Two Disjoint and Infinite Sets of Solutions for an Elliptic Equation Involving Critical Hardy-Sobolev Exponents
Tóm tắt
In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents
$$\left\{ {\matrix{{ - \Delta u = \mu |u{|^{{2^ * } - 2}}u + {{|u{|^{{2^ * }(s) - 2}}u} \over {|x{|^s}}} + a(x)|u{|^{q - 2}}u} \hfill & {{\rm{in}}\,\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm{on}}\,\,\partial \Omega,} \hfill \cr } } \right.$$
where Ω is a smooth bounded domain in ℝN with 0 ∈ ∂Ω and all the principle curvatures of ∂Ω at 0 are negative,
$$a \in {{\cal C}^1}(\bar \Omega,{\mathbb{R}^{ * + }})$$
, μ > 0, 0 < s < 2, 1< q < 2 and
$$N > 2{{q + 1} \over {q - 1}}$$
. By
$${2^ * }: = {{2N} \over {N - 2}}$$
and
$${2^ * }(s): = {{2(N - s)} \over {N - 2}}$$
we denote the critical Sobolev exponent and Hardy-Sobolev exponent, respectively.
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