2017, 16(5): 1531-1552. doi: 10.3934/cpaa.2017073

Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential

1. 

Concord University College, Fujian Normal University Fuzhou, 350117, China College of Mathematics and Computer Science Fujian Normal University, Fuzhou, 350108, China

2. 

College of Mathematics and Computer Science Fujian Normal University, Fuzhou, 350108, China

Received  October 2014 Revised  February 2017 Published  May 2017

Fund Project: This work is supported by NSF of China (No. 11371091) and the innovation group of `Nonlinear analysis and its applications' (No. 021337120)

In this paper, variational methods are used to establish some existence and multiplicity results and provide uniform estimates of extremal values for a class of elliptic equations of the form:
$-Δ u - {{λ}\over{|x|^2}}u = h(x) u^q + μ W(x) u^p,\ \ x∈Ω\backslash\{0\}$
with Dirichlet boundary conditions, where
$0∈ Ω\subset\mathbb{R}^N $
(
$N≥q 3 $
) be a bounded domain with smooth boundary
$\partial Ω $
,
$μ>0 $
is a parameter,
$0 < λ < Λ={{(N-2)^2}\over{4}}$, $0 < q < 1 < p < 2^*-1 $
,
$h(x)>0 $
and
$W(x) $
is a given function with the set
$\{x∈ Ω: W(x)>0\} $
of positive measure.
Citation: Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073
References:
[1]

A. Ambrosetti, H. Brézis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[2]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure and Applied Mathematics, Wiley Interscience Publications, 1984.

[3]

C. O. Alves, A. El Hamidi, Nehari manifold and existence of positive solutions to a class of quasilinear problems, Nonlinear Anal., 60 (2005), 611-624. doi: 10.1016/j.na.2004.09.039.

[4]

J. García-Azorero, I. Peral, A. Primo, A borderline case in elliptic problems involving weights of Caffarelli-Kohn-Nirenberg type, Nonlinear Anal., 67 (2007), 1878-1894. doi: 10.1016/j.na.2006.07.046.

[5]

P. Baras, J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139. doi: 10.2307/1999277.

[6]

H. Brézis, J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.

[7]

K. J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.

[8]

F. Gazzola, A. Malchiodi, Some remarks on the equation $ -Δ u = λ(1+u)^p$ for varying $λ, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845. doi: 10.1081/PDE-120002875.

[9]

Y. Sun, Estimates for extremal values of $ -Δ u = h(x)u^q +λ W(x)u^p$, Commun. Pure Appl. Anal., 9 (2010), 751-760. doi: 10.3934/cpaa.2010.9.751.

[10]

Y. Sun, S. Li, A nonlinear elliptic equation with critical exponent: Estimates for extremal values, Nonlinear Anal., 69 (2008), 1856-1869. doi: 10.1016/j.na.2007.07.030.

[11]

Y. Sun, S. Li, Some remarks on a superlinear-singular problem: Estimates of $λ ^* $, Nonlinear Anal., 69 (2008), 2636-2650. doi: 10.1016/j.na.2007.08.037.

[12]

J. L. Vázquez, E. Zuazua, The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

show all references

References:
[1]

A. Ambrosetti, H. Brézis, G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543. doi: 10.1006/jfan.1994.1078.

[2]

J. P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure and Applied Mathematics, Wiley Interscience Publications, 1984.

[3]

C. O. Alves, A. El Hamidi, Nehari manifold and existence of positive solutions to a class of quasilinear problems, Nonlinear Anal., 60 (2005), 611-624. doi: 10.1016/j.na.2004.09.039.

[4]

J. García-Azorero, I. Peral, A. Primo, A borderline case in elliptic problems involving weights of Caffarelli-Kohn-Nirenberg type, Nonlinear Anal., 67 (2007), 1878-1894. doi: 10.1016/j.na.2006.07.046.

