Sign-changing bubbling solutions for an exponential nonlinearity in $\mathbb{R}^2$

Very differently from those perturbative techniques of Deng-Musso in [26], we use the assumption of a $C^1$-stable critical point to construct positive or sign-changing solutions with arbitrary $m$ isolated bubbles to the boundary value problem $-\Delta u=\lambda u|u|^{p-2}e^{|u|^p}$ under homogeneous Dirichlet boundary condition in a bounded, smooth planar domain $\Omega$, when $0<p<2$ and $\lambda>0$ is a small but free parameter. We build a vanishing identity of first order and an identity of second order to prove that for any $0<p<1$ the delicate energy expansion of these bubbling solutions always converges to $4\pi m$ from below, but for any $1<p<2$ the energy always converges to $4\pi m$ from above, where the latter case sharply recurs a result of De Marchis-Malchiodi-Martinazzi-Thizy in [32] involving concentration and compactness properties at any critical energy level $4\pi m$ only for positive bubbling solutions. A sufficient condition on the intersection between the nodal line of these sign-changing solutions and the boundary of the domain is founded. Moreover, for $\lambda$ small enough, we prove that when $\Omega$ is an arbitrary bounded domain, this problem has not only at least two pairs of bubbling solutions which change sign exactly once and whose nodal lines intersect the boundary, but also a bubbling solution which changes sign exactly twice or three times; when $\Omega$ has an axial symmetry, this problem has a bubbling solution which alternately changes sign arbitrarily many times along the axis of symmetry through the domain.