Han, Pigong Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential. (English) Zbl 1173.35061 Proc. Am. Math. Soc. 135, No. 2, 365-372 (2007). Summary: Let \(\Omega\) be an open bounded domain in \(\mathbb{R}^N\) \((N\geq 3)\) with smooth boundary \(\partial\Omega\), \(0\in\Omega\). We are concerned with the asymptotic behavior of solutions for the elliptic problem: \[ -\Delta u-\frac{\mu u}{| x|^2}=f(x,u),\qquad u\in H^1_0(\Omega),\tag{1} \] where \( 0\leq\mu<\big(\frac{N-2}{2}\big)^2\) and \(f(x,u)\) satisfies suitable growth conditions. By Moser iteration, we characterize the asymptotic behavior of nontrivial solutions for problem (1). In particular, we point out that the proof of Proposition 2.1 in J.-Q. Chen, Proc. Am. Math. Soc. 132, No. 11, 3225–3229 (2004; Zbl 1096.35042), is wrong. Cited in 21 Documents MSC: 35J65 Nonlinear boundary value problems for linear elliptic equations 35B45 A priori estimates in context of PDEs 35J25 Boundary value problems for second-order elliptic equations 35J70 Degenerate elliptic equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces Citations:Zbl 1096.35042 PDFBibTeX XMLCite \textit{P. Han}, Proc. Am. Math. Soc. 135, No. 2, 365--372 (2007; Zbl 1173.35061) Full Text: DOI References: [1] J. P. García Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998), no. 2, 441 – 476. · Zbl 0918.35052 · doi:10.1006/jdeq.1997.3375 [2] Jianqing Chen, Exact local behavior of positive solutions for a semilinear elliptic equation with Hardy term, Proc. Amer. Math. Soc. 132 (2004), no. 11, 3225 – 3229. · Zbl 1096.35042 [3] Kai Seng Chou and Chiu Wing Chu, On the best constant for a weighted Sobolev-Hardy inequality, J. London Math. Soc. (2) 48 (1993), no. 1, 137 – 151. · Zbl 0739.26013 · doi:10.1112/jlms/s2-48.1.137 [4] Daomin Cao and Pigong Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations 205 (2004), no. 2, 521 – 537. · Zbl 1154.35346 · doi:10.1016/j.jde.2004.03.005 [5] L. Caffarelli, R. Kohn, and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math. 53 (1984), no. 3, 259 – 275. · Zbl 0563.46024 [6] Henrik Egnell, Elliptic boundary value problems with singular coefficients and critical nonlinearities, Indiana Univ. Math. J. 38 (1989), no. 2, 235 – 251. · Zbl 0666.35072 · doi:10.1512/iumj.1989.38.38012 [7] Ivar Ekeland and Nassif Ghoussoub, Selected new aspects of the calculus of variations in the large, Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 2, 207 – 265. · Zbl 1064.35054 [8] Alberto Ferrero and Filippo Gazzola, Existence of solutions for singular critical growth semilinear elliptic equations, J. Differential Equations 177 (2001), no. 2, 494 – 522. · Zbl 0997.35017 · doi:10.1006/jdeq.2000.3999 [9] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001 [10] N. Ghoussoub and C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5703 – 5743. · Zbl 0956.35056 [11] Enrico Jannelli, The role played by space dimension in elliptic critical problems, J. Differential Equations 156 (1999), no. 2, 407 – 426. · Zbl 0938.35058 · doi:10.1006/jdeq.1998.3589 [12] D. Ruiz and M. Willem, Elliptic problems with critical exponents and Hardy potentials, J. Differential Equations 190 (2003), no. 2, 524 – 538. · Zbl 1163.35383 · doi:10.1016/S0022-0396(02)00178-X [13] Didier Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities, Trans. Amer. Math. Soc. 357 (2005), no. 7, 2909 – 2938. · Zbl 1134.35348 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.