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### Open Access Journals

CPAA

In this paper, we consider the problem $(Q_\varepsilon)$ : $\Delta ^2 u=
u^9 +\varepsilon f(x)$ in $\Omega$, $u=\Delta u=0$ on $\partial\Omega$,
where $\Omega$ is a bounded and smooth domain in $R^5$, $\varepsilon$ is a
small positive parameter, and $f$ is a smooth function. Our main
purpose is to characterize the solutions with some assumptions on
the energy. We prove that these solutions blow up at a critical
point of a function depending on $f$ and the regular part of the
Green's function. Moreover, we construct families of solutions of
$(Q_\varepsilon)$ which satisfy the conclusions of the first part.

CPAA

In this paper we study the nonlinear elliptic problem involving
nearly critical exponent $ (P_\varepsilon ): -\Delta u= $ $ |u|^{4/(n-2)+\varepsilon}u$ in $\Omega$, $u = 0$
on $\partial \Omega $, where $\Omega $ is a smooth bounded
domain in $\mathbb R^n $, $n \geq 3 $ and $\varepsilon$ is a
positive real parameter. We show that, for $\varepsilon$ small,
$(P_\varepsilon) $ has no sign-changing solutions with low energy
which blow up at two points. Moreover, we prove that there is no
sign-changing solutions which blow up at three points. We also
show that $(P_\varepsilon)$ has no bubble-tower sign-changing
solutions.

DCDS

We consider the existence and multiplicity of riemannian metrics of
prescribed mean curvature and zero boundary mean curvature on the
three dimensional half sphere $(S^3_+,g_c)$ endowed with its
standard metric $g_c$. Due to Kazdan-Warner type obstructions,
conditions on the function to be realized as a scalar curvature have
to be given. Moreover the existence of

*critical point at infinity*for the associated Euler Lagrange functional makes the existence results harder to be proved. However it turns out that such noncompact orbits of the gradient can be treated as a usual critical point once a*Morse Lemma at infinity*is performed. In particular their topological contribution to the level sets of the functional can be computed. In this paper we prove that, under generic conditions on $K$, this*topology at infinity*is a lower bound for the number of metrics in the conformal class of $g_c$ having prescribed scalar curvature and zero boundary mean curvature.
CPAA

In this paper we study the problem $(P_{\varepsilon}):
\Delta^2 u_\varepsilon = u_\varepsilon^{\frac{n+4}{n-4}}$, $ u_\varepsilon >0$ in $A_\varepsilon$;
$u_\varepsilon= \Delta u_\varepsilon= 0$ on $ \partial
A_\varepsilon $, where $\{A_\varepsilon \subset \mathbb R^n, \varepsilon >0 \}$ is a
family of bounded annulus shaped domains such that $A_\varepsilon$ becomes
"thin" as $\varepsilon\to 0$. Our main result is the following: Assume
$n\geq 6$ and let $C>0$ be a constant. Then there exists
$\varepsilon_0>0$ such that for any $\varepsilon <\varepsilon_0$,
the problem
$ (P_{\varepsilon})$ has no solution $u_\varepsilon$, whose energy,
$\int_{A_\varepsilon}|\Delta u_\varepsilon |^2$ is less than $C$.
Our proof involves a rather delicate analysis of asymptotic profiles
of solutions $u_\varepsilon$ when $\varepsilon\to 0$.

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