Abstract
This paper is devoted to the existence of solutions for a class of Kirchhoff type systems involving critical exponents. The proof of the main results is based on concentration compactness principle related to critical elliptic systems due to Kang combined with genus theory.
Highlights
The study of This type of equations is motivated by its various applications, for example, the case h(s) = s was used to model the time evolution of the condensate wave function in superfluid film, and is called the superfluid film equation in fluid mechanics by
One of the main difficulties of the quasilinear problem with nonhomogeneous term [∆(u2)]u is that there is no suitable space on which the energy functional is well defined
There have been several approaches used in recent years to overcome the difficulties such as minimizations [19,24], the Nehari or Pohozaev manifold [20,26], and change of variables [1,8,21,27]
Summary
This last inequality shows that {(wn, zn)} is bounded in X. Passing to a subsequence if necessary, we may assume that wn ⇀ w in H01(Ω) wn → w a.e. in Ω zn ⇀ z in H01(Ω) zn → z a.e. in Ω. Since {(f 2(wn), f 2(zn))} is bounded in X, we deduce that (f 2(wn), f 2(zn)) ⇀ (f 2(w), f 2(z)) in X and. In the sense of measures, where μ and ν are nonnegative bounded measures on RN. In view of Lemma 1.1 (f5), Holder’s inequality and Lebesgue’s dominated convergence theorem, lim sup 1 + 2f 2(wn)f (wn)∇wn∇φjεdx n→+∞ Ω. Up to subsequence lim lim ε→0 n→+∞ Ω and we have 1 + 2f 2(wn)f (wn)∇wn∇φjεdx = 0,.
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