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

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

References:

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[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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