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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 |
2. | College of Mathematics and Computer Science, Fujian Normal University, Fuzhou, 350108, China |
$-Δ u - {{λ}\over{|x|^2}}u = h(x) u^q + μ W(x) u^p,\ \ x∈Ω\backslash\{0\}$ |
$0∈ Ω\subset\mathbb{R}^N $ |
$N≥q 3 $ |
$\partial Ω $ |
$μ>0 $ |
$0 < λ < Λ={{(N-2)^2}\over{4}}$, $0 < q < 1 < p < 2^*-1 $ |
$h(x)>0 $ |
$W(x) $ |
$\{x∈ Ω: W(x)>0\} $ |
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. |
[2] |
J. P. Aubin and I. Ekeland,
Applied Nonlinear Analysis, Pure and Applied Mathematics, Wiley Interscience Publications, 1984. |
[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. |
[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. |
[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. |
[6] |
H. Brézis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[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. |
[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. |
[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 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. |
[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. |
[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. |
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. |
[2] |
J. P. Aubin and I. Ekeland,
Applied Nonlinear Analysis, Pure and Applied Mathematics, Wiley Interscience Publications, 1984. |
[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. |
[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. |
[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. |
[6] |
H. Brézis and J. L. Vázquez,
Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid, 10 (1997), 443-469.
|
[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. |
[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. |
[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 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. |
[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. |
[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. |
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