2018, 23(1): 29-43. doi: 10.3934/dcdsb.2018003

Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data

Faculty of Mathematics and Computer Science, University of Ƚódź, Banacha 22, 90-238 Ƚódź, Poland

Received  November 2016 Revised  April 2017 Published  January 2018

In the paper we consider a boundary value problem involving a differential equation with the fractional Laplacian $ (-Δ)^{α/2}$ for $ \mathit{\alpha }\in (1,2\rm{)}$ and some superlinear and subcritical nonlinearity $ G_{z}$ provided with a nonhomogeneous Dirichlet exterior boundary condition. Some sufficient conditions under which the set of weak solutions to the boundary value problem is nonempty and depends continuously in the Painlevé-Kuratowski sense on distributed parameters and exterior boundary data are stated. The proofs of the existence results rely on the Mountain Pass Theorem.

Citation: Dorota Bors. Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 29-43. doi: 10.3934/dcdsb.2018003
References:
[1]

D. Applebaum, Lévy processes -from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.

[2]

J-P. Aubin and H. Frankowska, Set-Valued Analysis Birkhäuser, Boston, 2009. doi: 10.1007/978-0-8176-4848-0.

[3]

G. Autuori, 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.

[4]

B. Barrios, E. Colorado, A. de Pablo, U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.

[5]

A. Bermudez, C. Saguez, Optimal control of a Signorini problem, SIAM J. Control Optim., 25 (1987), 576-582. doi: 10.1137/0325032.

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K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondracek, Potential Theory of Stable Processes and its Extensions Lecture Notes in Mathematics 1980, Springer, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-02141-1.

[7]

M. Bonforte, J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Arch. Ration. Mech. Anal., 218 (2015), 317-362. doi: 10.1007/s00205-015-0861-2.

[8]

D. Bors, Superlinear elliptic systems with distributed and boundary controls, Control Cybernet., 34 (2005), 987-1004.

[9]

D. Bors, Stability of nonlinear Dirichlet BVPs governed by fractional Laplacian Sci. World J. 2014 (2014), Article ID 920537, 10 pages. doi: 10.1155/2014/920537.

[10]

D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstr. Appl. Anal. 2013 (2013), Art. ID 240863, 10 pp.

[11]

X. Cabré, 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.

[12]

X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0.

[13]

L. A. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[14]

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

[15]

L. A. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[16]

Z.-Q. Chen, R. Song, Two-sided eigenvalue estimate for subordinate Brownian motion in bounded domain, J. Funct. Anal., 226 (2005), 90-113. doi: 10.1016/j.jfa.2005.05.004.

[17]

E. Di Nezza, G. Palatucci, 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.

[18]

M. Felsinger, M. Kassmann, P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809. doi: 10.1007/s00209-014-1394-3.

[19]

A. Fiscella, R. Servadei, E. Valdinoci, A resonance problem for non-local elliptic operators, Z. Anal. Anwend., 32 (2013), 411-431. doi: 10.4171/ZAA/1492.

[20]

D. Idczak, A. Rogowski, On a generalization of Krasnoselskii's theorem, J. Austral. Math. Soc. Ser. B, 72 (2002), 389-394.

[21]

T. Kulczycki, R. Stańczy, Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2581-2591. doi: 10.3934/dcdsb.2014.19.2581.

[22]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. doi: 10.1007/978-1-4757-2061-7.

[23]

G. Molica Bisci, R. Servadei, A bifurcation result for non-local fractional equations, Anal. Appl., 13 (2015), 371-394. doi: 10.1142/S0219530514500067.

[24]

G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems Encyclopedia of Mathematics and its Applications, 162 Cambridge University Press, 2016. doi: 10.1017/CBO9781316282397.

[25]

G. Molica Bisci, D. Repovs, R. Servadei, Nontrivial solutions of superlinear non-local problems, Forum Math., 28 (2016), 1095-1110. doi: 10.1515/forum-2015-0204.

[26]

X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.

[27]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.

[28]

R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[29]

R. Servadei, 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.

[30]

R. Servadei, E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non) local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126. doi: 10.4171/RMI/750.

[31]

R. Servadei, E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A., 144 (2014), 831-855. doi: 10.1017/S0308210512001783.

[32]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44.

[33]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.

show all references

References:
[1]

D. Applebaum, Lévy processes -from probability to finance and quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.

[2]

J-P. Aubin and H. Frankowska, Set-Valued Analysis Birkhäuser, Boston, 2009. doi: 10.1007/978-0-8176-4848-0.

[3]

G. Autuori, 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.

[4]

B. Barrios, E. Colorado, A. de Pablo, U. Sanchez, On some critical problems for the fractional Laplacian operator, J. Differential Equations, 252 (2012), 6133-6162. doi: 10.1016/j.jde.2012.02.023.

