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 and 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, J. Funct Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[2]

C. 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. doi: 10.1017/S0308210511000175.

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B. Barrios, E. Colorado, R. Servadei and F. Sorai, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900. doi: 10.1016/j.anihpc.2014.04.003.

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K. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Equ., 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.

[6]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[8]

D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $R^N$, Pro. Roy. Soc. Edinburgh, 126 (1996), 443-463. doi: 10.1017/S0308210500022836.

[9]

E. Colorado, A. De Pablo and U. Sánches, Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85. doi: 10.2140/pjm.2014.271.65.

[10]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for the fractional Laplacians in $R$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.

[11]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Lapacian, arXiv:1302.2652, (2013).

[12]

N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equation, Nonlinear Anal., 29 (1997), 889-901. doi: 10.1016/S0362-546X(96)00176-9.

[13]

T. Hsu and H. Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $R^N$, J. Math. Anal. Appl., 365 (2010), 758-775. doi: 10.1016/j.jmaa.2009.12.004.

[14]

H. Lin, Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\R^N$, Bound. value probl. 2012, 24 (2012), 17pp. doi: 10.1186/1687-2770-2012-24.

[15]

P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.

[16]

P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.

[17]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam, 29 (2006), 1091-1126. doi: 10.4171/RMI/750.

[18]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[19]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 5 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105.

[20]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.

[21]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.

[22]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.

[23]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304.

[24]

H. Wang, Palais-Smale approaches to semilinear elliptic equations in unbounded domains, Electron J. Diff. Equ., Monogragh 06 (2004), 142pp.

[25]

T. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270. doi: 10.1016/j.jmaa.2005.05.057.

[26]

X, Yu, The Nehari manifold for elliptic equation involving the square root of the Laplacian, J. Diff. Equ., 252 (2012), 1283-1308. doi: 10.1016/j.jde.2011.09.015.

[27]

X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Diff. Equ., 92 (1991), 163-178. doi: 10.1016/0022-0396(91)90045-B.

show all references

References:
[1]

A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[2]

C. 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. doi: 10.1017/S0308210511000175.

[3]

B. Barrios, E. Colorado, R. Servadei and F. Sorai, A critical fractional equation with concave-convex power nonlinearities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 875-900. doi: 10.1016/j.anihpc.2014.04.003.

[4]

K. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Equ., 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.

[5]

X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians I: regularity, maximum principles, and hamiltonian estimates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 23-53. doi: 10.1016/j.anihpc.2013.02.001.

[6]

X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.

[7]

L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.

[8]

D. Cao and H. Zhou, Multiple positive solutions of nonhomogeneous semilinear elliptic equations in $R^N$, Pro. Roy. Soc. Edinburgh, 126 (1996), 443-463. doi: 10.1017/S0308210500022836.

[9]

E. Colorado, A. De Pablo and U. Sánches, Perturbations of a critical fractional equation, Pacific J. Math., 271 (2014), 65-85. doi: 10.2140/pjm.2014.271.65.

[10]

R. Frank and E. Lenzmann, Uniqueness of non-linear ground states for the fractional Laplacians in $R$, Acta Math., 210 (2013), 261-318. doi: 10.1007/s11511-013-0095-9.

[11]

R. Frank, E. Lenzmann and L. Silvestre, Uniqueness of radial solutions for the fractional Lapacian, arXiv:1302.2652, (2013).

[12]

N. Hirano, Existence of entire positive solutions for nonhomogeneous elliptic equation, Nonlinear Anal., 29 (1997), 889-901. doi: 10.1016/S0362-546X(96)00176-9.

[13]

T. Hsu and H. Lin, Four positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $R^N$, J. Math. Anal. Appl., 365 (2010), 758-775. doi: 10.1016/j.jmaa.2009.12.004.

[14]

H. Lin, Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in $\R^N$, Bound. value probl. 2012, 24 (2012), 17pp. doi: 10.1186/1687-2770-2012-24.

[15]

P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 109-145.

[16]

P. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. II, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), 223-283.

[17]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Iberoam, 29 (2006), 1091-1126. doi: 10.4171/RMI/750.

[18]

R. Servadei and E. Valdinoci, Mountain Pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[19]

R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 5 (2013), 2105-2137. doi: 10.3934/dcds.2013.33.2105.

[20]

R. Servadei and E. Valdinoci, The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. Soc., 367 (2015), 67-102.

[21]

R. Servadei and E. Valdinoci, A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (2013), 2445-2464. doi: 10.3934/cpaa.2013.12.2445.

[22]

J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 36 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.

[23]

G. Tarantello, On nonhomogeneous elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992), 281-304.

[24]

H. Wang, Palais-Smale approaches to semilinear elliptic equations in unbounded domains, Electron J. Diff. Equ., Monogragh 06 (2004), 142pp.

[25]

T. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270. doi: 10.1016/j.jmaa.2005.05.057.

[26]

X, Yu, The Nehari manifold for elliptic equation involving the square root of the Laplacian, J. Diff. Equ., 252 (2012), 1283-1308. doi: 10.1016/j.jde.2011.09.015.

[27]

X. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Diff. Equ., 92 (1991), 163-178. doi: 10.1016/0022-0396(91)90045-B.

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