Electron. J. Diff. Equ., Vol. 2016 (2016), No. 203, pp. 1-9.

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

Pawan Kumar Mishra, Konijeti Sreenadh

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.

Submitted March 5, 2016. Published July 27, 2016.
Math Subject Classifications: 35A15, 35R11
Key Words: Fractional Laplacian; bifurcation; exponential growth.

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Pawan Kumar Mishra
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khaz, New Delhi-16, India
email: pawanmishra31284@gmail.com
Konijeti Sreenadh
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khaz, New Delhi-16, India
email: sreenadh@gmail.com

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