Existence of sign-changing solution for a problem involving the fractional Laplacian with critical growth nonlinearities

ABSTRACT

In this paper, we study the existence of sign-changing solution for a non-local problem, involving the fractional Laplacian operator and critical growth nonlinearities, namely $\begin{array}{rl}& (-\mathrm{\Delta }{)}^{s}u=\lambda f(x,u)+|u{|}^{2_s^{\ast }-2}u\mathrm{in}\mathrm{\Omega },\\ & u=0\mathrm{in}{\mathbb{R}}^N\setminus \mathrm{\Omega },\end{array}$ where Ω is a bounded smooth domain of ${\mathbb{R}}^N$, $s\in (0,1)$, $2_s^{\ast }$ is the fractional critical Sobolev exponent and λ is a positive parameter. Under certain assumptions on f, we show that the problem has a least-energy sign-changing solution for λ large. The proof is based on constrained minimization in a subset of Nehari manifold, containing all the possible solutions which change sign of this equation.

COMMUNICATED BY:

• T. Bartsch

KEYWORDS:

• Fractional Laplacian
• sign-changing solutions
• critical exponents
• variational methods

AMS SUBJECT CLASSIFICATIONS:

• 49J35
• 35S15
• 35B33
• 35A15

Acknowledgment

The authors would like to thank the anonymous referee for corrections and valuable suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

R. F. Gabert was financed in part by CAPES – Finance Code 001.