Critical exponent for half-Laplacian in the whole space

We study the existence of {weak} solutions for fractional elliptic equations of the type, \begin{equation*} (-\Delta)^{\frac{1}{2}} u+ V(x) u= h(u), u> 0 ;$\text{in}$ ;\mathbb R, \end{equation*} %where $1<q<2,p>2,1<\beta \le 2,\lambda >0,K(x)>0,f$ is continuous and sign changing. where $h$ is a real valued function that behaves like $e^{u^2}$ as $u\to \mathrm{\infty }$ and $V(x)$ is a positive, continuous unbounded function. Here $(-\mathrm{\Delta }{)}^{\frac{1}{2}}$ is the fractional Laplacian operator. We show the existence of mountain-pass solution when the nonlinearity is superlinear near $t=0$. We also study the corresponding critical exponent problem for the Kirchhoff equation $$m({\int }_{\mathbb{R}}\mathrm{\mid }(-\mathrm{\Delta }{)}^{\frac{1}{2}}u{\mathrm{\mid }}^{2}dx+{\int }_{\text{mb}}R{u}^{2}V(x)dx)((-\mathrm{\Delta }{)}^{\frac{1}{2}}u+V(x)u)=f(u)\text{in}\mathbb{R}$$ where $f(u)$ behaves like $e^{u^2}$ as $u\to \mathrm{\infty }$ and $f(u)\sim u^3$ as $u\to 0$.