The Existence of Nodal Solutions for the Half-Quasilinear p-laplacian Problems

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1 Journal of Mathematical Research with Applications Mar., 017, Vol. 37, No., pp. 4 5 DOI:13770/j.issn: The Existence of Nodal Solutions for the Half-Quasilinear p-laplacian Problems Wenguo SHEN Department of Basic Courses, Lanzhou Institute of Technology, Gansu , P. R. China Abstract In this paper, we study the existence of nodal solutions for the following problem: (φ p (x )) = α(t)φ p (x + ) + β(t)φ p (x ) + ra(t)f(x), 0 < t < 1, x(0) = x(1) = 0, where φ p(s) = s p s, a C([0, 1], (0, )), x + = max{x, 0}, x = min{x, 0}, α(t), β(t) C[0, 1]; f C(R, R), sf(s) > 0 for s 0, and, (0, ), where = lim f(s)/φ p (s), = lim f(s)/φ p (s). s 0 s + We use bifurcation techniques and the approximation of connected components to prove our main results. Keywords Bifurcation; Half-Quasilinear problems; Nodal solutions; p-laplacian MR(010) Subject Classification 53C4; 53B30; 53C50 1. Introduction Let E be a real Banach space with the norm. Consider the operator equation u = λbu + H(λ, u), (1.1) where B is a compact linear operator and H : R E E is compact with H = o( u ) at u = 0 uniformly on bounded λ intervals. Krasnoselsii [1] has shown that all characteristic values of B which are of odd multiplicity are bifurcation points. Rabinowitz [] has extended this result by showing that bifurcation has global consequences. Furthermore, if the characteristic value µ of B has multiplicity 1 and S = {(λ, u) : (λ, u) satisfies (1.1) and u 0} R E. Dancer [3] has shown that there are two distinct unbounded continua C µ + and Cµ, consisting of the bifurcation branch C µ of S emanating from (µ, 0), which satisfy either C µ + and Cµ are both unbounded or C + µ C µ {(µ, 0)}. In the past ten years, some authors [4 9] studied the existence of Nodal solutions for the problems by applying Rabinowitz bifurcation techniques []. By using bifurcation techniques of Received January 5, 016; Accepted November 3, 016 Supported by the National Natural Science Foundation of China (Grant No ) and the Natural Science Foundation of Gausu Province (Grant No. 145RJZA087). address: shenwg369@163.com

2 The existence of nodal solutions for the Half-quasilinear p-laplacian problems 43 Dancer [3], Dai [10,11] also considered the p-laplacian problems. Recently, the problem involving non-differentiable nonlinearities have also been investigated by applying bifurcation techniques, see [1 16] and references therein. Meanwhile, Half-linear or Half-quasilinear problems have attracted the attention of many specialists in different equations because of their interesting applications [1,14 17]. Especially, Dai [16] considered the existence of nodal solutions for the following half-quasilinear eigenvalue problem (φ p (x )) = α(t)φ p (x + ) + β(t)φ p (x ) + ra(t)f(x), 0 < t < 1, x(0) = x(1) = 0, where φ p (s) = s p s, x + = max{x, 0}, x = min{x, 0}, α(t), β(t) C[0, 1] and f C(R, R). Under the following assumptions: where (A1) a C([0, 1], (0, )). (A) sf(s) > 0 for s (A3), (0, + ), they obtained the following result: f(s) = lim s 0φ p (s), = lim s + f(s) φ p (s), Theorem 1.1 Let (A1), (A), and (A3) hold. For some N and ν = +,, either λν < r < λν or λν < r < λν. has exactly 1 zeros in (0, 1) and is negative near Where λ ν be given in Lemma.1. Of course, the natural question is what would happen if (0, + ) or (0, + ). (1.) Obviously, the previous results cannot deal with this case. The purpose of this wor is to establish several results similar to those of [16]. The main methods used in this wor are unilateral global bifurcation techniques and the approximation of connected components. Moreover, we consider the cases of, (0, ), while the authors of [14 16] only studied the cases of, (0, ). Remar 1. We also note that, in high-dimensional case, there are also a lot of fundamental papers on the global bifurcation for p-laplacian [18 1]. Remar 1.3 For the abstract unilateral global bifurcation theory, we refer the reader to [3,10,14,16,,3] and the references therein. The rest of this paper is arranged as follows. In Section, we give some preliminaries. In Section 3, we investigate the existence of nodal solutions for a class of the half-quasilinear p-laplacian problems under the conditions (H1) (H8), respectively.. Preliminaries Let Y = C[0, 1] with the norm x = max t [0,1] x(t). Let E = {x C 1 [0, 1] x(0) = x(1) = 0} with the norm x = max t [0,1] x(t) + max t [0,1] x (t). Let E = R E under the

