  We establish a series of concentration and oscillation estimates for elliptic
equations with exponential nonlinearity $e^{u^p}$ in a disc. Especially, we
show various new results on the supercritical case $p>2$ which are left open in
the previous works. We begin with the concentration analysis of blow-up
solutions by extending the scaling and pointwise techniques developed in the
previous studies. A striking result is that we detect an infinite sequence of
bubbles in the supercritical case $p>2$. The precise characterization of the
limit profile, energy, and location of each bubble is given. Moreover, we
arrive at a natural interpretation, the infinite sequence of bubbles causes the
infinite oscillation of the solutions. Based on this idea and our concentration
estimates, we next carry out the oscillation analysis. The results allow us to
estimate intersection points and numbers between blow-up solutions and singular
functions. Applying this, we finally demonstrate the infinite oscillations of
the bifurcation diagrams of supercritical equations. In addition, we also
discuss what happens on the sequences of bubbles in the limit cases $p\to 2^+$
and $p\to \infty$ respectively. As above, the present work discovers a direct
path connecting the concentration and oscillation analyses. It leads to a
consistent and straightforward understanding of concentration, oscillation, and
bifurcation phenomena on blow-up solutions of supercritical problems.
