# American Institute of Mathematical Sciences

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 , $ q>1 $, $ 0\leq\theta<1 $, $ \lambda $is a measure which is concentrated on a set with zero $ r $capacity $ (p
.
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:
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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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##### 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
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