# American Institute of Mathematical Sciences

April  2019, 12(2): 151-170. doi: 10.3934/dcdss.2019011

## 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

* Corresponding author: Giovanni Molica Bisci

Dedicated to Professor Vicenţiu Rǎdulescu with deep esteem and admiration

Received  May 2017 Revised  December 2017 Published  August 2018

In this paper we discuss the existence and non-existence of weak solutions to parametric fractional equations involving the square root of the Laplacian
 $A_{1/2}$
in a smooth bounded domain
 $Ω\subset \mathbb{R}^{n}$
(
 $n≥2$
) and with zero Dirichlet boundary conditions. Namely, our simple model is the following equation
 \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}
The existence of at least two non-trivial
 $L^{∞}$
-bounded weak solutions is established for large value of the parameter
 $λ$
, requiring that the nonlinear term
 $f$
is continuous, superlinear at zero and sublinear at infinity. Our approach is based on variational arguments and a suitable variant of the Caffarelli-Silvestre extension method.
Citation: Vincenzo Ambrosio, Giovanni Molica Bisci, Dušan Repovš. Nonlinear equations involving the square root of the Laplacian. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 151-170. doi: 10.3934/dcdss.2019011
##### References:
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Molica Bisci, A pseudo-index approach to fractional equations, Expo. Math., 33 (2015), 502-516. doi: 10.1016/j.exmath.2014.12.001. Google Scholar [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. Google Scholar [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. Google Scholar [9] H. Brézis, Analyse Fonctionelle, Théorie et applications, Masson, Paris, 1983. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] G. Molica Bisci and D. Repovš, Existence and localization of solutions for nonlocal fractional equations, Asymptot. Anal., 90 (2014), 367-378. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] R. Musina and A. Nazarov, On fractional Laplacians, Commun. Partial Differential Equations, 39 (2014), 1780-1790. doi: 10.1080/03605302.2013.864304. Google Scholar [37] P. Piersanti and P. Pucci, Existence theorems for fractional $p$-Laplacian problems, Anal. Appl., 15 (2017), 607-640. doi: 10.1142/S0219530516500020. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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##### 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. Google Scholar [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. Google Scholar [3] V. Ambrosio and G. Molica Bisci, Periodic solutions for a fractional asymptotically linear problem, Proc. Edinb. Math. Soc. Sect. A, in press.Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [9] H. Brézis, Analyse Fonctionelle, Théorie et applications, Masson, Paris, 1983. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] G. Molica Bisci and D. Repovš, Existence and localization of solutions for nonlocal fractional equations, Asymptot. Anal., 90 (2014), 367-378. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [36] R. Musina and A. Nazarov, On fractional Laplacians, Commun. Partial Differential Equations, 39 (2014), 1780-1790. doi: 10.1080/03605302.2013.864304. Google Scholar [37] P. Piersanti and P. Pucci, Existence theorems for fractional $p$-Laplacian problems, Anal. Appl., 15 (2017), 607-640. doi: 10.1142/S0219530516500020. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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