Quasi-radial solutions for the Lane–Emden problem in the ball

Abstract

We consider the semilinear elliptic problem where B is the unit ball of $${\mathbb {R}}^2$$ centered at the origin and $$p\in (1,+\infty )$$. We prove the existence of sign-changing solutions to ($${{\mathcal {E}}}_p$$) having 2 nodal domains, whose nodal line does not touch $$\partial B$$ and which are non-radial. We call these solutions quasi-radial. The result is obtained for any p sufficiently large, considering least energy nodal solutions in spaces of functions invariant under suitable dihedral groups of symmetry and proving that they fulfill the required qualitative properties. We also show that these symmetric least energy solutions are instead radial for p close enough to 1, thus displaying a breaking of symmetry phenomenon in dependence on the exponent p. We then investigate the nonradial bifurcation at certain values of p from the sign-changing radial least energy solution of ($${{\mathcal {E}}}_p$$). The bifurcation result gives again, with a different approach and for values of p close to the ones at which the bifurcations appear, the existence of non-radial but quasi-radial nodal solutions.