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**

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 $\operatorname{supp}f$.

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. A similar statement holds if $F(u^*)<0$ for some $u^*<0$ (replace $u$ with $-u$).