Local bifurcation from the second eigenvalue of the Laplacian in a square

In this work we study local bifurcation from the branch of trivial solutions for a class of semilinear elliptic equations, at the second eigenvalue $\lambda_2$ of a square. We find that the bifurcation set can be locally described as the union of exactly four bifurcation branches of nontrivial solutions which cross the bifurcation point $(\lambda_2,0)$. We also compute the Morse index of the solutions in the four branches.