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 - u= u|u|^p-2e^|u|ᵖ with homogeneous Dirichlet boundary condition in a bounded, smooth planar domain, when 00 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 small enough, we prove that when 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 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.