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**precursor**$ \mu (x)$ for the activator $A$ in $\mathbb{R}^1$:

$A_t=$ε^{2}$A^{''}- \mu (x) A+\frac{A^2}{H} \mbox{ in } (-1, 1),$

$\tau H_t=D H^{''}-H+ A^2 \mbox{ in } (-1, 1),$

$ A' (-1)= A' (1)= H' (-1) = H' (1) =0.$

Such an equation
exhibits a typical Turing bifurcation of the
** second kind**, i.e., homogeneous uniform
steady states do not exist in the system.

We establish the existence and stability of
$N-$peaked steady-states in terms of the
precursor $\mu(x)$ and the diffusion
coefficient $D$. It is shown that $\mu (x)$
plays an essential role for both existence and
stability of spiky patterns. In particular, we
show that precursors can ** give rise to
instability**. This is a ** new effect** which
is not present in the homogeneous case.

$\Delta u-u + u^p=0 \ \mbox{in} \ \R^{N-1} \times (0, L),$

$ u>0, \frac{\partial u}{\partial \nu}=0 \ \mbox{on} \ \partial (\R^{N-1} \times (0, L)) $

where $ 1< p\leq \frac{N+2}{N-2}$. When $ 1 < p <\frac{N+2}{N-2}$, it is shown that there exists a unique L _{*} >0 such that for L $\leq $L _{*} , the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for L >L _{*} , the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers L _{*} < L _{**} such that the least energy solution is trivial when L $\leq$L _{*}, the least energy solution is nontrivial when L $\in$(L _{*},L _{**}], and the least energy solution does not exist when L >L _{**}. A connection with Delaunay surfaces in CMC theory is also made.

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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 $p > p_c (\alpha)$.

*non-degenerate*and

*non-minimal*with respect to the curve length. For any given integer $N\ge 2$ and for small $\varepsilon$ away from certain critical numbers, we construct a solution exhibiting $N$ interior layers at mutual distances $O(\varepsilon|\ln\varepsilon|)$ whose center of mass collapse onto $\Gamma$ at speed $O(\varepsilon^{1+\mu})$ for small positive constant $\mu$ as $\varepsilon\to 0$. Asymptotic location of these layers is governed by a Toda system.

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