Concentration and oscillation analysis of positive solutions to semilinear elliptic equations with exponential growth in a disc

Abstract

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.