Bifurcation and Nodal Solutions for the Half-Linear Problems with Nonasymptotic Nonlinearity at 0 and ∞

By applying the bifurcation techniques of Rabinowitz [1, 2], Im et al. [3] studied the existence of positive solutions of the problems and some authors [4–7] studied the existence of nodal solutions of the problems. Recently, Dai and Ma [8] established the unilateral global bifurcation theory for the problems by applying the bifurcation techniques of Dancer [9]. Later, Dai [10] also considered the existence of nodal solutions for the problems with nonasymptotic nonlinearity at 0 or∞ by applying the unilateral global bifurcation theory of [8]. For the abstract unilateral global bifurcation theory, we refer the reader to [1, 2, 8, 9, 11–13] and the references therein. On the other hand, problems involving nondifferentiable nonlinearity have also been investigated by applying bifurcation techniques; see [13–17]. In particular, Berestycki [14] established the Rabinowitz-type global interval bifurcation result and Ma and Dai [13] established the unilateral global interval bifurcation theorem. Meanwhile, half-linear or halfquasilinear boundary value problems have attracted the attention of many specialists in different equations because of their interesting applications; see [13, 14, 16–18]. Among them, Berestycki [14] (or see [13]) established the spectrum for the following half-linear eigenvalue problem:

Of course, the natural question is that of what would happen if  0 ∉ (0, +∞) or  ∞ ∉ (0, +∞).Obviously, the previous results cannot deal with this case.The purpose of this work is to establish several results similar to those of [13].The main methods used in this work are global bifurcation techniques and the approximation of connected components.Moreover, we consider the cases of  0 ,  ∞ ∉ (0, ∞), while the authors of [13,16,17] only studied the cases of  0 ,  ∞ ∈ (0, ∞).
In this paper, we will investigate the existence of nodal solutions for problem (4), where  satisfies condition (1).Throughout this paper, we assume that  satisfies (2) and the following assumptions: The rest of this paper is arranged as follows.In Section 2, we gave some preliminaries.In Section 3, we give the interval for the parameter  which ensure the existence of single or multiple nodal solutions for half-linear problem (4) under assumptions (2) and (1)-(8) for the nonlinearity .
By Lemma 4, we obtain the following result that will be used later.
Let   ∈  be a solution of the equation Then   must change sign on  as  → +∞.
By simple computation, we can show that After taking a subsequence if necessary, we may assume that as  → +∞, where   is th eigenvalue of the following problem: Set  = ( − ) + , () = V(( − ) + ).By some simple computations, we can show Let   be the corresponding eigenfunction of   .Since   → ∞ as  → ∞, Lemma 2.1 of [4] implies that   must change sign on .Note that the conclusion of Lemma 4 also is valid if  =  = 0. Using these facts and Lemma 4, we can obtain the desired result.
We start this section by studying the following eigenvalue problem: where  > 0 is a parameter.Firstly, under the conditions (1)-(3), let  ∈ (R, R) be such that with lim || → 0 (()/) = 0. Let us consider as a bifurcation problem from the trivial solution  ≡ 0. By using Theorem 3 of [13] or Ma and Dai [13, Lemma 4.1] obtained the following Lemma.
Proof.We only prove the case of (i) since the proofs of the cases for (ii), (iii), and (iv) can be given similarly.
In view of the proof to prove [13, Theorem 4.1], we only need to show that D ]  joins for some positive constant  0 not depending on .
We consider the equation Let   =   /‖  ‖, and   should be the solutions of problem Since   is bounded in  2 [0, 1], choosing a subsequence and relabelling if necessary, we have that   →  for some  ∈  and ‖‖ = 1.
Furthermore, from (26) and the fact that  is nondecreasing, we have that lim  This contradicts ‖()‖ = 1.
Proof.We will only prove the case of (i) since the proofs of the cases for (ii), (iii), and (iv) are completely analogous.
Applying a similar method used in the proof of Theorem 12, we obtain an unbounded connected component Similar to the method of the proof of Theorem 10, we can obtain that (∞, ∞) ∈ D ]  .It follows that the result is obtained.
Applying a similar method used in the proof of Theorem 12, we obtain an unbounded connected component In the following, we can show that (0, ∞) ∈ D ]  .Assume that there exists a sequence (  ,   ) ∈ D ]  such that (51) We divide the proof into two steps.
If {  } is bounded, then there exists a constant  1 not depending on  such that ‖  ‖ ≤ We claim that there exists  0 ∈ {0, 1, . . ., } such that Otherwise, we have This is a contradiction.
Similar to the proof of Step 1, we can get a contradiction.
Applying a similar method used in the proof of Theorem 15, we obtain an unbounded connected component D ]  ⊂ S ]  with (∞, 0) ∈ D ]  .Similar to the proof of Theorem 10, we can show that (∞, ∞) ∈ D ]  .