A Local Bifurcation Theorem for Degenerate Elliptic Equations With Radial Symmetry

Abstract

In this work we provide local bifurcation results for equations involving the p-Laplacian in balls. We analyze the continua C n of radial solutions emanating from (l n, p , 0), {l n, p } being the radial eigenvalues of − D p . First, we show that the only nontrivial solutions close to (l n, p , 0) lie on a continuous curve, thus extending the Crandall-Rabinowitz theorem. Second, it is proved that C n 0 {(l n, p , 0)} splits into two unbounded connected pieces, characterized by their nodal properties thus sharpening previous results. © 2002 Elsevier Science

Bifurcations of Positive and Negative Continua in Quasilinear Elliptic Eigenvalue Problems

“…Last but not least, for the radially symmetric problem (1.1) in a ball Ω ⊂ R N , a local bifurcation result of Crandall-Rabinowitz-type [9, Theorem 1.7, p. 325] has been obtained in García-Melián and Sabina de Lis [21,Theorem 2,p. 30].…”

Positive solutions of a degenerate elliptic equation with logistic reaction

“…However, they have given only an estimate dist(O λ , ∂Ω) ≤ Cλ 1/p as λ → 0, where C is a constant independent of λ, without explicit information about C and any estimate of dist(O λ , ∂Ω) from below. In virtue of an exact estimate for O λ , García-Melián and Sabina de Lis [9] have utilized the solutions for N = 1, whose dependence on λ is understood well, to make upper and lower solutions and concluded that (1.3) also holds true in the case N ≥ 2. The subdiffusive case p > q can also be investigated in the same way as the equidiffusive case.…”

“…One can observe that there exists a unique solution u λ for every λ > 0 and that as the equidiffusive case, O λ (u λ ) is nonempty for sufficiently small λ > 0 and it grows as in (1.3). See the author and Yamada [19] for N = 1 and [9] with its Remarks 2.2 b for N ≥ 2. For uniqueness, see also Diaz and Saa [5].…”

“…As for (a), we can prove the following theorem by the method of upper and lower solutions: We will state our result on (b). The proof essentially consists of constructing suitable upper and lower solutions by the idea of García-Melián and Sabina de Lis [8,9] and the one-dimensional result in [19].…”

p-Laplacian problems with jumping nonlinearities

“…For instance, the papers [1,5,[10][11][12][13][16][17][18][19]21,26,27] all obtain existence results for problems of the above form, although most of these papers impose considerable additional assumptions, of various types, on f . In this paper we extend most of the results obtained in these papers to the more general setting described above (in fact, we extend the results for the case p = 2, under fairly general condition on f , to the case p = 2).…”