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

• 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.
Mathematics Subject Classification: Primary: 35R11; Secondary: 35J20, 35J60.

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•  [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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