[5]

A. Bermudez, C. Saguez, Optimal control of a Signorini problem, SIAM J. Control Optim., 25 (1987), 576-582. doi: 10.1137/0325032.

[6]

K. Bogdan, T. Byczkowski, T. Kulczycki, M. Ryznar, R. Song and Z. Vondracek, Potential Theory of Stable Processes and its Extensions Lecture Notes in Mathematics 1980, Springer, Berlin, Heidelberg, 2009. doi: 10.1007/978-3-642-02141-1.

[7]

M. Bonforte, J. L. Vázquez, A priori estimates for fractional nonlinear degenerate diffusion equations on bounded domains, Arch. Ration. Mech. Anal., 218 (2015), 317-362. doi: 10.1007/s00205-015-0861-2.

[8]

D. Bors, Superlinear elliptic systems with distributed and boundary controls, Control Cybernet., 34 (2005), 987-1004.

[9]

D. Bors, Stability of nonlinear Dirichlet BVPs governed by fractional Laplacian Sci. World J. 2014 (2014), Article ID 920537, 10 pages. doi: 10.1155/2014/920537.

[10]

D. Bors, Global solvability of Hammerstein equations with applications to BVP involving fractional Laplacian, Abstr. Appl. Anal. 2013 (2013), Art. ID 240863, 10 pp.

[11]

X. Cabré, 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.

[12]

X. Cabré, Y. Sire, Nonlinear equations for fractional Laplacians Ⅱ: Existence, uniqueness, and qualitative properties of solutions, Trans. Amer. Math. Soc., 367 (2015), 911-941. doi: 10.1090/S0002-9947-2014-05906-0.

[13]

L. A. Caffarelli, S. Salsa, L. Silvestre, Regularity estimates for the solution and the free boundary of the obstacle problem for the fractional Laplacian, Invent. Math., 171 (2008), 425-461. doi: 10.1007/s00222-007-0086-6.

[14]

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

[15]

L. A. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[16]

Z.-Q. Chen, R. Song, Two-sided eigenvalue estimate for subordinate Brownian motion in bounded domain, J. Funct. Anal., 226 (2005), 90-113. doi: 10.1016/j.jfa.2005.05.004.

[17]

E. Di Nezza, G. Palatucci, 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.

[18]

M. Felsinger, M. Kassmann, P. Voigt, The Dirichlet problem for nonlocal operators, Math. Z., 279 (2015), 779-809. doi: 10.1007/s00209-014-1394-3.

[19]

A. Fiscella, R. Servadei, E. Valdinoci, A resonance problem for non-local elliptic operators, Z. Anal. Anwend., 32 (2013), 411-431. doi: 10.4171/ZAA/1492.

[20]

D. Idczak, A. Rogowski, On a generalization of Krasnoselskii's theorem, J. Austral. Math. Soc. Ser. B, 72 (2002), 389-394.

[21]

T. Kulczycki, R. Stańczy, Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2581-2591. doi: 10.3934/dcdsb.2014.19.2581.

[22]

J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, 1989. doi: 10.1007/978-1-4757-2061-7.

[23]

G. Molica Bisci, R. Servadei, A bifurcation result for non-local fractional equations, Anal. Appl., 13 (2015), 371-394. doi: 10.1142/S0219530514500067.

[24]

G. Molica Bisci, V. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems Encyclopedia of Mathematics and its Applications, 162 Cambridge University Press, 2016. doi: 10.1017/CBO9781316282397.

[25]

G. Molica Bisci, D. Repovs, R. Servadei, Nontrivial solutions of superlinear non-local problems, Forum Math., 28 (2016), 1095-1110. doi: 10.1515/forum-2015-0204.

[26]

X. Ros-Oton, J. Serra, The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary, J. Math. Pures Appl., 101 (2014), 275-302. doi: 10.1016/j.matpur.2013.06.003.

[27]

R. Servadei, Infinitely many solutions for fractional Laplace equations with subcritical nonlinearity, Contemp. Math., 595 (2013), 317-340. doi: 10.1090/conm/595/11809.

[28]

R. Servadei, E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.

[29]

R. Servadei, 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.

[30]

R. Servadei, E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non) local operators, Rev. Mat. Iberoam., 29 (2013), 1091-1126. doi: 10.4171/RMI/750.

[31]

R. Servadei, E. Valdinoci, On the spectrum of two different fractional operators, Proc. Roy. Soc. Edinburgh Sect. A., 144 (2014), 831-855. doi: 10.1017/S0308210512001783.

[32]

E. Valdinoci, From the long jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl. SeMA, 49 (2009), 33-44.

[33]

J. L. Vázquez, Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885. doi: 10.3934/dcdss.2014.7.857.

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