3 44 Wenguo SHEN product topology. Let S + denote the set of functions in E which have exactly 1 interior nodal (i.e., nondegenerate) zeros in (0, 1) and are positive near t = Set S = S+, and S = S + S. They are disjoint and open in E. Finally, let Φ± = R S± and Φ = R S. Let S denote the closure of the set of nontrivial solutions of (1.) in R E, and S ν subset of with u S ± and S = S + S. denote the In [16], Dai established an important global bifurcation theorem from intervals for a class of second-order p-laplacian problems involving non-differentiable nonlinearity. Furthermore, Dai established the spectrum for the following Half-quasilinear problem (φ p (x )) = λa(t)φ p (x) + α(t)φ p (x + ) + β(t)φ p (x ), 0 < t < 1, x(0) = x(1) = 0, and obtained the following result. Lemma.1 ([16]) Let (A1) hold. There exist two sequences of simple half-eigenvalues for (.1) and λ + 1 < λ+ < < λ+ <, lim λ+ = + λ 1 < λ < < λ <, lim λ = +. The corresponding Half-quasilinear solutions are in {λ + } S+ and {λ } S. Further, aside from these solutions and the trivial ones, there are no other solutions of (.1). In order to prove our main results, by [16], we have Lemma. ([16]) If (λ, x) is a nontrivial solution of (1.) under assumptions (A1), (A), and (A3) and x has a double zero, then x Lemma.3 ([16]) Let b (t) max{b 1 (t), b 1 (t) + α(t) + β(t), b 1 (t) α(t) β(t)} for t (0, 1) and b i (t) C(0, 1), i = 1,. Also let u 1, u be solutions of the following differential equations (φ p (u )) + b i (t)φ p (u) + α(t)φ p (u + ) + β(t)φ p (u ), 0 < t < 1, u(0) = u(1) = 0, respectively. If (c, d) (0, 1), and u 1 (c) = u 1 (d) = 0, u 1 (t) 0 in (c, d), then either there exists τ (c, d) such that u (τ) = 0 or b = b 1 or b = b 1 + α + β or b = b 1 α β and u (t) = µu 1 (t) for some constant µ Lemma.4 ([16]) Let I = (a, b) be such that I (0, 1) and Let g n C((0, 1), R) be such that meas(i) > lim g n(t) = + uniformly on I. n + Let y n E be a solution of the equation { (φ p (y n)) + b i (t)φ p (y n ) + α(t)φ p (y + n ) + β(t)φ p (y n ) = 0, 0 < t < 1, y n (0) = y n (1) = (.1)

4 The existence of nodal solutions for the Half-quasilinear p-laplacian problems 45 Then y n must change sign on I as n +. In this section, we need the following topological lemma. Lemma.5 ([4]) connected subsets of X. Assume that Let X be a Banach space and {C n n = 1,,...} be a family of closed (i) There exist z n C n, n = 1,,... and z X, such that z n z ; (ii) r n = sup{ x x C n } = ; (iii) For all R > 0, ( n=1c n ) B R is a relative compact set of X, where B R = {x X x R}. Then there exists an unbounded component C in D and z C, where D := lim sup C n = {x X {n i } N and x ni C ni, such that x ni x} (see [5]). n 3. Nodal solutions for half quasilinear p-laplacian problems In the section, we shall investigate the existence of nodal solutions for the problem (1.), where a satisfies the condition (A1). Throughout this paper, we assume that f satisfies (A) and the following assumptions: (H1) (0, ) and = 0; (H) = 0 and (0, ); (H3) = and (0, ); (H4) (0, ) and = ; (H5) = and = 0; (H6) = 0 and = ; (H7) = 0 and = 0; (H8) = and =. We start this section by studying the following eigenvalue problem (φ p (x )) = α(t)φ p (x + ) + β(t)φ p (x ) + λra(t)f(x), 0 < t < 1, x(0) = x(1) = 0, (3.1) where λ > 0 is a parameter. Let ζ C(R, R) be such that with lim x 0 ζ(x) φ p(x) = Let us consider f(x) = φ p (x) + ζ(x) (φ p (x )) = λra(t) φ p (x) + α(t)φ p (x + ) + β(t)φ p (x ) + λra(t)ζ(x), 0 < t < 1, x(0) = x(1) = 0, (3.) as a bifurcation problem from the trivial solution x Dai [16] obtained the following Lemma. Lemma 3.1 ([16, Lemma 4.1]) Let (A1), (A), and (A3) hold. For some N and ν {+, }, ( λν r, 0) is a bifurcation point for problem ((3.). Moreover, there exists an unbounded continuum

