February  2018, 38(2): 675-695. doi: 10.3934/dcds.2018029

Nonexistence results for elliptic differential inequalities with a potential in bounded domains

Politecnico di Milano, Via Bonardi, 9, Milano, 20133, Italy

* Corresponding author:D. D. Monticelli

Received  June 2017 Published  February 2018

In this paper we are concerned with a class of elliptic differential inequalities with a potential in bounded domains both of $\mathbb{R}^m$ and of Riemannian manifolds. In particular, we investigate the effect of the behavior of the potential at the boundary of the domain on nonexistence of nonnegative solutions.

Citation: Dario D. Monticelli, Fabio Punzo. Nonexistence results for elliptic differential inequalities with a potential in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 675-695. doi: 10.3934/dcds.2018029
References:
[1]

H. BerestyckiI. Capuzzo Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023.  Google Scholar

[2]

I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Royal Soc. Edinburgh, Sect. A, 128 (1998), 1217-1247.  doi: 10.1017/S0308210500027293.  Google Scholar

[3]

L. D'Ambrosio and V. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities, Adv. Math., 224 (2010), 967-1020.  doi: 10.1016/j.aim.2009.12.017.  Google Scholar

[4]

L. D'Ambrosio and S. Lucente, Nonlinear Liouville theorems for Grushin and Tricomi operators, J. Diff. Eq., 193 (2003), 511-541.  doi: 10.1016/S0022-0396(03)00138-4.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order Springer-Verlag, Berlin, 2001.  Google Scholar

[6]

A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Am. Math. Soc., 36 (1999), 135-249.  doi: 10.1090/S0273-0979-99-00776-4.  Google Scholar

[7]

A. Grigor'yan and V. A. Kondratiev, On the existence of positive solutions of semilinear elliptic inequalities on Riemannian manifolds, in Around the research of Vladimir Maz'ya. Ⅱ, volume 12 of Int. Math. Ser. (N. Y. ), Springer, New York, (2010), 203-218. doi: 10.1007/978-1-4419-1343-2_8.  Google Scholar

[8]

A. Grigor'yan and Y. Sun, On non-negative solutions of the inequality $Δ u + u^σ ≤q 0$ on Riemannian manifolds, Comm. Pure Appl. Math., 67 (2014), 1336-1352.  doi: 10.1002/cpa.21493.  Google Scholar

[9]

P. MastroliaD. D. Monticelli and F. Punzo, Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds, Calc. Var. Part. Diff. Eq., 54 (2015), 1345-1372.  doi: 10.1007/s00526-015-0827-0.  Google Scholar

[10]

P. MastroliaD. D. Monticelli and F. Punzo, Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds, Math. Ann., 367 (2017), 929-963.  doi: 10.1007/s00208-016-1393-2.  Google Scholar

[11]

V. Mitidieri and S. I. Pohozev, Absence of global positive solutions of quasilinear elliptic inequalities, Dokl. Akad. Nauk, 359 (1998), 456-460.   Google Scholar

[12]

V. Mitidieri and S. I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems in $\mathbb R^N$, Tr. Mat. Inst. Steklova, 227 (1999), 192-222.   Google Scholar

[13]

V. Mitidieri and S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.   Google Scholar

[14]

V. Mitidieri and S. I. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129-162.  doi: 10.1007/s00032-004-0032-7.  Google Scholar

[15]

D. D. Monticelli, Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, J. Eur. Math. Soc., 12 (2010), 611-654.  doi: 10.4171/JEMS/210.  Google Scholar

[16]

S. I. Pohozaev and A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities, Ann. Inst. H. Poinc. Anal. Non Lin., 21 (2004), 487-502.  doi: 10.1016/j.anihpc.2003.06.002.  Google Scholar

[17]

F. Punzo, Blow-up of solutions to semilinear parabolic equations on Riemannian manifolds with negative sectional curvature, J. Math. Anal. Appl., 387 (2012), 815-827.  doi: 10.1016/j.jmaa.2011.09.043.  Google Scholar

[18]

F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 21 (2010), 359-396.  doi: 10.4171/RLM/578.  Google Scholar

[19]

Y. Sun, Uniqueness results for nonnegative solutions of semilinear inequalities on Riemannian manifolds, J. Math. Anal. Appl., 419 (2014), 646-661.  doi: 10.1016/j.jmaa.2014.05.011.  Google Scholar

[20]

