Optimal Wloc2,2-regularity, Pohozhaev's identity, and nonexistence of weak solutions to some quasilinear elliptic equations

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:252

Article

Il'yasov, Yavdat Sh²  Takac, Peter¹ 

[1] Univ Rostock, Inst Math, D-18051 Rostock, Germany

[2] Russian Acad Sci, Inst Math, Ufa Sci Ctr, Ufa 450077, Russia

关键词: Degenerate or singular elliptic problem;    Optimal regularity;    Pohozhaev's identity;    Nonexistence of phase transition solutions;   

DOI  :  10.1016/j.jde.2011.10.020

来源: Elsevier

【 摘 要 】

We begin by establishing a sharp (optimal) W-loc(2,2)-regularity result for bounded weak solutions to a nonlinear elliptic equation with the p-Laplacian. Delta(p)u =(def) div(vertical bar Delta u vertical bar(p-2)del u), 1 < p < infinity. We develop very precise, optimal regularity estimates on the ellipticity of this degenerate (for 2 < p < infinity) or singular (for 1 < p < 2) problem. We apply this regularity result to prove Pohozhaev's identity for a weak solution u is an element of W-1.p(Omega) of the elliptic Neumann problem -Delta(p)u + W'(u) = f(x) in Omega; partial derivative u/partial derivative v = 0 on partial derivative Omega. (P) Here, Omega is a bounded domain in R-N whose boundary partial derivative Omega is a C-2-manifold, v v(x(0)) denotes the outer unit normal to partial derivative Omega at x(0) is an element of partial derivative Omega, x = (x1, ... x(N)) is a generic point in Omega, and f is an element of L-infinity(Omega) boolean AND W-1.1(Omega). The potential W : R -> R is assumed to be of class C-1 and of the typical double-well shape of type W(s) = vertical bar 1 - vertical bar s vertical bar(beta)vertical bar(alpha) for s is an element of R, where alpha, beta > 1 are some constants. Finally, we take an advantage of the Pohozhaev identity to show that problem (P) with f = 0 in Omega has no phase transition solution u is an element of W-1.p(Omega) (1 < p <= N), such that -1 <= u <= 1 in Omega with u equivalent to -1 in Omega(-1) and u equivalent to 1 in Omega(1), where both Omega(-1) and Omega(1) are some nonempty subdomains of Such a scenario for u is possible only if N = 1 and Omega(-1), Omega(1) are finite unions of suitable subintervals of the open interval Omega (C) 2011 Elsevier Inc. All rights reserved.

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