5 46 Wenguo SHEN D ν of solutions of problem (3.), such that D ν (Φν {( λν r, 0)}). Remar 3. Any solution of (3.1) of the form (1, x) yields a solution x of (1.). In order to prove our main results, one will only show that D ν crosses the hyperplane {1} E in R E. Clearly, (A) implies f(0) = Hence, x = 0 is always the solution of (1.). Applying Lemma 3.1 (or [16, Lemma 4.1]), we establish the existence of nodal solutions of (1.) as follows. Theorem 3.3 Let (A1), (A), and (H1) hold. For some N and ν {+, }, assume that one (i) r ( λν, + ) for λ ν > 0; (ii) r (, λ ) ( λ+, + ) for νλ ν > 0; (iii) r (, λ+ ) ( λ, + ) for νλ ν < 0; (iv) r (, λν ) for λ ν < zeros in (0, 1) and is negative near 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 [16, Theorem 1.3], we only need to show that D ν joins ( λν r, 0) to (, ). To do this, it is enough to prove that [ λν r, + ) Proj R D ν. Assume on the contrary that sup{λ (λ, u) D ν } < +, then there exists a sequence (µ n, x n ) D ν such that lim n x n = +, µ n c 0 for some positive constant c 0 independent of n. By (H1), let f(x) = max 0 s x f(s). Then f is nondecreasing and We consider the equation lim x + f(x) = (3.3) xp 1 (φ p (x n)) = α(t)φ p (x + n ) + β(t)φ p (x n ) + λra(t)f(x n ), 0 < t < 1, x n (0) = x n (1) = 0, Dividing the above problem by x n p 1, let y n = x n x n, y n should be the solutions of problem (φ p (y n)) = α(t)φ p (y + n ) + β(t)φ p (y n ) + µ n ra(t) f(x n) x n p 1, y n (0) = y n (1) = (3.4) Since y n is bounded in C [0, 1], choosing a subsequence and relabeling if necessary, we have that y n y for some y E and y = 1.

6 The existence of nodal solutions for the Half-quasilinear p-laplacian problems 47 Since Furthermore, from (3.3) and the fact that f is nondecreasing, we have that lim x + f(x) f( x ) x p 1 x p 1 f( x ) x p 1 f(x) = (3.5) x p 1 f( x ) 0, x p 1 by (3.4), (3.5) and the compactness of L 1, we obtain that x +, (φ p (y )) = α(t)φ p (y + ) + β(t)φ p (y ), t (0, 1), y(0) = y(1) = By y(0) = y(1), there exists ξ (0, 1) such that y (ξ) = Furthermore, applying the similar method to prove [16, Lemma.] with obvious changes, we may obtain y(t) 0, t [0, 1]. This contradicts y(t) = 1. Theorem 3.4 Let (A1), (A) and (H) hold. For some N and ν {+, }, assume that one (i) r ( λν, + ) for λ ν > 0; (ii) r ( λ+, + ) (, λ ) for νλ ν > 0; (iii) r ( λ, + ) (, λ+ ) for νλ ν < 0; (iv) r (, λν ) for λ ν < zeros in (0, 1) and is negative near Proof We only prove the case of (i) since the proof of (ii) (iv) can be given similarly. If (λ, x) is any nontrivial solution of problem (1.), dividing problem (1.) by x (p 1) and setting y = Define x x yields (φ p (y )) = α(t)φ p (y + ) + β(t)φ p (y f(x) ) + λra(t), 0 < t < 1, x (p 1) y(0) = y(1) = f(y) := Evidently, problem (3.6) is equivalent to { y (p 1) f( y y ), if y 0, 0, if y = (φ p (y )) = α(t)φ p (y + ) + β(t)φ p (y ) + λra(t) f(y), 0 < t < 1, y(0) = y(1) = (3.6) (3.7) It is obvious that (λ, 0) is always the solution of problem (3.7). By simple computation, we can show that = and =. Now, applying Theorem 3.3, there exists an unbounded continuum C ν of solutions of the problem (3.7) emanating from (0, 0), such that C ν (Φν {(0, 0)}), and C ν joins (0, 0) to ( λν r, ).

