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$-Δ u=λ e^u$ |

$\Delta u=\lambda [\frac{1}{u^p}-\frac{1}{u^q}]$ in $B$, $u=\kappa \in (0,(\frac{p-1}{q-1})^{-1/(p-q)} ]$ on $\partial B$, $0 < u < \kappa$

in $B$, where $p > q > 1$ and $B$ is the unit ball in $\mathbb R^N$ ($N \geq 2$). We show that there exists $\lambda_\star>0$ such that for $0<\lambda <\lambda_\star$, the maximal solution is the only positive radial solution. Furthermore, if $2 \leq N < 2+\frac{4}{p+1} (p+\sqrt{p^2+p})$, the branch of positive radial solutions must undergo infinitely many turning points as the maxima of the radial solutions on the branch go to 0. The key ingredient is the use of a monotonicity formula.

$\Delta u=\lambda |x|^\alpha f(u)$ 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 $p > p_c (\alpha)$.

$-\Delta u=\frac{\lambda}{(1-u)^2}$ in $\Omega$, $u=0$ on $\partial \Omega$, $0 < u < 1$ in $\Omega$

where $\Omega$ is a rather symmetric domain in $\mathbb R^2$. We prove that there exists a $\lambda_\star>0$ such that for $\lambda \in (0, \lambda_\star)$ the minimal solution is unique. Then we analyze the asymptotic behavior of touch-down solutions, i.e., solutions with max$_\Omega u_i (0) \to 1$. We show that after a rescaling, the solution will be asymptotically symmetric. As a consequence, we show that the branch of positive solutions must undergo infinitely many bifurcations as the maximums of the solutions on the branch go to 1 (possibly only changes of direction). This gives a positive answer to some open problems in [12]. Our result is new even in the radially symmetric case. Central to our analysis is the monotonicity formula, one-dimensional Sobloev inequality, and classification of solutions to a supercritical problem

$ \Delta U=\frac{1}{U^2}\quad$ in $\mathbb R^2, U(0)=1, U(z) \geq 1.$

By a new type of comparison principle for a fourth order elliptic problem in general domains, we investigate the structure of positive solutions to Navier boundary value problems of a perturbed fourth order elliptic equation with negative exponent, which arises in the study of the deflection of charged plates in electrostatic actuators in the modeling of electrostatic micro-electromechanical systems (MEMS). It is seen that the structure of solutions relies on the boundary values. The global branches of solutions to the Navier boundary value problems are established. We also show that the behaviors of these branches are relatively "stable" with respect to the Navier boundary values.

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

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

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