We prove new results concerning the nonlinear scalar field equation \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = g(u)&\quad \hbox{in }\mathbb{R}^N,\; N\geq 3,\\ u\in H^1(\mathbb{R}^N)& \end{array} \right. \end{equation*} with a nonlinearity $g$ satisfying the general assumptions due to Berestycki and Lions. In particular, we find at least one nonradial solution for any $N\geq 4$ minimizing the energy functional on the topological Pohozaev constraint. If in addition $N\neq 5$, then there are infinitely many nonradial solutions. The results give a partial answer to an open question posed by Berestycki and Lions in [5,6]. Moreover, we build a critical point theory on a topological manifold, which enables us to solve the above equation as well as to treat new elliptic problems.

中文翻译：

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非线性标量场方程的非径向解

我们证明了关于非线性标量场方程的新结果 \begin{equation*} \left\{ \begin{array}{ll} -\Delta u = g(u)&\quad \hbox{in }\mathbb{R} ^N,\; N\geq 3,\\ u\in H^1(\mathbb{R}^N)& \end{array} \right。\end{equation*} 具有非线性 $g$，满足 Berestycki 和 Lions 的一般假设。特别是，我们找到了任何 $N\geq 4$ 的至少一个非径向解，最小化拓扑 Pohozaev 约束上的能量泛函。如果另外$N\neq 5$，则有无穷多个非径向解。结果部分回答了 Berestycki 和 Lions 在 [5,6] 中提出的一个开放性问题。此外，我们在拓扑流形上建立了临界点理论，这使我们能够求解上述方程以及处理新的椭圆问题。