{as $|x|\to\infty$.}
\end{aligned}
\right.
$$ 
Here, $f\in \mathscr C\_c^\infty(\mathbb R)$ is real-valued, $0\notin\operatorname{supp}f$, and $F(u)= \int\_0^u f(v)\,dv\geq0$ for all $u\in\mathbb R$.


Clearly, $\Delta u = f(u)$ is the Euler-Lagrange equation of the
functional
$$
 I(u) = \frac12\,\|\nabla u\|\_{L^2}^2 + \int\_{\mathbb R^N} F(u(x))\,dx,
$$
and also $\inf I(u)=0$ is attained at the trivial solution $u\equiv0$
(whatever the function space considered for $u$).


**Comment** 


1. Any solution $u$ is harmonic outside a large enough ball. Thus, by Liouville's theorem, in order to have a nontrivial solution $u$ the range of $u$ has to meet $\operatorname{supp}f$.
2. If $F(u^\*)<0$ for some $u^\*>0$, then there is a positive, spherically symmetric, and decreasing (with respect to $|x|$) solution $u$, see Theorem 4 in [H. Berestycki and P.-L. Lions](http://link.springer.com/article/10.1007%2FBF00250555). A similar statement holds if $F(u^\*)<0$ for some $u^\*<0$ (replace $u$ with $-u$).



Nontrivial solutions of a semilinear elliptic equation
Meanwhile, I have found an answer myself. It is reasonable to assume that $u\in L^\infty(\mathbb R^N)$ and $\nabla u\in L^2(\mathbb R^N;\mathbb R^N)$. Then $F(u)\in L^1(\mathbb R^N)$ as $\operatorname{supp}F(u)$ is compact. 


From Pohozaev's identity  

$$
 \frac{N-2}{N}\,\left\|\nabla u\right\|\_{L^2}^2 + \int\_{\mathbb R^N}F(u(x))\,dx = 0
$$ 
and $F\geq0$, one then concludes that $u\equiv0$.


