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

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

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) $
.
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. BoccardoG. 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. BoccardoT. 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

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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. HuangT. SuX. 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. HuangQ. TianJ. Wang and J. Mu, Stability for noncoercive elliptic equations, Electron. J. Differential Equations, 2016 (2016), 1-11.   Google Scholar

[19]

H.-F. HuoQ. 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. BoccardoG. 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. BoccardoT. 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. HuangT. SuX. 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. HuangQ. TianJ. Wang and J. Mu, Stability for noncoercive elliptic equations, Electron. J. Differential Equations, 2016 (2016), 1-11.   Google Scholar

[19]

H.-F. HuoQ. 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

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