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Asymptotic behavior of solutions to semilinear elliptic equations with Hardy potential. (English) Zbl 1173.35061

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.

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
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References:

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