Abstract
We investigate the multiplicity of solutions for one-dimensional p-Laplacian Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems: The first states that there exists exactly one solution when nonlinearities cross no eigenvalue. The second guarantees that there exist exactly two solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross just the first eigenvalue. The third claims that there exist at least three solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross the first and second eigenvalues. We obtain the first and second theorem by considering the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem, and the contraction mapping principle in the p-Lebesgue space (when \(p\ge 2\)). We obtain the third result by Leray–Schauder degree theory.
Similar content being viewed by others
1 Introduction
The p-Laplacian boundary value problems with p-growth conditions arise in applications of nonlinear elasticity theory, electro-rheological fluids, and in non-Newtonian fluid theory of a porous medium (cf. [6, 7, 13]). A typical model of elliptic equation with p-growth conditions is
In particular, when \(\alpha =0\), the operator
is called the p-Laplacian. In general, when \(p=p(x)\), the \(p(x)\)-Laplacian problems are inhomogeneous, so they may have singular phenomena like \(\inf \varLambda =0\), where Λ is the set of the eigenvalues of the \(p(x)\)-Laplacian eigenvalue problem
where Ω is a bounded domain in \(R^{N}\), \(N\ge 1\), with a smooth boundary ∂Ω. But the eigenvalue problem when \(\alpha =0\), \(p(x)=p\) constant and \(1< p<\infty \) has no singular phenomena, i.e., \(\inf \varLambda >0\). It was proved in [9] that the eigenvalue problem when \(\alpha =0\), \(p(x)=p\) constant and \(1< p< \infty \) has a nondecreasing sequence of nonnegative eigenvalues \(\lambda _{j}\), obtained by the Ljusternik–Schnirelman principle, tending to ∞ as \(j\to \infty \), and the corresponding orthonomalized eigenfunctions \(\phi _{j}\), \(j=1, 2,\dots \), where the first eigenvalue \(\lambda _{1}\) is positive and simple and only eigenfunctions associated with \(\lambda _{1}\) do not change sign, the set of eigenvalues is closed, the first eigenvalue \(\lambda _{1}\) is isolated. Thus there are two sequences of eigenfunctions \((\phi _{j})_{j}\) and \((\mu _{j})_{j}\) corresponding to the eigenvalues \(\lambda _{j}\) such that the first eigenfunction \(\phi _{1}>0\) in the sequence \((\phi _{j})_{j}\) and the first eigenfunction \(\mu _{1}<0\) in the sequence \((\mu _{j})_{j}\).
Now in this paper, let \(\phi _{1}\) be the first positive orthonormalized eigenfunction corresponding to \(\lambda _{1}\).
In this paper we consider the multiplicity of solutions \(u\in W^{1,p}( \varOmega )\) for the following p-Laplacian Dirichlet boundary value problem with jumping nonlinearities when \(N=1\) and \(p(x)=p\);
where \(\varOmega =(c,d)\subset R\), \(c< d\), is an open interval, \(2\le p<\infty \) and \(p^{\prime }\) is given by \(\frac{1}{p}+\frac{1}{p ^{\prime }}=1\), \(u^{+}=\max \{u,0\}\), \(u^{-}=-\min \{u,0\}\), \(s\in R\), \(L^{p}(\varOmega )\) is the p-Lebesgue space, with its dual space \(L^{p^{\prime }}(\varOmega )\), and \(W^{1,p}(\varOmega )\) is the Lebesgue–Sobolev space.
Our problems are characterized as a jumping problem, which was first suggested in the suspension bridge equation as a model of the nonlinear oscillations in a differential equation
This equation represents a bending beam supported by cables under a load f. The constant b represents the restoring force if the cables stretch. The nonlinearity \(u^{+}\) models the fact that cables resist expansion but do not resist compression. Choi and Jung (cf. [2,3,4]) and McKenna and Walter (cf. [12]) investigated the existence and multiplicity of solutions for the single nonlinear suspension bridge equation with a Dirichlet boundary condition. In [1], the authors investigated the multiplicity of solutions of a semilinear equation
where Ω is a bounded domain in \(R^{n}\), \(n\ge 1\), with a smooth boundary ∂Ω, and A is a second-order linear partial differential operator when the forcing term is a multiple \(s\phi _{1}\), \(s\in R\), of the positive eigenfunction and the nonlinearity crosses eigenvalues.
Our main theorems are as follows:
Theorem 1.1
-
(i)
If \(2\le p<\infty \), \(a< b\), \(-\infty < a, b<\lambda _{1}\), then (1.1) has exactly one nontrivial solution for all s in a bounded interval. In particular, we have that if \(2\le p<\infty \), \(a< b\), \(-\infty < a, b<\lambda _{1}\) and \(s>0\), then \(u=( \frac{s}{\lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1}>0\) is a solution and if \(-\infty < a, b<\lambda _{1}\) and \(s<0\), then \(u=(\frac{s}{\lambda _{1}-a})^{\frac{1}{p-1}}\phi _{1}<0\) is a solution.
