一类带有退化强制项的奇异椭圆方程解的存在性

Abstract

Abstract: In this article, the authors consider the existence of solutions for the following elliptic boundary value problem with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient of the following form
???20170509???,
where Ω⊂R^N(N ≥ p) is a bounded domain, B,γ,θ>0 and p > 1, and f is a non-negative function belonging to some Lebesgue space L^m(Ω) with m ≥ 1. By combining the truncation methods with several delicate test functions, the existence and regularity of solution is proved. The results show that the lower order term has some regularizing effects on the solution, even if it is singular.

Key words: Degenerate coercivity, Singularity, Lower order term, Existence, Regularity

CLC Number:

O175.2

Li Qingwei, Gao Wenjie, Han Yuzhu. Existence of Solutions to Some Singular Elliptic Problems with Degenerate Coercivity[J].Acta mathematica scientia,Series A, 2017, 37(5): 877-894.

Trendmd

References

[1] Boccardo L, Dall'Aglio A, Orsina L. Existence and regularity results for some elliptic equations with degenerate coercivity. Atti Sem Mat Fis Univ Modena, 1998, 46:51-81
[2] Alvino A, Boccardo L, Ferone V, Orsina L, Trombetti G. Existence results for nonlinear elliptic equations with degenerate coercivity. Annali di Matematica Pura ed Applicata, 2003, 182:53-79
[3] Boccardo L. Quasilinear elliptic equations with natural growth terms:the regularizing effects of lower order terms. J Nonlin Conv Anal, 2006, 7:355-365
[4] Croce G. The regularizing effects of some lower order terms in an elliptic equation with degenerate coercivity. Rendiconti di Matematica, 2007, 27:299-314
[5] Croce G. An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term. Discrete and Continuous Dynamical Systems, Series S, AIMS, 2012, 5:507-530
[6] Porretta A. Uniqueness and homogenization for a class of noncoercive operators in divergence form. Atti Sem Mat Fis Univ Modena, 1998, 46:915-936
[7] Arcoya D, Carmona J, Leonori T, Martínez-Aparicio P J, Orsina L, Petitta F. Existence and nonexistence of solutions for singular quadratic quasilinear equations. J Differential Equations, 2009, 246:4006-4042
[8] Arcoya D, Boccardo L, Leonori T, Porretta A. Some elliptic problems with singular natural growth lower order terms. J Differential Equations, 2010, 249:2771-2795
[9] Boccardo L. Dirichlet problems with singular and gradient quadratic lower order terms. ESAIM:Control, Optimisation and Calculus of Variations, 2008, 14:411-426
[10] Yang H T. Multiplicity and asymptotic behavior of positive solutions for a singular semilinear elliptic problem. J Differential Equations, 2003, 189:487-512
[11] Boccardo L, Gallouët T. Strongly nonlinear elliptic equations having natural growth terms and L¹ data. Nonl Anal, 1992, 19:573-579
[12] Bensoussan A, Boccardo L, Murat F. On a nonlinear partial differential equation having natural growth terms and unbounded solutions. Ann Inst H Poincaré Anal Non Linéaire, 1988, 5:347-364
[13] Boccardo L, Murat F, Puel J P. L^∞ estimate for some nonlinear elliptic partial differential equation and application to an existence result. SIAM J Math Anal, 1992, 23:326-333
[14] Zheng S Z, Zheng X L, Feng Z S. Regularity for a class of degenerate elliptic equations with discontinous coefficients under natural growth. J Math Anal Appl, 2008, 346:359-373
[15] Leoni F, Pellacci B. Local estimates and global existence for strongly nonlinear parabolic equations with locally integrable data. J Evol Equa, 2006, 6:113-144

Existence of Solutions to Some Singular Elliptic Problems with Degenerate Coercivity

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Wang Jun, Wang Tianlu, Wen Yanhua, Zhou Xianfeng. Global Solutions of IVP for N-Dimensional Nonlinear Fractional Differential Equations with Delay [J]. Acta mathematica scientia,Series A, 2018, 38(5): 903-910.
[2]	Yao Shaowen, Cheng Zhibo. Positive Periodic Solution for Damped Duffing Equation with Singularity [J]. Acta mathematica scientia,Series A, 2018, 38(3): 543-548.
[3]	Zhao Jihong. Logarithmical Regularity Criteria in Terms of Pressure for the Three Dimensional Dissipative System Modeling Electro-Hydrodynamics [J]. Acta mathematica scientia,Series A, 2018, 38(3): 549-564.
[4]	Xing Chao, Ren Mengzhang, Luo Hong. The Existence of Global Weak Solutions to Thermohaline Circulation Equations [J]. Acta mathematica scientia,Series A, 2018, 38(2): 284-290.
[5]	Zhang Shuqin, Hu Lei. Initial Value Problem for Nonlinear Fractional Differential Equations with Variable-order Derivative and a Parameter [J]. Acta mathematica scientia,Series A, 2018, 38(2): 291-303.
[6]	Zhu Xincai. Existence of Constrained Minimizers for Schrödinger-Poisson Equations in R^N [J]. Acta mathematica scientia,Series A, 2018, 38(1): 61-70.
[7]	Su Xiao, Wang Shubin. Finite Time Blow-Up for the Damped Semilinear Wave Equations with Arbitrary Positive Initial Energy [J]. Acta mathematica scientia,Series A, 2017, 37(6): 1085-1093.
[8]	Li Qin, Yang Zuodong. Infinitely Many High Energy Solutions of p-Kirchhoff-Type System with Sign-Changing Weight [J]. Acta mathematica scientia,Series A, 2017, 37(5): 869-876.
[9]	Gao Hongya, Jia Miaomiao. Local Regularity and Local Boundedness for Solutions to Obstacle Problems [J]. Acta mathematica scientia,Series A, 2017, 37(4): 706-713.
[10]	Huang Decheng, Chen Shouxin. New Proof for the Existence of Minimizing Energy Solutions for the Ginzburg-Landau Equations [J]. Acta mathematica scientia,Series A, 2017, 37(2): 299-306.
[11]	Ba Na, Zheng Lie. Partial Schauder Estimates for the Heat Equation [J]. Acta mathematica scientia,Series A, 2017, 37(2): 307-312.
[12]	Qiu Hua, Zhang Yingshu, Fang Shaomei. The Global Existence Result of a Leray-α-Oldroyd Model to the Incompressible Viscoelastic Flow [J]. Acta mathematica scientia,Series A, 2017, 37(1): 102-112.
[13]	Yuan Hailong, Li Yanling. Existence and Stability of Coexistence States for a Lotka-Volterra Competition Model [J]. Acta mathematica scientia,Series A, 2017, 37(1): 173-184.
[14]	Zhang Hongwei, Liu Gongwei, Hu Qingying. Initial Boundary Value Problem for a Class Wave Equation with Logarithmic Source Term [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1124-1136.
[15]	Liu Yang. Potential Well and Application to Non-Newtonian Filtration Equations at Critical Initial Energy Level [J]. Acta mathematica scientia,Series A, 2016, 36(6): 1211-1220.