  We consider the Lane-Emden problem on planar domains. When the exponent is
large, the existence and multiplicity of solutions strongly depend on the
geometric properties of the domain, which also deeply affect their qualitative
behavior. Remarkably, a wide variety of solutions, both positive and
sign-changing, have been found when the exponent is sufficiently large. In this
paper, we focus on this topic and fine new sign-changing solutions that exhibit
an unexpected concentration phenomenon as the exponent approaches infinity.
