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Nonlinear equations involving the square root of the Laplacian
1. | Dipartimento di Scienze Pure e Applicate (DiSPeA), Università degli Studi di Urbino 'Carlo Bo' Piazza della Repubblica, 13, 61029 Urbino, Pesaro e Urbino, Italy |
2. | Dipartimento PAU, Università degli Studi 'Mediterranea' di Reggio Calabria, Salita Melissari - Feo di Vito, 89100 Reggio Calabria, Italy |
3. | Faculty of Education, and Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia |
$A_{1/2}$ |
$Ω\subset \mathbb{R}^{n}$ |
$n≥2$ |
$\left\{ \begin{align} &{{A}_{1/2}}u = \lambda f(u) \\ &u = 0 \\ \end{align} \right.\begin{array}{*{35}{l}} {}&\text{in}\ \Omega \\ {}&\text{on }\partial \Omega . \\\end{array}$ |
$L^{∞}$ |
$λ$ |
$f$ |
References:
[1] |
V. Ambrosio,
Periodic solutions for a pseudo-relativistic Schrödinger equation, Nonlinear Anal. TMA, 120 (2015), 262-284.
doi: 10.1016/j.na.2015.03.017. |
[2] |
V. Ambrosio and G. Molica Bisci,
Periodic solutions for nonlocal fractional equations, Comm. Pure Appl. Anal., 16 (2017), 331-344.
doi: 10.3934/cpaa.2017016. |
[3] |
V. Ambrosio and G. Molica Bisci, Periodic solutions for a fractional asymptotically linear problem,
Proc. Edinb. Math. Soc. Sect. A, in press. |
[4] |
G. Autuori and P. Pucci,
Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, J. Differential Equations, 255 (2013), 2340-2362.
doi: 10.1016/j.jde.2013.06.016. |
[5] |
B. Barrios, E. Colorado, A. De Pablo and U. Sánchez,
On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.
doi: 10.1016/j.jde.2012.02.023. |
[6] |
R. Bartolo and G. Molica Bisci,
A pseudo-index approach to fractional equations, Expo. Math., 33 (2015), 502-516.
doi: 10.1016/j.exmath.2014.12.001. |
[7] |
R. Bartolo and G. Molica Bisci,
Asymptotically linear fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 196 (2017), 427-442.
doi: 10.1007/s10231-016-0579-2. |
[8] |
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. |
[9] |
H. Brézis,
Analyse Fonctionelle, Théorie et applications, Masson, Paris, 1983. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
L. Caffarelli and A. Vasseur,
Drift diffusion equation with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[14] |
A. Capella,
Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662.
doi: 10.3934/cpaa.2011.10.1645. |
[15] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[16] |
A. Kristály,
Multiple solutions of a sublinear Schrödinger equation, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 291-301.
doi: 10.1007/s00030-007-5032-1. |
[17] |
A. Kristály and V. Rǎdulescu,
Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden-Fowler equations, Studia Math., 191 (2009), 237-246.
doi: 10.4064/sm191-3-5. |
[18] |
A. Kristály and D. Repovš,
Multiple solutions for a Neumann system involving subquadratic nonlinearities, Nonlinear Anal., 74 (2011), 2127-2132.
doi: 10.1016/j.na.2010.11.018. |
[19] |
A. Kristály and D. Repovš,
On the Schrödinger-Maxwell system involving sublinear terms, Nonlinear Anal. Real World Appl., 13 (2012), 213-223.
doi: 10.1016/j.nonrwa.2011.07.027. |
[20] |
A. Kristály and I. J. Rudas,
Elliptic problems on the ball endowed with Funk-type metrics, Nonlinear Anal., 119 (2015), 199-208.
doi: 10.1016/j.na.2014.09.015. |
[21] |
A. Kristály and Cs. Varga,
Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity, J. Math. Anal. Appl., 352 (2009), 139-148.
doi: 10.1016/j.jmaa.2008.03.025. |
[22] |
T. Kuusi, G. Mingione and Y. Sire,
Nonlocal self-improving properties, Analysis & PDE, 8 (2015), 57-114.
doi: 10.2140/apde.2015.8.57. |
[23] |
T. Kuusi, G. Mingione and Y. Sire,
Nonlocal equations with measure data, Communications in Mathematical Physics, 337 (2015), 1317-1368.
doi: 10.1007/s00220-015-2356-2. |
[24] |
M. Marinelli and D. Mugnai,
The generalized logistic equation with indefinite weight driven by the square root of the Laplacian, Nonlinearity, 27 (2014), 2361-2376.
doi: 10.1088/0951-7715/27/9/2361. |
[25] |
J. Mawhin and G. Molica Bisci,
A Brezis-Nirenberg type result for a nonlocal fractional operator, J. Lond. Math. Soc., 95 (2017), 73-93.
doi: 10.1112/jlms.12009. |
[26] |
G. Molica Bisci and V. Rǎdulescu,
Multiplicity results for elliptic fractional equations with subcritical term, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 721-739.
