On dual cone theory for Euclidean Bosonic equations

In this paper we get the existence of nodal solutions and we will study some bifurcation properties for the following class of nonlocal problems P \(\begin{aligned} \left\{ \begin{array}{lcl} -\Delta e^{-c\Delta }u+u =g(x,u), \text{ in } \mathbb {R}^N\\ \lim _{|x|\rightarrow \infty }u(x)=0,\quad \\ \end{array} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }{e}^{-c\mathrm{\Delta }}u+u=g(x,u),\text{in}{\mathbb{R}}^N\hfill \\ {lim}_{|x|\to \infty }u(x)=0,\hfill \end{array}\right)\end{array}$$ where \(N\ge 3\) $N\ge 3$ , \(c>0\) $c>0$ , \(g: \mathbb {R}^N\times \mathbb {R} \rightarrow \mathbb {R}\) $g:{\mathbb{R}}^N\times \mathbb{R}\to \mathbb{R}$ is a \(C^1-\) $C^1-$ function, \(\Delta \) $\mathrm{\Delta }$ is the euclidean Laplacian and the linear operator \(e^{-c\Delta }\) $e^{-c\mathrm{\Delta }}$ is defined by the Fourier transform on a certain Hilbert space. This class of problems arises as models in the mathematical physics literature.