Download Critical Point Theory and Its Applications by Wenming Zou PDF

By Wenming Zou

This booklet offers many of the newest examine in severe aspect concept, describing tools and providing the most recent purposes. assurance contains extrema, even valued functionals, susceptible and double linking, signal altering strategies, Morse inequalities, and cohomology teams. functions defined comprise Hamiltonian platforms, Schrödinger equations and structures, leaping nonlinearities, elliptic equations and platforms, superlinear difficulties and beam equations.

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Liu-T. Weth [38], S. Li-Z. Q. Wang [217] and W. Zou [391], sign-changing high energy solutions were obtained. We will address this topic later in this book. Several authors considered the existence of high energy solutions with a perturbation from symmetry. For instance, the special case —Ai^ = \u\^~'^u -\-p{x) in O with 1^ = 0 on dQ was first studied by A. Bahri-H. Berestycki [25] and M. Struwe [348] independently (see also A. Bahri [24] and A. Bahri-P. L. Lions [27]). In P. Rabinowitz [293, 296] and K.

Let ak : = sup \\u\\p, ueZk,\\u\\=i then ak ^ 0 diS k ^ 00. Indeed, suppose t h a t this is not t h e case. Then there is an £0 and {uj} C E with Uj ± Wk--i, \\uj\\ = 1, \\uj\\p > SQ, where kj -^ 00 as j ^ 00. For any v e E, we may find a Wj G Wk^-i such t h a t Wj ^ i; as j ^ 00. , Uj -^ 0 weakly in E. Hence, Uj ^ 0 in I / ^ ( R ^ ) , a contradiction. 2. HIGH ENERGY SOLUTIONS 47 estimates: $AW > l\\uf-^Ml-^\\u\\; > \hf-c\\u\\i > \hf-cal\\ur. If we choose rk := (4cpQ^^)^/^^~^\ then for u e Zk with ||i^|| = r/c, we get that ^xiu)>iAcpalf/(^-''\\-^):=h.

G. A. Ambrosetti-P. Rabinowitz [19]). The readers may find some variants of it in M. Struwe [352]. 3. SMALL ENERGY SOLUTIONS 49 high energy solutions via SMP and the Fountain Theorem (cf. P. Rabinowitz [293], T. Bartsch [30], T. Bartsch-M. Wihem [48], M. Struwe [352], and also M. Willem [377], etc). In particular, in T. Bartsch-Z. Liu-T. Weth [38], S. Li-Z. Q. Wang [217] and W. Zou [391], sign-changing high energy solutions were obtained. We will address this topic later in this book. Several authors considered the existence of high energy solutions with a perturbation from symmetry.

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