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$\Delta u=\lambda |x|^\alpha f(u)$ in $\Omega, u=1$ on $\partial \Omega\qquad$ (1)

where $\lambda>0$, $\alpha \geq 0$, $\Omega$ is a bounded smooth domain in $R^N$ ($N \geq 2$) containing $0$ and $f$ is a $C^1$ function satisfying $\lim_{s \to 0^+} s^p f(s)=1$. We show that for each $\alpha \geq 0$, there is a critical power $p_c (\alpha)>0$, which is decreasing in $\alpha$, such that the branch of positive solutions possesses infinitely many bifurcation points provided $p > p_c (\alpha)$ or $p > p_c (0)$, and this relies on the shape of the domain $\Omega$. We get some important estimates of the Morse index of the regular and singular solutions. Moreover, we also study the radial solution branch of the related problems in the unit ball. We find that the branch possesses infinitely many turning points provided that $p>p_c (\alpha)$ and the Morse index of any radial solution (regular or singular) in this branch is finite provided that $0 < p \leq p_c (\alpha)$. This implies that the structure of the radial solution branch of (1) changes for $0 < p \leq p_c (\alpha)$ and $p > p_c (\alpha)$.

$- \Delta u + \frac{\mathit{\mu }}{{|x{|^2}}}u = {u^p}\;\;\;{\rm{in }}\;\;\;\Omega \setminus \{ 0\} ,\;\;\;u = 0\;\;\;{\rm{on}}\;\;\;\partial \Omega .\;\;\;\;\;\;\;\;\;\;\;\;\left( 1 \right)$ |

$-\epsilon^2\Delta u= u (u-a(x))(1-u) \mbox{ in } \Omega, u|_{\partial\Omega}=\infty$

has solutions with sharp interior layers and spikes, apart from boundary layers. We also determine the location of these layers and spikes.

$\Delta u+
|u|^{p-1}u+$ε^{1/2} f = 0 in Ω

u=ε^{1/2} g on $\partial$Ω

in a bounded smooth domain $\Omega \subset \R^N$ $(N\geq 3)$, when the exponent $p$ is supercritical and close enough to $\frac{N+2}{N-2}$. As $p\rightarrow \frac{N+2}{N-2}$, the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves Green's function. As applications, we will give some existence results, in particular, when $\O$ are symmetric domains perforated with the small hole and when $f=0$ and $g=0$.

We consider the asymptotic dynamics of a damped wave equations on a time-dependent domains with homogeneous Dirichlet boundary condition, the nonlinearity is allowed to have a cubic growth rate which is referred to as the critical exponent. To this end, we establish the existence and uniqueness of strong and weak solutions satisfying energy inequality under the assumption that the spatial domains $\mathcal{O}_{t}$ in $\mathbb{R}^{3}$ are obtained from a bounded base domain $\mathcal{O}$ by a $C^{3}$-diffeomorphism $r(·, t)$. Furthermore, we establish the pullback attractor under a slightly weaker assumption that the measure of the spatial domains are uniformly bounded above.

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