We consider the problem -epsilon(2) Delta u + u = vertical bar u vertical bar(p-1) u in Omega, partial derivative u/partial derivative nu = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, 1 < p < +infinity if N = 2, 1 < p < (N + 2)/(N - 2) if N >= 3 and epsilon is a parameter. We show that if the mean curvature of partial derivative Omega is not constant then, for epsilon small enough, such a problem has always a nodal solution u(epsilon) with one positive peak xi(epsilon)(1) and one negative peak xi(epsilon)(2) on the boundary. Moreover, H(xi(epsilon)(1)) and H(xi(epsilon)(2)) converge to max(partial derivative Omega) H and min(partial derivative Omega) H, respectively, as epsilon goes to zero. Here, H denotes the mean curvature of partial derivative Omega. Moreover, if Omega is a ball and N >= 3, we prove that for epsilon small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily many negative peaks on the boundary.
On the Multiplicity of Nodal Solutions to a Singularly Perturbed Neumann Problem / Anna Maria, Micheletti; Pistoia, Angela. - In: MEDITERRANEAN JOURNAL OF MATHEMATICS. - ISSN 1660-5446. - 5:3(2008), pp. 285-294. [10.1007/s00009-008-0150-5]
On the Multiplicity of Nodal Solutions to a Singularly Perturbed Neumann Problem
PISTOIA, Angela
2008
Abstract
We consider the problem -epsilon(2) Delta u + u = vertical bar u vertical bar(p-1) u in Omega, partial derivative u/partial derivative nu = 0 on partial derivative Omega, where Omega is a bounded smooth domain in R-N, 1 < p < +infinity if N = 2, 1 < p < (N + 2)/(N - 2) if N >= 3 and epsilon is a parameter. We show that if the mean curvature of partial derivative Omega is not constant then, for epsilon small enough, such a problem has always a nodal solution u(epsilon) with one positive peak xi(epsilon)(1) and one negative peak xi(epsilon)(2) on the boundary. Moreover, H(xi(epsilon)(1)) and H(xi(epsilon)(2)) converge to max(partial derivative Omega) H and min(partial derivative Omega) H, respectively, as epsilon goes to zero. Here, H denotes the mean curvature of partial derivative Omega. Moreover, if Omega is a ball and N >= 3, we prove that for epsilon small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily many negative peaks on the boundary.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.