Problems With Perturbations From Symmetric For Some Resonant Elliptic Equations

In this paper we consider multiplicity of solutions with perturbations from symmetricfor some resonant problem with lim sup infinite. Let ft be a bounded domain in Rn(n > 3) with smooth boundary ds for every is distinct eigenvalue of the problemvk is the eigenfunctions corresponding to kandFirstly, we obtain multiplicity of solutions for symmetric resonant problemby symmetric version of the mountain pass theorem. If k = 0,the problem becomestheorem 9.38 of reference [1], so this section is extension of reference [1].Secondly, we obtain multiplicity of solutions for resonant problem with perturbationsfrom symmetricby lemma1, lemma2 and lemma3 of reference [3]. Apart from continuity, we do not assume any condition on the nonlinear term, where 0 < e < 1 is a parameter. If e = 0, the problem becomes symmetric problem and e = 0, k = 0, we obtain theorem 9.38 of reference [1], while if k = 0, the Problem becomes result of reference [3], so the section is extend for references [1] [3] and symmetric problem .Thirdly, we obtain multiplicity of solutions for resonant non-homogeneous boundary perturbations from symmetric problem without parameterby a new perturbation method introduced by Bolle in reference [4], applied in references [5] [6] and extend in reference [7]. References [5] [6] have considered some exceptive case while the section consider general case. If k= 0,g = 0, the problem becomes theorem 10.4 of reference [1].Lastly, The problemis perturbations from symmetric problem with non-homogeneous boundary problem withparameter. For the problem , we obtain multiplicity of solutions by method used in reference [3].