Infinitely many global continua bifurcating from a single solution of an elliptic problem with concave convex nonlinearity

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:433

Article

Bartsch, Thomas¹  Mandel, Rainer² 

[1] Univ Giessen, Math Inst, D-35392 Giessen, Germany

[2] Scuola Normale Super Pisa, I-56126 Pisa, Italy

关键词: Global bifurcation;    Concave-convex nonlinearity;   

DOI  :  10.1016/j.jmaa.2015.07.055

来源: Elsevier

【 摘 要 】

We study the bifurcation of solutions of semilinear elliptic boundary value problems of the form [GRAPHICS] on an annulus Omega subset of R-N, with a concave convex nonlinearity, a special case being the nonlinearity first considered by Ambrosetti, Brezis and Cerami: f lambda(vertical bar x vertical bar, u, vertical bar del u vertical bar) = lambda(vertical bar u vertical bar(q-2) u + vertical bar u vertical bar(p-2)u with 1 < q < 2 < p. Although the trivial solution u(0) equivalent to 0 is nondegenerate if lambda = 0 we prove that (lambda(0), u(0)) = (0,0) is a bifurcation point. In fact, the bifurcation scenario is very singular: We show that there are infinitely many global continua of radial solutions C-j(+/-) subset of R x C-1 ((Omega) over bar), j is an element of N-0 that bifurcate from the trivial branch R x {0} at (lambda(0), u(0)) = (0,0) and consist of solutions having precisely j + 1 nodal annuli. A detailed study of these continua shows that they accumulate at R->= 0 x {0} so that every (A, 0) with A > 0 is a bifurcation point. Moreover, adding a point at infinity to C-1((Omega) over bar) they also accumulate at R x {infinity}, so there is bifurcation from infinity at every lambda is an element of R. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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