New solutions for the Lane-Emden problem in planar domains

We consider the Lane-Emden problem −Δu=|u|p−1uinΩ,u=0on∂Ω, where Ω⊂R2 is a smooth bounded domain. When the exponent p 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 p is sufficiently large. In this paper, we focus on this topic and find new sign-changing solutions that exhibit an unexpected concentration phenomenon as p approaches +∞.