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

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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 $u\equiv0$.

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