American Institute of Mathematical Sciences

Advanced
March  2020, 28(1): 165-182. doi: 10.3934/era.2020011

Non-existence of solutions to some degenerate coercivity elliptic equations involving measures data

Maoji Ri 1, , Shuibo Huang 1,2,, and  Canyun Huang 3,

1. 

School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou, Gansu 730030, China
2. 

Key Laboratory of Streaming Data Computing Technologies and Application, Northwest Minzu University, Lanzhou, Gansu 730030, China
3. 

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

* Corresponding author: Shuibo Huang

Received  October 2019 Revised  February 2020 Published  March 2020

Fund Project: This research was partially supported by the National Natural Science Foundation of China (No. 11761059), Program for Yong Talent of State Ethnic Affairs Commission of China(No. XBMU-2019-AB-34), Fundamental Research Funds for the Central Universities (No.31920200036) and Key Subject of Gansu Province

In this paper, we main consider the non-existence of solutions
$ u $
 by approximation to the following quasilinear elliptic problem with principal part having degenerate coercivity:
$ \begin{align*} \left \{ \begin{array}{rl} -\text{div}\left(\frac{|\nabla u|^{p-2}\nabla u}{(1+|u|)^{(p-1)\theta}}\right)+|u|^{q-1}u = \lambda, \; &x\in\Omega, \\ u = 0, \; &x\in\partial\Omega, \end{array} \right. \end{align*} $
provided
$ \begin{align*} q>\frac{r(p-1)[1+\theta(p-1)]}{r-p}, \end{align*} $
where
$ \Omega $
 is a bounded smooth subset of
$ \mathbb{R}^N(N>2) $
,
$ 1<p<N $
,
$ q>1 $
,
$ 0\leq\theta<1 $
,
$ \lambda $
 is a measure which is concentrated on a set with zero
$ r $
 capacity
$ (p<r\leq N) $
.
Keywords: Degenerate coercivity, measures data, non-existence.
Mathematics Subject Classification: 35R06, 5J70, 35A01.
Citation: Maoji Ri, Shuibo Huang, Canyun Huang. Non-existence of solutions to some degenerate coercivity elliptic equations involving measures data. Electronic Research Archive, 2020, 28 (1) : 165-182. doi: 10.3934/era.2020011
References:
[1]

A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura. Appl. (4), 182 (2003), 53-79. doi: 10.1007/s10231-002-0056-y.  Google Scholar

[2]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An L1 theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273.  Google Scholar

[3]

P. Bénilan, H. Brézis and M. Crandall, A semilinear equation in $L^1(\mathbb{R}^N)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 523-555.  Google Scholar

[4]

L. Boccardo, Some elliptic problems with degenerate coercivity, Adv. Nonlinear Stud., 6 (2006), 1-12.  doi: 10.1515/ans-2006-0101.  Google Scholar

[5]

L. Boccardo, Some cases of weak continuity in nonlinear Dirichlet problems, J. Funct. Anal., 277 (2019), 3673-3687.  doi: 10.1016/j.jfa.2019.05.020.  Google Scholar

[6]

L. Boccardo and H. Brézis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 6 (2003), 521-530.  Google Scholar

[7]

L. Boccardo, G. Croce and L. Orsina, Nonlinear degenerate elliptic problems with $W^{1, 1}_0(\Omega)$ solutions, Manuscripta Math., 137 (2012), 419-439.  doi: 10.1007/s00229-011-0473-6.  Google Scholar

[8]

L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539-551.  doi: 10.1016/S0294-1449(16)30113-5.  Google Scholar

[9]

H. Brézis, Nonlinear elliptic equations involving measures, in Contributions to Nonlinear Partial Differential Equations, Res. Notes Math., 89, Pitman, Boston, MA, 1983, 82–89.  Google Scholar

[10]

G. Cirmi, On the existence of solutions to non-linear degenerate elliptic equations with measures data, Ricerche Mat., 42 (1993), 315-329.   Google Scholar

[11]

G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions for elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 741-808.  Google Scholar

[12]

