Bifurcation and multiplicity of solutions for the fractional Laplacian with critical exponential nonlinearity

Abstract

We study the fractional elliptic equation $$\displaylines{ (-\Delta)^{1/2} u = \lambda u+|u|^{p-2}ue^{u^2} ,\quad\text{in } (-1,1),\cr u=0\quad\text{in } \mathbb{R}\setminus(-1,1), }$$ where $\lambda$ is a positive real parameter, p>2 and $(-\Delta)^{1/2}$ is the fractional Laplacian operator. We show the multiplicity of solutions for this problem using an abstract critical point theorem of literature in critical point theory. Precisely, we extended the result of Cerami, Fortuno and Struwe [5] for the fractional Laplacian with exponential nonlinearity.

Keywords

• Fractional Laplacian
• bifurcation
• exponential growth