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1.
In 1988, A. Bahri and P.L. Lions [A. Bahri, P.L. Lions, Morse-index of some min–max critical points. I. Application to multiplicity results, Comm. Pure Appl. Math. 41 (1988) 1027–1037] studied the following elliptic problem: where Ω is a bounded smooth domain of , 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 Δ2u=|u|p−2u+f(x,u) with both Dirichlet and Navier boundary conditions, as well as with strongly coupled elliptic systems. 相似文献
2.
The aim of this paper is to discuss the positive solutions of the p-Laplace problem
−div(|u|p−2u)+g(u)|u|p=λuq,