September  2016, 15(5): 1671-1688. doi: 10.3934/cpaa.2016008

Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$

1. 

School of Mathematics and Computer Science, Wuhan Polytechnic University, Wuhan, 430023, China

Received  September 2015 Revised  April 2016 Published  July 2016

In this paper, we investigate the existence of positive solutions of the following equation \begin{eqnarray} (-\Delta)^s v +\lambda v=f(x) v^{p-1}+h(x)v^{q-1}, \ x\in R^N,\\ v\in H^s(R^N), \end{eqnarray} where $1\leq q< 2 < p < 2_s^*=\frac{2N}{N-2s}$, $0 < s < 1$, $N>2s$ and $\lambda>0$ is a parameter. Since the concave and convex nonlinearities are involved, the variational functional of the equation has different properties. Via variational method, we show that the equation admits a positive ground state solution for all $\lambda>0$ strictly larger than a threshold value. Moreover, under certain conditions on $f$ and for sufficiently large $\lambda>0$, we also prove that there are at least $k+1$ ($k$ is a positive integer) positive solutions of the equation.
Citation: Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008
References:
[1]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, \emph{J. Funct Anal.}, 122 (1994), 519. doi: 10.1006/jfan.1994.1078. Google Scholar

[2]

C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 143 (2013), 39. doi: 10.1017/S0308210511000175. Google Scholar

[3]

B. Barrios, E. Colorado, R. Servadei and F. Sorai, A critical fractional equation with concave-convex power nonlinearities,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 32 (2015), 875. doi: 10.1016/j.anihpc.2014.04.003. Google Scholar

[4]

K. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,, \emph{J. Diff. Equ.}, 193 (2003), 481. doi: 10.1016/S0022-0396(03)00121-9. Google Scholar

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and hamiltonian estimates,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 31 (2014), 23. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar

[6]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[8]

D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $R^N$,, \emph{Pro. Roy. Soc. Edinburgh}, 126 (1996), 443. doi: 10.1017/S0308210500022836. Google Scholar

[9]

E. Colorado, A. De Pablo and U. Sánches, Perturbations of a critical fractional equation,, \emph{Pacific J. Math.}, 271 (2014), 65. doi: 10.2140/pjm.2014.271.65. Google Scholar

[10]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for the fractional Laplacians in $R$,, \emph{Acta Math.}, 210 (2013), 261. doi: 10.1007/s11511-013-0095-9. Google Scholar

[11]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Lapacian,, \arXiv{1302.2652}, (2013). Google Scholar

[12]

N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equation,, \emph{Nonlinear Anal.}, 29 (1997), 889. doi: 10.1016/S0362-546X(96)00176-9. Google Scholar

[13]

T. Hsu and H. Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $R^N$,, \emph{J. Math. Anal. Appl.}, 365 (2010), 758. doi: 10.1016/j.jmaa.2009.12.004. Google Scholar

[14]

H. Lin, Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\R^N$,, \emph{Bound. value probl. 2012}, 24 (2012). doi: 10.1186/1687-2770-2012-24. Google Scholar

[15]

P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 109. Google Scholar

[16]

P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223. Google Scholar

[17]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, \emph{Rev. Mat. Iberoam}, 29 (2006), 1091. doi: 10.4171/RMI/750. Google Scholar

[18]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators,, \emph{J. Math. Anal. Appl.}, 389 (2012), 887. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar

[19]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, \emph{Discrete Contin. Dyn. Syst.}, 5 (2013), 2105. doi: 10.3934/dcds.2013.33.2105. Google Scholar

[20]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, \emph{Trans. Amer. Math. Soc.}, 367 (2015), 67. Google Scholar

[21]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2445. doi: 10.3934/cpaa.2013.12.2445. Google Scholar

[22]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 36 (2011), 21. doi: 10.1007/s00526-010-0378-3. Google Scholar

[23]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 9 (1992), 281. Google Scholar

[24]

H. Wang, Palais-Smale approaches to semilinear elliptic equations in unbounded domains,, \emph{Electron J. Diff. Equ.}, 06 (2004). Google Scholar

[25]

T. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,, \emph{J. Math. Anal. Appl.}, 318 (2006), 253. doi: 10.1016/j.jmaa.2005.05.057. Google Scholar

[26]

X, Yu, The Nehari manifold for elliptic equation involving the square root of the Laplacian,, \emph{J. Diff. Equ.}, 252 (2012), 1283. doi: 10.1016/j.jde.2011.09.015. Google Scholar

