Sign-changing bubbling solutions for an exponential nonlinearity in R²

Quite differently from those perturbation techniques in {DM}, we use the assumption of a C¹-stable critical point to construct positive or sign-changing bubbling solutions to the boundary value problem -Δ u=λ u|u|ᵖ⁻²e|u|ᵖ with homogeneous Dirichlet boundary condition in a bounded, smooth planar domain Ω, when $0<p<2$ and λ>0 is a small parameter. 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 λ small enough, we prove that when Ω 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 Ω 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.