-
(ii)
If \(2\le p<\infty \), \(a< b\), \(\lambda _{j}< a, b<\lambda _{j+1}\), \(s\in R\) is bounded, \(j=1, 2,\dots \), (1.1) has exactly one nontrivial solution for all s in a bounded interval.
Theorem 1.2
-
(i)
If \(1< p<\infty \), \(a< b\), \(-\infty < a<\lambda _{1}<b< \lambda _{2}\), \(s\in R\) and \(s>0\), then (1.1) has no solution,
-
(ii)
If \(1< p<\infty \), \(a< b\), \(-\infty < a<\lambda _{1}<b<\lambda _{2}\), \(s\in R\) and \(s=0\), then (1.1) has exactly one solution \(u=0\).
-
(iii)
If \(2\le p<\infty \), \(a< b\), \(-\infty < a<\lambda _{1}<b<\lambda _{2}\), \(s\in R\), then there exists \(s_{1}<0\) such that for any s with \(s_{1}\le s<0\), (1.1) has exactly two solutions.
Theorem 1.3
-
(i)
If \(1< p<\infty \), \(a< b\), \(-\infty < a<\lambda _{1}\), \(\lambda _{2}< b<\lambda _{3}\), \(s\in R\) and \(s>0\), then (1.1) has no solution,
-
(ii)
If \(1< p<\infty \), \(a< b\), \(-\infty < a<\lambda _{1}\), \(\lambda _{2}< b< \lambda _{3}\), \(s\in R\) and \(s=0\), then (1.1) has exactly one solution \(u=0\).
-
(iii)
If \(2\le p<\infty \), \(a< b\), \(-\infty < a<\lambda _{1}\), \(\lambda _{2}< b<\lambda _{3}\), \(s\in R\), then there exists \(s_{1}<0\) such that for any s with \(s_{1}\le s<0\), (1.1) has at least three solutions.
For the proofs of (i) and (ii) of Theorem 1.1, (iii) of Theorem 1.2 and (iii) of Theorem 1.3, we use the contraction mapping principle under the condition \(p\ge 2\) and direct computations. The outline of the proofs of Theorems 1.1, 1.2 and 1.3 is as follows: In Sect. 2, we introduce some preliminaries and prove Theorem 1.1 by direct computations using the first eigenfunction and the contraction mapping principle. In Sect. 3, we prove Theorem 1.2 by a similar method to that of Theorem 1.1. In Sect. 4, we prove Theorem 1.3 by direct computations using the first eigenfunction, contraction mapping principle and Leray–Schauder degree theory under the condition \(p\ge 2\).
2 Preliminaries and proof of Theorem 1.1
Let \(L^{p}(\varOmega )\) be the Lebesgue space defined by
and \(W^{1,p}(\varOmega )\) be the Lebesgue–Sobolev space defined by
We introduce norms on \(L^{p}(\varOmega )\) and \(W^{1,p}(\varOmega )\), respectively, by
Let us define the operator \(-L_{p}\) by
We first consider the problem:
where \(1< p<\infty \) and \(f\in L^{r}(\varOmega )\), \(r>1\). Then (2.1) has a unique solution \(u\in C^{1}(\bar{\varOmega })\) which is of the form
where \(g_{p}(t)=|t|^{p-2}t\) for \(t\neq 0\), \(g_{p}(0)=0\) and its inverse \(g^{-1}_{p}\) is \(g^{-1}_{p}(t)=t^{\frac{1}{p-1}}\) if \(t>0\) and \(g^{-1}_{p}(t)=-|t|^{\frac{1}{p-1}}\) if \(t<0\), and \(c_{f}\) is the unique constant such that \(u(d)=0\). By [10, Lemma 2.1] or [11, Lemma 4.2], the solution operator S is such that \(S:L^{p}(\varOmega )\to C^{1} (\bar{ \varOmega })\) is continuous and, by [5, Corollary 2.3], the embedding \(S:L^{p}(\varOmega )\to C(\bar{\varOmega })\) is continuous and compact. By [8], we also have a Poincaré-type inequality.
Lemma 2.1
Let \(1< p<\infty \). Then the embedding
is continuous and compact, and for every \(u\in C^{\infty }_{0}(\varOmega )\) we have
for a positive constant C independent of u.
By Lemma 2.1, we obtain the following:
Lemma 2.2
Assume that \(1< p<\infty \), \(f(x,u)\in L^{p}(\varOmega )\). Then the solutions of the problem
belong to \(W^{1,p}(\varOmega )\).
