A Bahri-Lions theorem revisited

摘要

In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min-max critical points. 1. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027-1037] studied the following elliptic problem: -Delta u = vertical bar u vertical bar(p-2)u + f (x, u), u is an element of H-0(1)(Omega), where Omega is a bounded smooth domain of R-N, 2 < p < (2N - 2)/(N - 2) and f(x, u) is not assumed to be odd in u. They proved the existence of infinitely many solutions under an appropriate growth restriction on f. In the present paper, we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign-changing solutions. In addition we derive an estimate on the number of their nodal domains. We also deal with the corresponding fourth order equation Delta(2)u = vertical bar u vertical bar(p-2)u + f (x, u) with both Dirichlet and Navier boundary conditions, as well as with strongly coupled elliptic systems.