January  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.   Google Scholar

[2]

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

[3]

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

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B. BarriosE. ColoradoA. de Pablo and 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.  Google Scholar

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

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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.  Google Scholar

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M. Bonforte and 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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

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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

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X. Cabré and 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.  Google Scholar

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L. A. CaffarelliS. Salsa and 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.  Google Scholar

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L. A. 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

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L. A. Caffarelli and 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.  Google Scholar

[16]

Z.-Q. Chen and 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.  Google Scholar

[17]

E. Di NezzaG. 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

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

[19]

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

[20]

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

[21]

T. Kulczycki and 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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[26]

X. Ros-Oton and 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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[30]

R. Servadei and 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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

B. BarriosE. ColoradoA. de Pablo and 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.  Google Scholar

[5]

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

[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.  Google Scholar

[7]

M. Bonforte and 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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[11]

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

[12]

X. Cabré and 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.  Google Scholar

[13]

L. A. CaffarelliS. Salsa and 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.  Google Scholar

[14]

L. A. 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

[15]

L. A. Caffarelli and 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.  Google Scholar

[16]

Z.-Q. Chen and 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.  Google Scholar

[17]

E. Di NezzaG. 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

[18]

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

[19]

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

[20]

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

[21]

T. Kulczycki and 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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[26]

X. Ros-Oton and 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.  Google Scholar

[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.  Google Scholar

[28]

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

[29]

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

[30]

R. Servadei and 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.  Google Scholar

[31]

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

[32]

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

[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.  Google Scholar

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