On the existence and profile of nodal solutions for a two-dimensional elliptic problem with large exponent in nonlinearity

We study the existence of nodal solutions to the boundary value problem $-\Delta u=|u|^{p-1 } u$ in a bounded, smooth domain $\Omega$ in $\mathbb{R}^2,$ with homogeneous Dirichlet boundary condition, when $p$ is a large exponent. We prove that, for $p$ large enough, there exist at least two pairs of solutions which change sign exactly once and whose nodal lines intersect the boundary of $\Omega.$