[5]

P. Baras, J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139. doi: 10.2307/1999277.

[6]

H. Brézis, J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.

[7]

K. J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499. doi: 10.1016/S0022-0396(03)00121-9.

[8]

F. Gazzola, A. Malchiodi, Some remarks on the equation $ -Δ u = λ(1+u)^p$ for varying $λ, p$ and varying domains, Comm. Partial Differential Equations, 27 (2002), 809-845. doi: 10.1081/PDE-120002875.

[9]

Y. Sun, Estimates for extremal values of $ -Δ u = h(x)u^q +λ W(x)u^p$, Commun. Pure Appl. Anal., 9 (2010), 751-760. doi: 10.3934/cpaa.2010.9.751.

[10]

Y. Sun, S. Li, A nonlinear elliptic equation with critical exponent: Estimates for extremal values, Nonlinear Anal., 69 (2008), 1856-1869. doi: 10.1016/j.na.2007.07.030.

[11]

Y. Sun, S. Li, Some remarks on a superlinear-singular problem: Estimates of $λ ^* $, Nonlinear Anal., 69 (2008), 2636-2650. doi: 10.1016/j.na.2007.08.037.

[12]

J. L. Vázquez, E. Zuazua, The Hardy inequality and the asymptotic behavior of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153. doi: 10.1006/jfan.1999.3556.

[1]

Soohyun Bae. Classification of positive solutions of semilinear elliptic equations with Hardy term. Conference Publications, 2013, 2013 (special) : 31-39. doi: 10.3934/proc.2013.2013.31

[2]

Yinbin Deng, Qi Gao. Asymptotic behavior of the positive solutions for an elliptic equation with Hardy term. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 367-380. doi: 10.3934/dcds.2009.24.367

[3]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033

[4]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527

[5]

Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145

[6]

Alfonso Castro, Rosa Pardo. A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 783-790. doi: 10.3934/dcdsb.2017038

[7]

Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Multiple solutions for nonlinear elliptic equations with an asymmetric reaction term. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2469-2494. doi: 10.3934/dcds.2013.33.2469

[8]

Xing Liu, Yijing Sun. Multiple positive solutions for Kirchhoff type problems with singularity. Communications on Pure & Applied Analysis, 2013, 12 (2) : 721-733. doi: 10.3934/cpaa.2013.12.721

[9]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83

[10]

Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363

[11]

Lucio Boccardo, Luigi Orsina, Ireneo Peral. A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 513-523. doi: 10.3934/dcds.2006.16.513

[12]

Monica Musso, Donato Passaseo. Multiple solutions of Neumann elliptic problems with critical nonlinearity . Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 301-320. doi: 10.3934/dcds.1999.5.301

[13]

Jinggang Tan. Positive solutions for non local elliptic problems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (2) : 837-859. doi: 10.3934/dcds.2013.33.837

[14]

Yijing Sun. Estimates for extremal values of $-\Delta u= h(x) u^{q}+\lambda W(x) u^{p}$. Communications on Pure & Applied Analysis, 2010, 9 (3) : 751-760. doi: 10.3934/cpaa.2010.9.751

[15]

J. R. L. Webb. Multiple positive solutions of some nonlinear heat flow problems. Conference Publications, 2005, 2005 (Special) : 895-903. doi: 10.3934/proc.2005.2005.895

[16]

Asadollah Aghajani. Regularity of extremal solutions of semilinear elliptic problems with non-convex nonlinearities on general domains. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3521-3530. doi: 10.3934/dcds.2017150

[17]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439

[18]

Rong Xiao, Yuying Zhou. Multiple solutions for a class of semilinear elliptic variational inclusion problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 991-1002. doi: 10.3934/jimo.2011.7.991

[19]

Shujie Li, Zhitao Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity . Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 489-493. doi: 10.3934/dcds.1999.5.489

[20]

Qingfang Wang. Multiple positive solutions of fractional elliptic equations involving concave and convex nonlinearities in $R^N$. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1671-1688. doi: 10.3934/cpaa.2016008

2016 Impact Factor: 0.801

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]