D. Giachetti and M. Porzio, Exitence results for some nonuniformly elliptic equations with irregular data, J. Math. Anal. Appl., 257 (2001), 100-130.  doi: 10.1006/jmaa.2000.7324.  Google Scholar

[13]

D. Giachetti and M. Porzio, Elliptic equations with degenerate coercivity: Gradient regularity, Acta. Math. Sin. (Engl. Ser.), 19 (2003), 349-370.  doi: 10.1007/s10114-002-0235-1.  Google Scholar

[14]

S. Huang, Quasilinear elliptic equations with exponential nonlinearity and measure data, Math. Methods Appl. Sci., 43 (2020), 2883-2910.  doi: 10.1002/mma.6088.  Google Scholar

[15]

S. Huang, T. Su, X. Du and X. Zhang, Entropy solutions to noncoercive nonlinear elliptic equations with measure data, Electron. J. Differential Equations, 2019 (2019), 1-22.   Google Scholar

[16]

S. Huang and Q. Tian, Marcinkiewicz estimates for solution to fractional elliptic Laplacian equation, Comput. Math. Appl., 78 (2019), 1732-1738.  doi: 10.1016/j.camwa.2019.04.032.  Google Scholar

[17]

S. Huang and Q. Tian, Harnack-type inequality for fractional elliptic equations with critical exponent, Math. Methods Appl. Sci., (2020), 1–18. doi: 10.1002/mma.6280.  Google Scholar

[18]

S. Huang, Q. Tian, J. Wang and J. Mu, Stability for noncoercive elliptic equations, Electron. J. Differential Equations, 2016 (2016), 1-11.   Google Scholar

[19]

H.-F. Huo, Q. Yang and H. Xiang, Dynamics of an edge-based SEIR model for sexually transmitted diseases, Math. Biosci. Eng., 17 (2020), 669-699.  doi: 10.3934/mbe.2020035.  Google Scholar

[20]

J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.  doi: 10.24033/bsmf.1617.  Google Scholar

[21]

X. Li and S. Huang, Stability and bifurcation for a single-species model with delay weak kernel and constant rate harvesting, Complexity, 2019 (2019). doi: 10.1155/2019/1810385.  Google Scholar

[22]

L. Orsina and A. Prignet, Non-existence of solutions for some nonlinear elliptic equations involving measures, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 167-187.  doi: 10.1017/S0308210500000093.  Google Scholar

[23]

L. Orsina and A. Porretta, Strong stability results for nonlinear elliptic equations with respect to very singular perturbation of the data, Commum. Contemp. Math., 3 (2001), 259-285.  doi: 10.1142/S0219199701000378.  Google Scholar

[24]

L. Orsina and A. Prignet, Strong stability results for solutions of elliptic equations with power-like lower order terms and measure data, J. Funct. Anal., 189 (2002), 549-566.  doi: 10.1006/jfan.2001.3846.  Google Scholar

[25]

M. M. Porzio and F. Smarrazzo, Radon measure-valued solutions for some quasilinear degenerate elliptic equations, Ann. Mat. Pura. Appl. (4), 194 (2015), 495-532. doi: 10.1007/s10231-013-0386-y.  Google Scholar

[26]

Q. Tian and Y. Xu, Effect of the domain geometry on the solutions to fractional Brezis-Nirenberg problem, J. Funct. Spaces, 2019 (2019), 4pp. doi: 10.1155/2019/1093804.  Google Scholar

[27]

Y. Ye, H. Liu, Y. Wei, M. Ma and K. Zhang, Dynamic study of a predator-prey model with weak Allee effect and delay, Adv. Math. Phys., 2019 (2019), 15pp. doi: 10.1155/2019/7296461.  Google Scholar

show all references

References:
[1]

A. Alvino, L. Boccardo, V. Ferone, L. Orsina and G. Trombetti, Existence results for nonlinear elliptic equations with degenerate coercivity, Ann. Mat. Pura. Appl. (4), 182 (2003), 53-79. doi: 10.1007/s10231-002-0056-y.  Google Scholar

[2]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vázquez, An L1 theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4), 22 (1995), 241-273.  Google Scholar