7 48 Wenguo SHEN Under the inversion y y y = x, we obtain C ν D ν such that D ν λν joins ( r, 0) to (0, ). being solutions of the problem (1.) Theorem 3.5 Let (A1), (A) and (H3) hold. For some N and ν {+, }, assume that one (i) r (0, λν ) for λ ν > 0; (ii) r (0, λ+ ) ( λ, 0) for νλ ν > 0; (iii) r (0, λ ) ( λ+, 0) for νλ ν < 0; (iv) r ( λν, 0) for λ ν < zeros in (0, 1) and is negative near Proof Inspired by the idea of [6] or see [11], we define the cut-off function of f as follows nφ p (s), s [ 1 n, 1 n [ ], f( f [n] n (s) := ) ] 1 n (ns ) + f( p n ), s ( 1 n, n ), [ f( n ) + ] 1 n (ns + ) + f( p n ), s ( n, 1 n ), f(s), s (, n ] [ n, + ). We consider the following problem (φ p (x )) = α(t)φ p (x + ) + β(t)φ p (x ) + λra(t)f [n] (x), 0 < t < 1, x(0) = x(1) = Clearly, we can see that lim n + f [n] (s) = f(s), (f [n] ) 0 = n and (f [n] ) =. Similarly to the proof of [16, Theorem 1.3], by Lemma 3.1 and Remar 3., there exists an unbounded continuum D ν[n] of solutions of the problem (3.8) emanating from ( λν rn, 0), such that D ν[n] (Φ ν {( λν rn, 0)}), and D ν[n] joins ( λν rn, 0) to ( λν r, ). Taing z n = ( λν rn, 0) and z = (0, 0), we have that z n z. So condition (i) in Lemma.5 is satisfied with z = (0, 0). Obviously r n = sup{λ + x (λ, x) D ν[n] } =, and accordingly, (ii) in Lemma.5 holds. (iii) in Lemma.5 can be deduced directly from the Arezela-Ascoli Theorem and the definition of f [n]. Therefore, by Lemma.5, lim sup n D ν[n] contains an unbounded connected component D ν with (0, 0) D ν. From lim n + f [n] (s) = f(s), (3.8) can be converted to the equivalent equation (3.1). Since D ν[n] Φ ν, we conclude D ν Φν. Moreover, D ν S ν by (1.). Similarly to the method of the proof of case of [16, Theorem 1.3], we can obtain that ( λν r, ) D ν. Theorem 3.6 Let (A1), (A) and (H4) hold. For some N and ν {+, }, assume that one (3.8)

8 The existence of nodal solutions for the Half-quasilinear p-laplacian problems 49 (i) r (0, λν ) for λ ν > 0; (ii) r (0, λ+ ) ( λ, 0) for νλ ν > 0; (iii) r (0, λ ) ( λ+, 0) for νλ ν < 0; (iv) r ( λν, 0) for λ ν < zeros in (0, 1) and is negative near Proof By the method similar to the proofs of Theorems 3.4 and 3.5 with obvious changes, we can obtain the result. Theorem 3.7 Let (A1), (A) and (H5) hold. For some N and ν {+, }, assume that one (i) r (0, + ) for λ ν > 0 or (ii) r (0, + ) (, 0) for νλ ν > 0, or νλν < 0; (iii) r (, 0) for λ ν < zeros in (0, 1) and is negative near Proof Define 1 n φ p(s), s (, n] [n, + ), nφ p (n)+f( n) n (s + n) + f( n), s ( n, n), nφ p (n) f(n) n (s n) + f(n), s (n, n), f [n] (s) := f(s), s [ n, n ] [ n, n], We consider the following problem [ f( n ) + ] 1 n (ns + ) + f( p n ), s ( n, 1 n [ ), f( n ) ] 1 n (ns ) + f( p n ), s ( 1 n, n ), nφ p (s), s [ 1 n, 1 n ]. (φ p (x )) = α(t)φ p (x + ) + β(t)φ p (x ) + λra(t)f [n] (x), 0 < t < 1, x(0) = x(1) = Clearly, we can see that lim n + f [n] (s) = f(s), (f [n] ) 0 = n and (f [n] ) = 1 n. Applying the similar method used in the proof of Theorem 3.5, we obtain an unbounded connected component D ν S ν with (0, 0) D ν. Similarly to the proof of Theorem 3.3, we get (, ) D ν, and the result is obtained. Theorem 3.8 Let (A1), (A) and (H6) hold. For some N and ν {+, }, assume that one (i) r (0, + ) for λ ν > 0;

