## Topological Methods in Nonlinear Analysis

### Multiplicity of positive solutions for fractional Laplacian equations involving critical nonlinearity

#### Abstract

In this paper, we consider the following problem involving fractional Laplacian operator \begin{equation*} (-\Delta)^{s} u=\lambda f(x)|u|^{q-2}u+|u|^{2^{*}_{s}-2}u\quad \text{in } \Omega,\qquad u=0\quad \text{on } \partial\Omega, \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{N}$, $0<s<1$, $2^*_{s}={2N}/({N-2s})$, and $(-\Delta)^{s}$ is the fractional Laplacian. We will prove that there exists $\lambda_{*}>0$ such that the problem has at least two positive solutions for each $\lambda\in (0,\lambda_{*})$. In addition, the concentration behavior of the solutions are investigated.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 53, Number 1 (2019), 151-182.

Dates
First available in Project Euclid: 20 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1550631833

Digital Object Identifier
doi:10.12775/TMNA.2018.043

Mathematical Reviews number (MathSciNet)
MR3939152

#### Citation

Zhang, Jinguo; Liu, Xiaochun; Jiao, Hongying. Multiplicity of positive solutions for fractional Laplacian equations involving critical nonlinearity. Topol. Methods Nonlinear Anal. 53 (2019), no. 1, 151--182. doi:10.12775/TMNA.2018.043. https://projecteuclid.org/euclid.tmna/1550631833

#### References

• R.A. Adams, Sobolev Space, Pure Appl. Math. vol 65, Academic Press, New York, London, 1975.
• D. Applebaum, Lévy process-from probability to finance and quantum groups, Notices Amer. Math. Soc. 51 (2004), 1336–1347.
• B. Barrios, E. Colorado, A. de Pablo and U. Sánchez, On some critical problems for the fractional Laplacian operator, J. Differential Equation 252 (2012), 6133–6126.
• B. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh. Sect. A 143 (2013), 39–71.
• H. Brezis and L. Nirengerg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.
• X. Cabré and J. Solá-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math. 58 (2005), 1678–1732.
• X. Cabré and J. Tan, Positive solutions for nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.
• L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), 1245–1260.
• D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $\mathbb{R}^{N}$, Proc. Roy. Soc. Edinburgh Sect. A 126 (1996), 443–463.
• W. Chi, S. Kim and K. Lee, Asymptotic behavior of solutions for nonlinear elliptic problems with the fractional Laplacian, J. Functional Analysis 266 (2014), 6531–6598.
• J. Dávila, M. de Pino and J. Wei, Concentration standing wave for the fractional nonlinear Schrödinger equation, J. Differential Equations 256 (2014), 858–892.
• E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), 521–573.
• P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237–1262.
• R.L. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Laplacian, Comm. Pure Appl. Math. 69 (2016), no. 9, 1671–1726.
• A. Garroni and S. Müller, $\Gamma$-limit of a phase-field model of dislocations, SIMA J. Math. Anal.36 (2005), 1943–1964.
• L. Lions and E. Magenes, Problémes aux Limits Non Homogénes et Applications, vol. 1, Trav. et Rech. Math., vol. 17, Dunod, Paris, 1968.
• G. Palatucci and A. Pisante, Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces, Calc. Var. 50 (2014), 799–829.
• G. Palatucci and A. Pisante, A global compactness type results for Palais–Smale sequences in fractional Sobolev space, Nonlinear Anal. 117 (2015), 1–7; (2014), 799–829.
• R. Servadei and E. Valdinoci, The Brezis–Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc. 367 (2015), 67–102.
• R. Servadei and E. Valdinoci, A Brezis–Nirenberg result for nonlocal critical equations in low dimension, Comm. Pure Appl. Anal. 12 (2013), no. 6, 2445–2464.
• R. Servadei and E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), 831–855.
• X. Shang, J. Zhang and Y. Yang, Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent, Comm. Pure Appl. Anal. 13 (2014), no. 2, 567–584.
• L. Silvestre, Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math. 60 (2007), 67–112.
• J. Tan, The Brezis–Nirenberg type problem involving the square root of the Laplacian, Calc. Var. 42 (2011), 21–41.
• C. Tarantello, On nonhomogeneous elliptic involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire 9 (1992), 281–304.
• T.F. Wu, On semilinear elliptic equations involving critical Sobolev exponents and sign-changing weight function, Comm. Pure Appl. Anal. 7 (2008), 383–405.
• J. Zhang and X. Liu, The Nehari manifold for a semilinear elliptic problem with the nonlinear boundary condition, J. Math. Anal. Appl. 400 (2013), 100–119.
• J. Zhang and X. Liu, Three solutions for a fractional elliptic problems with critical and supercritical growth, Acta Math. Sci. 36,B (2016), no. 6, 1–13.