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September  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

Yaoping Chen 1,2, and  Jianqing Chen 2,

1. 

Concord University College, Fujian Normal University, Fuzhou, 350117, 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.
Keywords: Elliptic problems, Hardy term, multiple positive solutions, extremal values, uniform estimates.
Mathematics Subject Classification: Primary: 35J20, 35J60.
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 and 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.  Google Scholar

[2]

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

[3]

C. O. Alves and 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.  Google Scholar

[4]

J. García-Azorero, I. Peral and 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.  Google Scholar

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

[6]

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

[7]

K. J. Brown and 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.  Google Scholar

[8]

F. Gazzola and 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.  Google Scholar

[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.  Google Scholar

[10]

Y. Sun and 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.  Google Scholar

[11]

Y. Sun and 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.  Google Scholar

[12]

J. L. Vázquez and 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.  Google Scholar

show all references

References:
[1]

A. Ambrosetti, H. Brézis and 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.  Google Scholar

[2]

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

[3]

C. O. Alves and 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.  Google Scholar

[4]

J. García-Azorero, I. Peral and 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.  Google Scholar

[5]

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

[6]

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

[7]

K. J. Brown and 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.  Google Scholar

[8]

F. Gazzola and 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.  Google Scholar

[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.  Google Scholar

[10]

Y. Sun and 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.  Google Scholar

[11]

Y. Sun and 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.  Google Scholar

[12]

J. L. Vázquez and 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.  Google Scholar

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