[27]

X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation,, \emph{J. Diff. Equ.}, 92 (1991), 163. doi: 10.1016/0022-0396(91)90045-B. Google Scholar

show all references

References:
[1]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems,, \emph{J. Funct Anal.}, 122 (1994), 519. doi: 10.1006/jfan.1994.1078. Google Scholar

[2]

C. Brändle, E. Colorado, A. de Pablo and U. Sánchez, A concave-convex elliptic problem involving the fractional Laplacian,, \emph{Proc. Roy. Soc. Edinburgh Sect. A}, 143 (2013), 39. doi: 10.1017/S0308210511000175. Google Scholar

[3]

B. Barrios, E. Colorado, R. Servadei and F. Sorai, A critical fractional equation with concave-convex power nonlinearities,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 32 (2015), 875. doi: 10.1016/j.anihpc.2014.04.003. Google Scholar

[4]

K. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,, \emph{J. Diff. Equ.}, 193 (2003), 481. doi: 10.1016/S0022-0396(03)00121-9. Google Scholar

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and hamiltonian estimates,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 31 (2014), 23. doi: 10.1016/j.anihpc.2013.02.001. Google Scholar

[6]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, \emph{Adv. Math.}, 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. Google Scholar

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, \emph{Comm. Partial Differential Equations}, 32 (2007), 1245. doi: 10.1080/03605300600987306. Google Scholar

[8]

D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $R^N$,, \emph{Pro. Roy. Soc. Edinburgh}, 126 (1996), 443. doi: 10.1017/S0308210500022836. Google Scholar

[9]

E. Colorado, A. De Pablo and U. Sánches, Perturbations of a critical fractional equation,, \emph{Pacific J. Math.}, 271 (2014), 65. doi: 10.2140/pjm.2014.271.65. Google Scholar

[10]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for the fractional Laplacians in $R$,, \emph{Acta Math.}, 210 (2013), 261. doi: 10.1007/s11511-013-0095-9. Google Scholar

[11]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Lapacian,, \arXiv{1302.2652}, (2013). Google Scholar

[12]

N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equation,, \emph{Nonlinear Anal.}, 29 (1997), 889. doi: 10.1016/S0362-546X(96)00176-9. Google Scholar

[13]

T. Hsu and H. Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $R^N$,, \emph{J. Math. Anal. Appl.}, 365 (2010), 758. doi: 10.1016/j.jmaa.2009.12.004. Google Scholar

[14]

H. Lin, Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\R^N$,, \emph{Bound. value probl. 2012}, 24 (2012). doi: 10.1186/1687-2770-2012-24. Google Scholar

[15]

P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 109. Google Scholar

[16]

P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 1 (1984), 223. Google Scholar

[17]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators,, \emph{Rev. Mat. Iberoam}, 29 (2006), 1091. doi: 10.4171/RMI/750. Google Scholar

[18]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators,, \emph{J. Math. Anal. Appl.}, 389 (2012), 887. doi: 10.1016/j.jmaa.2011.12.032. Google Scholar

[19]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, \emph{Discrete Contin. Dyn. Syst.}, 5 (2013), 2105. doi: 10.3934/dcds.2013.33.2105. Google Scholar

[20]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian,, \emph{Trans. Amer. Math. Soc.}, 367 (2015), 67. Google Scholar

[21]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension,, \emph{Commun. Pure Appl. Anal.}, 12 (2013), 2445. doi: 10.3934/cpaa.2013.12.2445. Google Scholar

[22]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, \emph{Calc. Var. Partial Differential Equations}, 36 (2011), 21. doi: 10.1007/s00526-010-0378-3. Google Scholar

[23]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent,, \emph{Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire}, 9 (1992), 281. Google Scholar

[24]

H. Wang, Palais-Smale approaches to semilinear elliptic equations in unbounded domains,, \emph{Electron J. Diff. Equ.}, 06 (2004). Google Scholar

[25]

T. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,, \emph{J. Math. Anal. Appl.}, 318 (2006), 253. doi: 10.1016/j.jmaa.2005.05.057. Google Scholar

[26]

X, Yu, The Nehari manifold for elliptic equation involving the square root of the Laplacian,, \emph{J. Diff. Equ.}, 252 (2012), 1283. doi: 10.1016/j.jde.2011.09.015. Google Scholar

[27]

X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation,, \emph{J. Diff. Equ.}, 92 (1991), 163. doi: 10.1016/0022-0396(91)90045-B. Google Scholar

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