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báo cáo hóa học: " Sign-changing solutions for some nonlinear problems with strong resonance"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Sign-changing solutions for some nonlinear problems with strong resonance | Qian Boundary Value Problems 2011 2011 18 http www.boundaryvalueproblems.eom content 2011 1 18 o Boundary Value Problems a SpringerOpen Journal RESEARCH Open Access Sign-changing solutions for some nonlinear problems with strong resonance Aixia Qian Correspondence qaixia@amss.ac.cn School of Mathematic Sciences Qufu Normal University Qufu Shandong 273165 P. R. of China Springer Abstract By means of critical point and index theories we obtain the existence and multiplicity of sign-changing solutions for some elliptic problems with strong resonance at infinity under weaker conditions. 2000 Mathematics Subject Classification 35J65 58E05. Keywords critical point theory strong resonance index theory Cerami condition 1 Introduction In this article we consider the following equation i -Au f I u e H0 . where o is a bounded domain in R with smooth boundary do. In order to explain what we mean a brief description is necessary. We suppose that f is asymptotically linear i.e. lim - exists. If we set u TO u a lim f 1 2 u TO u then we can write f u au g u with g u - 0 as u 0. u We denote 11 12 . 1j . to be the distinct eigenvalues sequence of -A with the Dirichlet boundary conditions. We state that problem 1.1 is resonant at infinity if a in 1.2 is an eigenvalue 1k. The situation lim g u 0 and lim ỉ g t dt p e R u TO u TO 0 is what we call a strong resonance. 2011 Qian licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License http creativecommons.org licenses by 2.0 which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Qian Boundary Value Problems 2011 2011 18 http www.boundaryvalueproblems.eom content 2011 1 18 Page 2 of 9 Now we present some of the results of this article. We write 1.1 in the following form i Au XkU g u 0 u e H1 Q . We assume that g is a smooth function satisfying the following conditions. gi g t t 0 as t . g2 the real function G t