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

Abstract

We consider the semilinear elliptic problem −∆u = |u| p−1 u in B u = 0 on ∂B (Ep) where B is the unit ball of R 2 centered at the origin and p ∈ (1, +∞). We prove the existence of non-radial sign-changing solutions to (Ep) which are quasi-radial, namely solutions whose nodal line is the union of a finite number of disjoint simple closed curves, which are the boundary of nested domains contained in B.In particular the nodal line of these solutions doesn't touch ∂B.The result is obtained with two different approaches: via nonradial bifurcation from the least energy sign-changing radial solution up of (Ep) at certain values of p and by investigating the qualitative properties, for p large, of the least energy nodal solutions in spaces of functions invariant by the action of the dihedral group generated by the reflection with respect to the x-axis and the rotation about the origin of angle 2π k for suitable integers k. We also prove that for certain integers k the least energy nodal solutions in these spaces of symmetric functions are instead radial, showing in particular a breaking of symmetry phenomenon in dependence on the exponent p.

Sharp quantization for Lane–Emden problems in dimension two

“…We mention that very interesting complementary results were obtained recently by Kamburov and Sirakov [14]. At last, we also mention that, even if the situation is far from being as well understood in the nodal case, where we no longer assume that the u p 's are positive, some asymptotic-analysis [3,13], as well as some constructive [5,10,12] results were obtained. To conclude, as explained in De Marchis, Ianni and Pacella [4], the techniques to get the quantization result in [6] are not without similarity with the ones developed by Druet [7] to get the analogue quantization for 2D Moser-Trudinger critical problems.…”

On a singular eigenvalue problem and its applications in computing the Morse index of solutions to semilinear PDE’s

“…Observe that this result is true as far as one consider the eigenvalues which are strictly negative and cannot be true for the zero eigenvalue as observed in [22,Lemma 4.3] in the case of dimension 2. In dimension N ≥ 3 a similar relation holds also for the eigenvalue zero.…”

“…Proof. The first estimate in (3.18) has been proved for classical solutions in [20] and [22] in the case M = 2 and in Lemma 5.9 in [15] in the case M > 2. In both cases it is a consequence of the local estimate…”

“…Moreover the knowledge of the Morse index in symmetric spaces turns useful in the case of multiple bifurcation to distinguish solutions as can be seen in [23] and [5] dealing respectively with positive and nodal solutions for the Lane Emden problem in an annulus. It can also give existence of different solutions by minimizing the functional associated with (1.1) in that symmetric spaces, as done in [22], dealing with the Lane-Emden problem in the disk. Our approach discloses information on the symmetries of the corresponding eigenfunctions and hence on the Morse index and degeneracy in symmetric spaces of functions.…”

Global bifurcation for the Hénon problem

“…They are very different from the radial ones since their nodal surfaces intersect the boundary of the ball. Another interesting paper by Gladiali and Ianni [19] showed the existence of solutions to the Lane-Emden equation which are nonradial but "quasi-radial", in the sense that their nodal lines are the boundary of nested domains contained in the disc. Some of these quasi-radial solutions are produced as least energy nodal solutions in symmetric spaces, some others by bifurcation w.r.t.…”