[3]

P. Bénilan, H. Brézis and M. Crandall, A semilinear equation in $L^1(\mathbb{R}^N)$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 2 (1975), 523-555.  Google Scholar

[4]

L. Boccardo, Some elliptic problems with degenerate coercivity, Adv. Nonlinear Stud., 6 (2006), 1-12.  doi: 10.1515/ans-2006-0101.  Google Scholar

[5]

L. Boccardo, Some cases of weak continuity in nonlinear Dirichlet problems, J. Funct. Anal., 277 (2019), 3673-3687.  doi: 10.1016/j.jfa.2019.05.020.  Google Scholar

[6]

L. Boccardo and H. Brézis, Some remarks on a class of elliptic equations with degenerate coercivity, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 6 (2003), 521-530.  Google Scholar

[7]

L. Boccardo, G. Croce and L. Orsina, Nonlinear degenerate elliptic problems with $W^{1, 1}_0(\Omega)$ solutions, Manuscripta Math., 137 (2012), 419-439.  doi: 10.1007/s00229-011-0473-6.  Google Scholar

[8]

L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 539-551.  doi: 10.1016/S0294-1449(16)30113-5.  Google Scholar

[9]

H. Brézis, Nonlinear elliptic equations involving measures, in Contributions to Nonlinear Partial Differential Equations, Res. Notes Math., 89, Pitman, Boston, MA, 1983, 82–89.  Google Scholar

[10]

G. Cirmi, On the existence of solutions to non-linear degenerate elliptic equations with measures data, Ricerche Mat., 42 (1993), 315-329.   Google Scholar

[11]

G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions for elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), 741-808.  Google Scholar

[12]

D. Giachetti and M. Porzio, Exitence results for some nonuniformly elliptic equations with irregular data, J. Math. Anal. Appl., 257 (2001), 100-130.  doi: 10.1006/jmaa.2000.7324.  Google Scholar

[13]

D. Giachetti and M. Porzio, Elliptic equations with degenerate coercivity: Gradient regularity, Acta. Math. Sin. (Engl. Ser.), 19 (2003), 349-370.  doi: 10.1007/s10114-002-0235-1.  Google Scholar

[14]

S. Huang, Quasilinear elliptic equations with exponential nonlinearity and measure data, Math. Methods Appl. Sci., 43 (2020), 2883-2910.  doi: 10.1002/mma.6088.  Google Scholar

[15]

S. Huang, T. Su, X. Du and X. Zhang, Entropy solutions to noncoercive nonlinear elliptic equations with measure data, Electron. J. Differential Equations, 2019 (2019), 1-22.   Google Scholar

[16]

S. Huang and Q. Tian, Marcinkiewicz estimates for solution to fractional elliptic Laplacian equation, Comput. Math. Appl., 78 (2019), 1732-1738.  doi: 10.1016/j.camwa.2019.04.032.  Google Scholar

[17]

S. Huang and Q. Tian, Harnack-type inequality for fractional elliptic equations with critical exponent, Math. Methods Appl. Sci., (2020), 1–18. doi: 10.1002/mma.6280.  Google Scholar

[18]

S. Huang, Q. Tian, J. Wang and J. Mu, Stability for noncoercive elliptic equations, Electron. J. Differential Equations, 2016 (2016), 1-11.   Google Scholar

[19]

H.-F. Huo, Q. Yang and H. Xiang, Dynamics of an edge-based SEIR model for sexually transmitted diseases, Math. Biosci. Eng., 17 (2020), 669-699.  doi: 10.3934/mbe.2020035.  Google Scholar

[20]

J. Leray and J. L. Lions, Quelques résultats de Višik sur les problèmes elliptiques semi-linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107.  doi: 10.24033/bsmf.1617.  Google Scholar

[21]

X. Li and S. Huang, Stability and bifurcation for a single-species model with delay weak kernel and constant rate harvesting, Complexity, 2019 (2019). doi: 10.1155/2019/1810385.  Google Scholar

[22]