doi: 10.1007/s00030-014-0302-1. |
[27] |
G. Molica Bisci and V. Rǎdulescu,
Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.
doi: 10.1007/s00526-015-0891-5. |
[28] |
G. Molica Bisci and V. Rǎdulescu,
A sharp eigenvalue theorem for fractional elliptic equations, Israel Journal of Math., 219 (2017), 331-351.
doi: 10.1007/s11856-017-1482-2. |
[29] |
G. Molica Bisci, V. Rǎdulescu and R. Servadei,
Variational Methods for Nonlocal Fractional Problems. With a Foreword by Jean Mawhin, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 162 Cambridge, 2016.
doi: 10.1017/CBO9781316282397. |
[30] |
G. Molica Bisci and D. Repovš,
Existence and localization of solutions for nonlocal fractional equations, Asymptot. Anal., 90 (2014), 367-378.
|
[31] |
G. Molica Bisci and D. Repovš,
Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., 420 (2014), 167-176.
doi: 10.1016/j.jmaa.2014.05.073. |
[32] |
G. Molica Bisci and D. Repovš,
On doubly nonlocal fractional elliptic equations, Rend. Lincei Mat. Appl., 26 (2015), 161-176.
doi: 10.4171/RLM/700. |
[33] |
G. Molica Bisci, D. Repovš and R. Servadei,
Nontrivial solutions of superlinear nonlocal problems, Forum Math., 28 (2016), 1095-1110.
doi: 10.1515/forum-2015-0204. |
[34] |
G. Molica Bisci, D. Repovš and L. Vilasi,
Multipe solutions of nonlinear equations involving the square root of the Laplacian, Appl. Anal., 96 (2017), 1483-1496.
doi: 10.1080/00036811.2016.1221069. |
[35] |
D. Mugnai and D. Pagliardini,
Existence and multiplicity results for the fractional Laplacian in bounded domains, Adv. Calc. Var., 10 (2017), 111-124.
doi: 10.1515/acv-2015-0032. |
[36] |
R. Musina and A. Nazarov,
On fractional Laplacians, Commun. Partial Differential Equations, 39 (2014), 1780-1790.
doi: 10.1080/03605302.2013.864304. |
[37] |
P. Piersanti and P. Pucci,
Existence theorems for fractional $p$-Laplacian problems, Anal. Appl., 15 (2017), 607-640.
doi: 10.1142/S0219530516500020. |
[38] |
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. |
[39] |
R. Servadei and E. Valdinoci,
On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1-25.
doi: 10.1017/S0308210512001783. |
[40] |
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. |
[41] |
J. Tan,
Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-859.
doi: 10.3934/dcds.2013.33.837. |
show all references
Dedicated to Professor Vicenţiu Rǎdulescu with deep esteem and admiration
References:
[1] |
V. Ambrosio,
Periodic solutions for a pseudo-relativistic Schrödinger equation, Nonlinear Anal. TMA, 120 (2015), 262-284.
doi: 10.1016/j.na.2015.03.017. |
[2] |
V. Ambrosio and G. Molica Bisci,
Periodic solutions for nonlocal fractional equations, Comm. Pure Appl. Anal., 16 (2017), 331-344.
doi: 10.3934/cpaa.2017016. |
[3] |
V. Ambrosio and G. Molica Bisci, Periodic solutions for a fractional asymptotically linear problem,
Proc. Edinb. Math. Soc. Sect. A, in press. |
[4] |
G. Autuori and P. Pucci,
Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, J. Differential Equations, 255 (2013), 2340-2362.
doi: 10.1016/j.jde.2013.06.016. |
[5] |
B. Barrios, E. Colorado, A. De Pablo and U. Sánchez,
On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162.
doi: 10.1016/j.jde.2012.02.023. |
[6] |
R. Bartolo and G. Molica Bisci,
A pseudo-index approach to fractional equations, Expo. Math., 33 (2015), 502-516.
doi: 10.1016/j.exmath.2014.12.001. |
[7] |
R. Bartolo and G. Molica Bisci,
Asymptotically linear fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 196 (2017), 427-442.
doi: 10.1007/s10231-016-0579-2. |
[8] |
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. |
[9] |
H. Brézis,
Analyse Fonctionelle, Théorie et applications, Masson, Paris, 1983. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
L. Caffarelli and A. Vasseur,
Drift diffusion equation with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[14] |
A. Capella,
Solutions of a pure critical exponent problem involving the half-Laplacian in annular-shaped domains, Commun. Pure Appl. Anal., 10 (2011), 1645-1662.
doi: 10.3934/cpaa.2011.10.1645. |
[15] |
E. Di Nezza, G. Palatucci and E. Valdinoci,
Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573.
doi: 10.1016/j.bulsci.2011.12.004. |
[16] |
A. Kristály,
Multiple solutions of a sublinear Schrödinger equation, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 291-301.
doi: 10.1007/s00030-007-5032-1. |
[17] |
A. Kristály and V. Rǎdulescu,
Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden-Fowler equations, Studia Math., 191 (2009), 237-246.
