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