L. Orsina and A. Prignet, Non-existence of solutions for some nonlinear elliptic equations involving measures, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 167-187.  doi: 10.1017/S0308210500000093.  Google Scholar

[23]

L. Orsina and A. Porretta, Strong stability results for nonlinear elliptic equations with respect to very singular perturbation of the data, Commum. Contemp. Math., 3 (2001), 259-285.  doi: 10.1142/S0219199701000378.  Google Scholar

[24]

L. Orsina and A. Prignet, Strong stability results for solutions of elliptic equations with power-like lower order terms and measure data, J. Funct. Anal., 189 (2002), 549-566.  doi: 10.1006/jfan.2001.3846.  Google Scholar

[25]

M. M. Porzio and F. Smarrazzo, Radon measure-valued solutions for some quasilinear degenerate elliptic equations, Ann. Mat. Pura. Appl. (4), 194 (2015), 495-532. doi: 10.1007/s10231-013-0386-y.  Google Scholar

[26]

Q. Tian and Y. Xu, Effect of the domain geometry on the solutions to fractional Brezis-Nirenberg problem, J. Funct. Spaces, 2019 (2019), 4pp. doi: 10.1155/2019/1093804.  Google Scholar

[27]

Y. Ye, H. Liu, Y. Wei, M. Ma and K. Zhang, Dynamic study of a predator-prey model with weak Allee effect and delay, Adv. Math. Phys., 2019 (2019), 15pp. doi: 10.1155/2019/7296461.  Google Scholar

[1]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, 2021, 15 (1) : 159-183. doi: 10.3934/ipi.2020076
[2]

Ziang Long, Penghang Yin, Jack Xin. Global convergence and geometric characterization of slow to fast weight evolution in neural network training for classifying linearly non-separable data. Inverse Problems & Imaging, 2021, 15 (1) : 41-62. doi: 10.3934/ipi.2020077
[3]

Izumi Takagi, Conghui Zhang. Existence and stability of patterns in a reaction-diffusion-ODE system with hysteresis in non-uniform media. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020400
[4]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272
[5]

Michael Winkler, Christian Stinner. Refined regularity and stabilization properties in a degenerate haptotaxis system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4039-4058. doi: 10.3934/dcds.2020030
[6]

Giuseppina Guatteri, Federica Masiero. Stochastic maximum principle for problems with delay with dependence on the past through general measures. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020048
[7]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[8]

Haodong Yu, Jie Sun. Robust stochastic optimization with convex risk measures: A discretized subgradient scheme. Journal of Industrial & Management Optimization, 2021, 17 (1) : 81-99. doi: 10.3934/jimo.2019100
[9]

Juhua Shi, Feida Jiang. The degenerate Monge-Ampère equations with the Neumann condition. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020297
[10]

João Vitor da Silva, Hernán Vivas. Sharp regularity for degenerate obstacle type problems: A geometric approach. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1359-1385. doi: 10.3934/dcds.2020321
[11]

Paul E. Anderson, Timothy P. Chartier, Amy N. Langville, Kathryn E. Pedings-Behling. The rankability of weighted data from pairwise comparisons. Foundations of Data Science, 2021  doi: 10.3934/fods.2021002
[12]

Azmy S. Ackleh, Nicolas Saintier. Diffusive limit to a selection-mutation equation with small mutation formulated on the space of measures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1469-1497. doi: 10.3934/dcdsb.2020169
[13]

Michiel Bertsch, Danielle Hilhorst, Hirofumi Izuhara, Masayasu Mimura, Tohru Wakasa. A nonlinear parabolic-hyperbolic system for contact inhibition and a degenerate parabolic fisher kpp equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3117-3142. doi: 10.3934/dcds.2019226
[14]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[15]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026
[16]

Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286
[17]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365
[18]

Pierre Baras. A generalization of a criterion for the existence of solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 465-504. doi: 10.3934/dcdss.2020439
[19]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392
[20]

Denis Serre. Non-linear electromagnetism and special relativity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 435-454. doi: 10.3934/dcds.2009.23.435

 Impact Factor: 0.263

Tools

Article outline

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences