By Wenming Zou
This ebook provides a few of the most modern learn in severe aspect thought, describing tools and offering the latest purposes. assurance comprises extrema, even valued functionals, vulnerable and double linking, signal altering options, Morse inequalities, and cohomology teams. functions defined comprise Hamiltonian platforms, Schrödinger equations and platforms, leaping nonlinearities, elliptic equations and structures, superlinear difficulties and beam equations.
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Additional resources for Critical Point Theory and Its Applications
Example text
2. BOUNDED MINIMIZING SEQUENCES 29 Then it is easy t o check t h a t (/^(i^), ^ A * ( ^ ) ) ^ ^ ^ ^ ^ ||y^*|| < 1 for ah u e E. F u r t h e r m o r e , for u e B C DSQ{X), C{^) = 1 ^^^ {I',{U),V^{U))=^{U){I'^{U),VX{U)) II^A('^)II — ^o, we have > 1\\I',{U)\\ > ^£0. ^(0, u) = u for each u ^ E (note t h a t ^ vanishes on an open set containing t h e points where / ^ = 0)- It is well known t h a t there exists a unique solution r]{t,u) for t >0. un)) < lx{r]{0,un)) < IxA^n) < Mx^ + (A - A,) < A^A + ^ for n large enough.
Zou in [385]. 2, a " monotonicity method" is applied which was introduced by M. Struwe in [351, 352] for minimization problems and essentially developed by L. Jeanjean in [191, 193, 196] for one positive solution of mountain pass type. The readers may consult the papers of B. Buffoni-L. A. Stuart [74], D. G. de Figueiredo-P. L. Lions [163], L. Jeanjean [194], N. Hirano-S. Li-Z. Q. Wang [185], Z. Liu-S. Li-Z. Q. Wang [236], P. L. Lions [227], M. Schechter [316], M. Struwe [349] and other papers for results without the (PS) condition.
D Notes and Comments. 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.