American Institute of Mathematical Sciences

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

 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.
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:
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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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