Infinitely Many Solutions for a Nonlinear Elliptic PDE with Multiple Hardy–Sobolev Critical Exponents
Differential Equations and Dynamical Systems - Trang 1-21 - 2023
Tóm tắt
In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with multiple Hardy–Sobolev critical exponents
$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u=\mu \vert u \vert ^{2^{*}-2} u + \sum _{i=1}^{l} \frac{ \vert u \vert ^{2^{*}(s_{i})-2}u}{ \vert x \vert ^{s_{i}}}+ a(x) \vert u \vert ^{q-2} u &{} \; in \; \Omega , \\ u=0 &{} \; on \; \partial \Omega , \end{array}\right. \end{aligned}$$
where
$$\Omega $$
is a smooth bounded domain in
$${\mathbb {R}}^{N}$$
with
$$0\in \partial \Omega $$
and all the principle curvatures of
$$ \partial \Omega $$
at 0 are negative,
$$a \in {\mathcal {C}}^{1}({\bar{\Omega }}, \mathbb {R^{*}}^{+}),$$
$$ \mu > 0,$$
$$0 2\frac{q+1}{q -1}.$$
By
$$2^{*}:=\frac{2 N}{N-2}$$
and
$$2^{*}(s_{i}):=\frac{2 (N-s_{i})}{N-2}$$
we denote the critical Sobolev exponent and Hardy–Sobolev exponents, respectively.
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