Fractional Hardy-Sobolev elliptic problems

Jianfu Yang, Xiaohui Yu

DOI: http://dx.doi.org/10.12775/TMNA.2019.075

Abstract


In this paper, we study the following singular nonlinear elliptic problem \begin{equation} \begin{cases} \displaystyle (-\Delta)^{ \alpha/ 2} u=\lambda |u|^{r-2}u+\mu\frac{|u|^{q-2}u}{|x|^{s}} &\text{in }\Omega, \\ u=0 &\text{on } \partial\Omega, \end{cases} \tag{P} \end{equation} where $\Omega$ is a smooth bounded domain in $\mathbb R^N(N\geq 2)$ with $0\in \Omega$, $\lambda,\mu> 0$, $0< s\leq\alpha$, $(-\Delta)^{\alpha/ 2}$ is the spectral fractional Laplacian operator with $0< \alpha< 2$. We establish existence results and nonexistence results of problem \eqref{eq:1} for subcritical, Sobolev critical and Hardy-Sobolev critical cases.

Keywords


Critical Hardy-Sobolev exponent; decaying law; existence

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References


B. Barrios, E. Colorado, A. De Pablo and U. Sánchez, On some critical problems for the fractional Laplacian oprator, J. Differential Equations 252 (2012), 6133–6162.

C. Brändle, E. Colorado and A. De Pablo, A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A 143 (2013), 39–71.

H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math, Soc. 88 (1983), 486–490.

H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), 437–477.

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math. 224 (2010), 2052–2093.

A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non local semilinear equations, Comm. Partial Differential Equations 36 (2011), 1353–1384.

L. Caffarelli and L. Silvestre, An extention problem related to the fractional Laplacian, Comm. Partial Differentail Equations 32 (2007), 1245–1260.

D. Cao, X. He and S. Peng, Positive solutions for some singular critical growth nonlinear elliptic equations, Nonlinear Anal. 60 (2005), 589–609.

W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 (2006), 330–343.

W. Chen, S. Mosconi and M. Squassina, Nonlocal problems with critical Hardy nonlinearity, J. Funct. Anal. 275 (2018), 3065–3114.

A. Cotsiolis and N.K. Travoularis, Best constants for Sobolev inequalities for higher order fractional derivatives, J. Math. Anal. Appl. 295 (2004), 225–236.

E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 229 (2012), 521–573.

M.M. Fall and T. Weth, Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal. 263 (2012), 2205–2227.

R.L. Frank and R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities, J. Funct. Anal. 255 (2008), 3407–3430.

N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), 5703–5743.

T. Jin, Y. Li and J. Xiong, On a fractional Nirenberg problem, part I: Blow up analysis and compactness of solutions, J. Eur. Math. Soc. 16 (2014), 1111–1171.

D. Kang and S. Peng, Existence of solutions for elliptic problems with critical Sobolev–Hardy exponents, Israel J. Math. 143 (2004), 281–297.

D. Kang and S. Peng, Singular elliptic problems in RN with critical Sobolev–Hardy exponents, Nonlinear Anal. 68 (2008), no. 5, 1332–1345.

G. Li and S. Peng, Remarks on elliptic problems involving the Caffarelli–Kohn–Nirenberg inequalities, Proc. Amer. Math. Soc. 136 (2008), 1221–1228.

S. Lin and H. Wadade, Minimizing problems for the Hardy–Sobolev type inequality with the singularity on the boundary, Tohoku Math. J. 64 (2012), 79–103.

E.H. Lieb and M. Loss, Analysis, Graduate Studies in Mathematicas, vol. 14, Amer. Math. Soc., 2001.

P. Lions, The concentration compactness principle in the calculus of variations. The limit case (Parts 1 and 2), Rev. Mat. Iberoam. 1 (1985), 45–121, 145–201.

G. Lu and J.Zhu, Symmetry and regularity of extremals of an integral equation related to the Hardy–Sobolev inequality, Calc. Var. Partial Differential Equations 42 (2011), 563–577.

P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference, vol. 65, Amer. Math. Soc., Providence, R.I., 1986.

J. Tan, The Brezis–Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations 42 (2011), 21–41.

J. Tan and J. Xiong, A Harnack inequality for fractional Laplace equations with lower order terms, Discrete Contin. Dyn. Syst. 31 (2011), 975–983.

D. Yafaev, Sharp constants in the Hardy–Rellich inequalities, J. Funct. Anal. 168 (1999), 121–144.

J. Yang, Fractional Sobolev–Hardy inequality in RN , Nonlinear Anal. 119 (2015), 179–185.


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