Critical exponent for half-Laplacian in the whole space

Abstract

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 \;\textrm{in} \;\mathbb R, \end{equation*} %where $1<q<2,\;p>2,\;1<\beta\leq2\;, \lambda>0, K(x)>0, f$ is continuous and sign changing. where $h$ is a real valued function that behaves like $e^{u^2}$ as $u\rightarrow \infty$ and $V(x)$ is a positive, continuous unbounded function. Here $(-\Delta)^{\frac{1}{2}}$ 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 \[ m\left(\int_{\mathbb R}|(-\Delta)^{\frac{1}{2}}u|^2 dx+ \int_\mb R u^2 V(x)dx\right)\left((-\Delta)^{\frac{1}{2}} u+ V(x) u\right)= f(u)\;\, \text{in}\, \mathbb R \] where $f(u)$ behaves like $e^{u^2}$ as $u\rightarrow \infty$ and $f(u)\sim u^3$ as $u\rightarrow 0$.