Generic Fredholm alternative-type results for the one dimensional p-Laplacian

Abstract

In this paper we give a generic necessary and sufficient condition for the solvability of2000 Mathematics Subject Classification: 34B15, 47H12.

Basis properties of eigenfunctions of the 𝑝-Laplacian

“…The results of this paper appear to be useful in numerical approaches to boundary value problems for quasilinear equations involving the p-Laplacian, e.g., of the type considered in [3] and [4]. We hope to report on this elsewhere.…”

On lower and upper solutions without ordering on time scales

“…If f satisfies certain growth and limit conditions, we obtain the existence of a solution. The similar approach for the p-Laplacian can be found in Drábek et al [7].…”

On the Number and Structure of Solutions for a Fredholm Alternative with the p-Laplacian

“…In fact, we will see that, for any p=2; both types of solutions exist whenever z n 4z4z n and z=0; however, we are unable to verify if they are distinct also for z n oz4z # or z # 4zoz n : Solely in the case of one space dimension (N ¼ 1), i.e. when O ¼ ð0; aÞ is a bounded interval in R 1 ; Dra´bek et al [10,Theorem 1.3] succeeded to show that the two types of solutions are indeed distinct whenever z n ozoz n and z=0: Finally, given any d > 0; we show that the set of all weak solutions to problem (1.4) is bounded in C 1 ð % O OÞ uniformly for jzj5d and for z ¼ 0 as well. More precisely, we obtain a blow-up-like asymptotic behavior of every weak solution as z !…”

“…The orthogonality condition (1.6) is not assumed only in the recent work of Dra´bek et al [10] and Dra´bek and Holubova´ [11]. In [10,Theorem 1.3] it is shown that for O ¼ ð0; aÞ; p=2; and f ¼ zj 1 þ f T 2 C 1 ½0; a; problem (1.4) has at least one weak solution if and only if z n 4z4z n ; where À1oz n o0 oz n o1 are some numbers depending on f T ; and at least two distinct solutions provided z n ozoz n and z=0: (As already mentioned above, in our present work we are unable to show z # ¼ z n and z Last but not the least, we recall that nonuniqueness in problem (1.1) is not typical for the case l ¼ l 1 : It occurs already for 0olol 1 and even if N ¼ 1; see [15] (for 1opo2) and [27] …”