Download Analysis and Continuum Mechanics: A Collection of Papers by Professor Dr. Stuart S. Antman, Professor Dr. Haïm Brezis, PDF

By Professor Dr. Stuart S. Antman, Professor Dr. Haïm Brezis, Professor Dr. Bernard D. Coleman, Professor Dr. Martin Feinberg, Professor Dr. John A. Nohel, Professor Dr. William P. Ziemer (auth.)

ISBN-10: 3540509178

ISBN-13: 9783540509172

The 39 papers during this assortment are dedicated usually to the precise mathematical research of difficulties in continuum mechanics, but in addition to difficulties of a merely mathematical nature in most cases attached to partial differential equations from continuum physics. the entire papers are devoted to J. Serrin and have been initially released within the "Archive of Rational Mechanics and Analysis".

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Extra resources for Analysis and Continuum Mechanics: A Collection of Papers Dedicated to J. Serrin on His Sixtieth Birthday

Sample text

Letting k -+ 00, we obtain ()(. 2 fJ*. d. 7. 6, suppose in addition that f in E. 1) also. -Y. -M. NI Proof. 3»). 6 applies. d. 9) = < (n + 2)/(n - 2). For pur- u> 0 0, in R n , where K(x) > O. That is, we set f(x, u) = Q(x) uP, and h(x, u) = K(x) uP, both Q and K are bounded functions on R n , and Q(x) < K(x) for all Ixl > R, for some R > O. 7 are satisfied, and J(u) where = t I ul1 2 - f Qu~+1 _1_ p + lRn Mf = {UE E\ {O} IIIull 2 = JQU~+I} Mh = {UE E\ {O} IIIull 2= i KU~+I} lIull 2 = f(IVu12 +u 2 ), u+(x) = max{u(x),O}, Rn and E = HJ(Rn).

That is, the sequence {II Uk II}, Uk being a critical point of Jk obtained by use of the Mountain-Pass Lemma (as in § 2), is still uniformly bounded. Thus, by 35 A Semilinear Elliptic Equation passing to a subsequence if necessary, we may assume that Uk converges weakly to it in Er • Let B be a (fixed) ball in Rn. 2) (b) and standard interior Festimates and Schauder estimates for solutions Uk, we conclude that (passing to a subsequence again if necessary) Uk converges to Ii in the space C 2 (B).

5) Next, we wish to obtain similar bounds for rk's and uk(rk)'s for k EE S. We proceed as follows. For k EE S, we have uk(rk) > O. -Y. -M. NI ° ately that there is an R > such that rk ~ R for all k EE S. Therefore, there is a constant R such that rk ~ R for all k and uk(rk) > min {P, b} > for all k. By the compactness of B R , there is a subsequence rkJ that converges to r;;::; R and as j ~ 00 since Uk converges to u uniformly on BR • Thus, and =$ 0. 6. 2) hold. 1) has a positive radial solution in E,.

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