Y. Sun, On nonexistence of positive solutions of quasilinear inequality on Riemannian manifolds, Proc. Amer. Math. Soc., 143 (2015), 2969-2984.  doi: 10.1090/S0002-9939-2015-12705-0.  Google Scholar

show all references

References:
[1]

H. BerestyckiI. Capuzzo Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78.  doi: 10.12775/TMNA.1994.023.  Google Scholar

[2]

I. Birindelli and E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Royal Soc. Edinburgh, Sect. A, 128 (1998), 1217-1247.  doi: 10.1017/S0308210500027293.  Google Scholar

[3]

L. D'Ambrosio and V. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities, Adv. Math., 224 (2010), 967-1020.  doi: 10.1016/j.aim.2009.12.017.  Google Scholar

[4]

L. D'Ambrosio and S. Lucente, Nonlinear Liouville theorems for Grushin and Tricomi operators, J. Diff. Eq., 193 (2003), 511-541.  doi: 10.1016/S0022-0396(03)00138-4.  Google Scholar

[5]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order Springer-Verlag, Berlin, 2001.  Google Scholar

[6]

A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Am. Math. Soc., 36 (1999), 135-249.  doi: 10.1090/S0273-0979-99-00776-4.  Google Scholar

[7]

A. Grigor'yan and V. A. Kondratiev, On the existence of positive solutions of semilinear elliptic inequalities on Riemannian manifolds, in Around the research of Vladimir Maz'ya. Ⅱ, volume 12 of Int. Math. Ser. (N. Y. ), Springer, New York, (2010), 203-218. doi: 10.1007/978-1-4419-1343-2_8.  Google Scholar

[8]

A. Grigor'yan and Y. Sun, On non-negative solutions of the inequality $Δ u + u^σ ≤q 0$ on Riemannian manifolds, Comm. Pure Appl. Math., 67 (2014), 1336-1352.  doi: 10.1002/cpa.21493.  Google Scholar

[9]

P. MastroliaD. D. Monticelli and F. Punzo, Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds, Calc. Var. Part. Diff. Eq., 54 (2015), 1345-1372.  doi: 10.1007/s00526-015-0827-0.  Google Scholar

[10]

P. MastroliaD. D. Monticelli and F. Punzo, Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds, Math. Ann., 367 (2017), 929-963.  doi: 10.1007/s00208-016-1393-2.  Google Scholar

[11]

V. Mitidieri and S. I. Pohozev, Absence of global positive solutions of quasilinear elliptic inequalities, Dokl. Akad. Nauk, 359 (1998), 456-460.   Google Scholar

[12]

V. Mitidieri and S. I. Pohozaev, Nonexistence of positive solutions for quasilinear elliptic problems in $\mathbb R^N$, Tr. Mat. Inst. Steklova, 227 (1999), 192-222.   Google Scholar

[13]

V. Mitidieri and S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova, 234 (2001), 1-384.   Google Scholar

[14]

V. Mitidieri and S. I. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129-162.  doi: 10.1007/s00032-004-0032-7.  Google Scholar

[15]

D. D. Monticelli, Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators, J. Eur. Math. Soc., 12 (2010), 611-654.  doi: 10.4171/JEMS/210.  Google Scholar

[16]

S. I. Pohozaev and A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities, Ann. Inst. H. Poinc. Anal. Non Lin., 21 (2004), 487-502.  doi: 10.1016/j.anihpc.2003.06.002.  Google Scholar

[17]

F. Punzo, Blow-up of solutions to semilinear parabolic equations on Riemannian manifolds with negative sectional curvature, J. Math. Anal. Appl., 387 (2012), 815-827.  doi: 10.1016/j.jmaa.2011.09.043.  Google Scholar

[18]

F. Punzo and A. Tesei, On a semilinear parabolic equation with inverse-square potential, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 21 (2010), 359-396.  doi: 10.4171/RLM/578.  Google Scholar

[19]

Y. Sun, Uniqueness results for nonnegative solutions of semilinear inequalities on Riemannian manifolds, J. Math. Anal. Appl., 419 (2014), 646-661.  doi: 10.1016/j.jmaa.2014.05.011.  Google Scholar

[20]

Y. Sun, On nonexistence of positive solutions of quasilinear inequality on Riemannian manifolds, Proc. Amer. Math. Soc., 143 (2015), 2969-2984.  doi: 10.1090/S0002-9939-2015-12705-0.  Google Scholar

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