doi: 10.4064/sm191-3-5. |
[18] |
A. Kristály and D. Repovš,
Multiple solutions for a Neumann system involving subquadratic nonlinearities, Nonlinear Anal., 74 (2011), 2127-2132.
doi: 10.1016/j.na.2010.11.018. |
[19] |
A. Kristály and D. Repovš,
On the Schrödinger-Maxwell system involving sublinear terms, Nonlinear Anal. Real World Appl., 13 (2012), 213-223.
doi: 10.1016/j.nonrwa.2011.07.027. |
[20] |
A. Kristály and I. J. Rudas,
Elliptic problems on the ball endowed with Funk-type metrics, Nonlinear Anal., 119 (2015), 199-208.
doi: 10.1016/j.na.2014.09.015. |
[21] |
A. Kristály and Cs. Varga,
Multiple solutions for a degenerate elliptic equation involving sublinear terms at infinity, J. Math. Anal. Appl., 352 (2009), 139-148.
doi: 10.1016/j.jmaa.2008.03.025. |
[22] |
T. Kuusi, G. Mingione and Y. Sire,
Nonlocal self-improving properties, Analysis & PDE, 8 (2015), 57-114.
doi: 10.2140/apde.2015.8.57. |
[23] |
T. Kuusi, G. Mingione and Y. Sire,
Nonlocal equations with measure data, Communications in Mathematical Physics, 337 (2015), 1317-1368.
doi: 10.1007/s00220-015-2356-2. |
[24] |
M. Marinelli and D. Mugnai,
The generalized logistic equation with indefinite weight driven by the square root of the Laplacian, Nonlinearity, 27 (2014), 2361-2376.
doi: 10.1088/0951-7715/27/9/2361. |
[25] |
J. Mawhin and G. Molica Bisci,
A Brezis-Nirenberg type result for a nonlocal fractional operator, J. Lond. Math. Soc., 95 (2017), 73-93.
doi: 10.1112/jlms.12009. |
[26] |
G. Molica Bisci and V. Rǎdulescu,
Multiplicity results for elliptic fractional equations with subcritical term, NoDEA Nonlinear Differential Equations Appl., 22 (2015), 721-739.
doi: 10.1007/s00030-014-0302-1. |
[27] |
G. Molica Bisci and V. Rǎdulescu,
Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.
doi: 10.1007/s00526-015-0891-5. |
[28] |
G. Molica Bisci and V. Rǎdulescu,
A sharp eigenvalue theorem for fractional elliptic equations, Israel Journal of Math., 219 (2017), 331-351.
doi: 10.1007/s11856-017-1482-2. |
[29] |
G. Molica Bisci, V. Rǎdulescu and R. Servadei,
Variational Methods for Nonlocal Fractional Problems. With a Foreword by Jean Mawhin, Encyclopedia of Mathematics and its Applications, Cambridge University Press, 162 Cambridge, 2016.
doi: 10.1017/CBO9781316282397. |
[30] |
G. Molica Bisci and D. Repovš,
Existence and localization of solutions for nonlocal fractional equations, Asymptot. Anal., 90 (2014), 367-378.
|
[31] |
G. Molica Bisci and D. Repovš,
Higher nonlocal problems with bounded potential, J. Math. Anal. Appl., 420 (2014), 167-176.
doi: 10.1016/j.jmaa.2014.05.073. |
[32] |
G. Molica Bisci and D. Repovš,
On doubly nonlocal fractional elliptic equations, Rend. Lincei Mat. Appl., 26 (2015), 161-176.
doi: 10.4171/RLM/700. |
[33] |
G. Molica Bisci, D. Repovš and R. Servadei,
Nontrivial solutions of superlinear nonlocal problems, Forum Math., 28 (2016), 1095-1110.
doi: 10.1515/forum-2015-0204. |
[34] |
G. Molica Bisci, D. Repovš and L. Vilasi,
Multipe solutions of nonlinear equations involving the square root of the Laplacian, Appl. Anal., 96 (2017), 1483-1496.
doi: 10.1080/00036811.2016.1221069. |
[35] |
D. Mugnai and D. Pagliardini,
Existence and multiplicity results for the fractional Laplacian in bounded domains, Adv. Calc. Var., 10 (2017), 111-124.
doi: 10.1515/acv-2015-0032. |
[36] |
R. Musina and A. Nazarov,
On fractional Laplacians, Commun. Partial Differential Equations, 39 (2014), 1780-1790.
doi: 10.1080/03605302.2013.864304. |
[37] |
P. Piersanti and P. Pucci,
Existence theorems for fractional $p$-Laplacian problems, Anal. Appl., 15 (2017), 607-640.
doi: 10.1142/S0219530516500020. |
[38] |
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. |
[39] |
R. Servadei and E. Valdinoci,
On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A, 144 (2014), 1-25.
doi: 10.1017/S0308210512001783. |
[40] |
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. |
[41] |
J. Tan,
Positive solutions for non local elliptic problems, Discrete Contin. Dyn. Syst., 33 (2013), 837-859.
doi: 10.3934/dcds.2013.33.837. |
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