9 50 Wenguo SHEN (ii) r (0, + ) (, 0) for νλ ν > 0, or νλν < 0; (iii) r (, 0) for λ ν < zeros in (0, 1) and is negative near Proof By the method similar to the proofs of Theorems 3.4 and 3.7 with obvious changes, we can obtain the result. Theorem 3.9 Let (A1), (A) and (H7) hold. For some N and ν {+, }, assume that one (i) There exists a λ ν ν > 0 for λν > 0, such that r (λν ν, + ); (ii) There exists a νλ ν ν > 0 for νλν > 0, such that r (, λ ) (λ+ +, + ); (iii) There exists a νλ ν ν < 0 for νλν < 0, such that r (, λ+ + ) (λ, + ); (iv) There exists a λ ν ν < 0 for λν < 0, such that r (, λν ν ). zeros in (0, 1) and is negative near Proof Define 1 n φ p(s), s (, n] [n, + ), nφ p (n)+f( n) n (s + n) + f( n), s ( n, n), nφ p (n) f(n) n (s n) + f(n), s (n, n), f [n] (s) := f(s), s [ n, n ] [ n, n], We consider the following problem [ f( n ) + ] 1 n (ns + ) + f( p n ), s ( n, 1 n [ ), f( n ) ] 1 n (ns ) + f( p n ), s ( 1 n, n ), 1 n φ p(s), s [ 1 n, 1 n ]. (φ p (x )) = α(t)φ p (x + ) + β(t)φ p (x ) + λra(t)f [n] (x), 0 < t < 1, x(0) = x(1) = Clearly, we can see that lim n + f [n] (s) = f(s), (f [n] ) 0 = 1 n and (f [n] ) = 1 n. Applying the similar method used in the proof of Theorem 3.4, we obtain an unbounded connected component D ν S ν with (, 0) D ν. Similarly to the proof of Theorem 3.3, we can show that (, ) D ν. Theorem 3.10 Let (A1), (A) and (H8) hold. For some N and ν {+, }, assume that one (i) There exists a λ ν ν > 0 for λν > 0, such that r (λν ν, + ); (ii) There exists a νλ ν ν > 0 for νλν > 0, such that r (, λ ) (λ+ +, + ); (iii) There exists a νλ ν ν < 0 for νλν < 0, such that r (, λ+ + ) (λ, + );

10 The existence of nodal solutions for the Half-quasilinear p-laplacian problems 51 (iv) There exists a λ ν ν < 0 for λν < 0, such that r (, λν ν ). zeros in (0, 1) and is negative near Proof Define nφ p (s), s (, n] [n, + ), nφ p(n)+f( n) n (s + n) + f( n), s ( n, n), nφ p (n) f(n) n (s n) + f(n), s (n, n), f [n] (s) := f(s), s [ n, n ] [ n, n], We consider the following problem [ f( n ) + ] 1 n (ns + ) + f( p n ), s ( n, 1 n [ ), f( n ) ] 1 n (ns ) + f( p n ), s ( 1 n, n ), nφ p (s), s [ 1 n, 1 n ]. (φ p (x )) = α(t)φ p (x + ) + β(t)φ p (x ) + λra(t)f [n] (x), 0 < t < 1, x(0) = x(1) = Clearly, we can see that lim n + f [n] (s) = f(s), (f [n] ) 0 = n and (f [n] ) = n. Applying the similar method used in the proof of Theorem 3.5, we obtain an unbounded connected component D ν Φν with (0, 0) D ν. By Theorem 3.6, we can show that (0, ) D ν. Remar 3.11 Clearly, if p =, α = β = 0, the problem (1.) was studied in [6,7] under the conditions (A) and (H5) or (H6), and the results of Theorems 3.6 and 3.10 improve those of Theorems 1.1 and 1. in [6] or [7]. Remar 3.1 Note that if α = β 0, the results of Theorems are equivalent to those of Theorems.1.8 of [11], respectively. Hence, Theorems extend the corresponding results of [11]. Remar 3.13 The nonlinear term of (1.) is not necessarily homogeneous linearizable at the origin and infinity because of the influence of the term αx + + βx. So the bifurcation results of [1 3,10,11] cannot be applied directly to obtain our results. Remar 3.14 We consider the cases of, (0, ), while the authors of [16] only studied the cases of, (0, ). Hence, Theorems extend the Theorem 4.1 of [16]. References [1] M. A. KRASNOSELSKII. Topological Methods in the Theory of Nonlinear Integral Equations. The Macmillan Co., New Yor, [] P. H. RABINOWITZ. Some global results for nonlinear eigenvalue problems. J. Functional Analysis, 1971, 7:

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