Minimal nodal solutions of the pure critical exponent problem on a symmetric domain

Abstract

We establish existence of nodal solutions to the pure critical exponent problem −∆u = |u| 2 * −2 u in Ω, u = 0 on ∂Ω, where Ω a bounded smooth domain which is invariant under an orthogonal involution of R N . We extend previous results for positive solutions due to Coron, Dancer, Ding, and Passaseo to existence and multiplicity results for solutions which change sign exactly once. (2000): 35J65, 35J20 Mathematics Subject Classification

Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size

“…The first such result was pointed out by Marchi and Pacella [13] for domains with thin tunnels. For symmetric domains with a small hole existence of a sign changing solution was shown by Clapp and Weth in [5] and their method applies also to get the result of [13]. Recently Musso and Pistoia [14] proved that, if the domain has certain symmetries and a small hole whose boundary is a sphere, then the number of sign changing solutions becomes arbitrarily large as the radius of the sphere goes to zero.…”

“…Later a remarkable result was obtained by Bahri and Coron [1] who showed that the same holds for any R. In fact, they showed that (℘ ) has at least one positive solution in every domain having nontrivial homology with Z/2-coefficients. This last condition is, however, not necessary: examples of contractible domains for which problem (℘ ) has a solution have been given, for example, in [5,9,10,15,16].…”

The Ljapunov–Schmidt Reduction for Some Critical Problems

“…If the domain has several holes, Rey in [42] and Li-YanYang in [30] constructed solutions blowing-up at the centers of the holes as the size of the holes goes to zero. On the other hand, Clapp-Weth in [15] found a second solution to (C)ǫ, but they were unable to say if it was positive or changed sign. Clapp-Musso-Pistoia in [14] found positive and sign changing solutions to (C)ǫ blowing-up at the center of the hole and at one or more points inside the domain as ǫ goes to zero.…”

Energy bounds for entire nodal solutions of autonomous superlinear equations

“…This bound has been used in some papers to obtain existence and multiplicity results for solutions of (1.3), see e.g. [16,17,24]. The proof of Lemma 1.1 is easy, and it is given for instance in [39, p.185].…”

“…A fascinating correlation between the solvability of (1.3) and geometric properties of was discovered by Bahri and Coron [5,19], and their work stimulated immense further research. We refer the reader to [21,26,33] for classical results and to [6,9,17], including the references given therein, for more recent developments.…”