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Sign-changing concentration phenomena of an anisotropic sinh-Poisson type equation with a Hardy or H\'{e}non term
We consider the following anisotropic sinh-Poisson tpye equation with a Hardy
or H\'{e}non term:
{\begin{array}{ll} -\Div (a(x)\nabla u)+
a(x)u=\varepsilon^2a(x)|x-q|^{2\alpha}(e^u-e^{-u}) &\mbox{in $\Omega$,} \\
\frac{\partial u}{\partial n}=0, &\mbox{on $\Omega$,} \end{array} where
, , , is a smooth bounded domain, is the unit outward
normal vector of and is a smooth positive function
defined on . From finite dimensional reduction method, we proved
that the problem \eqref{115} has a sequence of sign-changing solutions with
arbitrarily many interior spikes accumulating to , provided is
a local maximizer of . However, if is a strict
local maximum point of and satisfies ,
we proved that \eqref{115} has a family of sign-changing solutions with
arbitrarily many mixed interior and boundary spikes accumulating to .
Under the same condition, we could also construct a sequence of blow-up
solutions for the following problem {\begin{array}{ll} -\Div (a(x)\nabla u)+
a(x)u=\varepsilon^2a(x)|x-q|^{2\alpha}e^u &\mbox{in ,} \\
\frac{\partial u}{\partial n}=0, &\mbox{on .} \end{array}$
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