For given \(v(u)\in L^{p}(\varOmega )\) and \(h(x)\in L^{p}(\varOmega )\), the equation
is equivalent to the equation
We observe that
Proof of Theorem 1.1
(i) We assume that \(-\infty < a, b<\lambda _{1}\). Let us choose \(\mu >0\) and \(\epsilon >0\) so that \(-\mu +\epsilon < a,b< \lambda _{1}-\epsilon \), and choose \(\tau =\frac{a+b}{2}\). Then problem (1.1) can be rewritten as
or equivalently,
We have that
Let us set the right-hand side of (2.3) as
Then \(T(u)\) satisfies
Since \(|\cdot |^{p-1}\) is continuous, for given \(u\in W^{1,p}(\varOmega )\), there exists \(v\in W^{1,p}(\varOmega )\) in a small neighborhood of u such that \(\frac{b-a}{2}|u|^{p-1}+s\phi _{1}^{p-1}\) and \(\frac{b-a}{2}|v|^{p-1}+s\phi _{1}^{p-1}\) have the same sign. We can also check that for any \(2\le p<\infty \), \(s<0\) and for any u and v, where v is in a small neighborhood of u such that \(\frac{b-a}{2}|u|^{p-1}+s \phi _{1}^{p-1}\) and \(\frac{b-a}{2}|v|^{p-1}+s\phi _{1}^{p-1}\) have the same sign, we have
Thus for any u and v, where v is in a small neighborhood of u such that \(\frac{b-a}{2}|u|^{p-1}+s\phi _{1}^{p-1}\) and \(\frac{b-a}{2}|v|^{p-1}+s\phi _{1}^{p-1}\) have the same sign, we have
Since \(\lambda _{1}-\tau >\frac{b-a}{2}\) and \(2\le p<\infty \), we have \(\frac{(\frac{b-a}{2})^{\frac{1}{p-1}}}{\lambda _{1}-\tau }<1\). Thus T is a contraction mapping on \(L^{p}(\varOmega )\). Thus T has a unique solution in \(L^{p}(\varOmega )\). Thus (1.1) has a unique solution.
(ii) We assume that \(\lambda _{j}< a, b<\lambda _{j+1}\), \(j=1, 2,\dots \) Let us choose \(\epsilon >0\) so that \(\lambda _{j}+ \epsilon < a, b<\lambda _{j+1}-\epsilon \). Let us set \(\tau = \frac{a+b}{2}\). Then (1.1) can be rewritten as
We observe that
By the same process as that of the proof of Theorem 1.1(i), the mapping on \(L^{p}(\varOmega )\) given by
satisfies
for any u and v, where v is in a small neighborhood of u such that \(\frac{b-a}{2}|u|^{p-1}+s\phi _{1}^{p-1}\) and \(\frac{b-a}{2}|v|^{p-1}+s \phi _{1}^{p-1}\) have the same sign, and \(2\le p<\infty \). Since \(\frac{\lambda _{j+1}-\lambda _{j}}{2}>\frac{b-a}{2}\) and \(2\le p< \infty \), we have \(\frac{2}{\lambda _{j+1}-\lambda _{j}}(\frac{b-a}{2})^{ \frac{1}{p-1}}<1\). It follows that N is a contraction mapping on \(L^{p}(\varOmega )\). Therefore N has a unique solution in \(L^{p}( \varOmega )\). Thus (1.1) has a unique solution. □
3 Proof of Theorem 1.2
(i) (For the case \(s>0\)) We assume that \(1< p<\infty \), \(a< b\), \(-\infty < a<\lambda _{1}<b<\lambda _{2}\) and \(s>0\). Then (3.1) can be rewritten as
Taking the inner product with \(\phi _{1}\), we have
The left-hand side of (3.1) is equal to 0. On the other hand, the right-hand side of (3.1) is positive because \(b-\lambda _{1}>0\), \(-(a-\lambda _{1})>0\) and \(s\phi _{1}^{p-1}>0\) for \(s>0\). Thus if \(s>0\), then there is no solution for (1.1).
(ii) (For the case \(s=0\)) If \(s=0\), then (3.2) is reduced to the equation
i.e.,
Since \(b-\lambda _{1}>0\) and \(-(a-\lambda _{1})>0\), the only possibility for (3.3) to hold is that \(u=0\).
(iii) (For the case \(s<0\)) We assume that \(2\le p<\infty \), \(a< b\), \(-\infty < a<\lambda _{1}<b<\lambda _{2}\) and \(s<0\). Let V be a subspace of \(L^{p}(\varOmega )\) spanned by \(\phi _{1}\) and W be the orthogonal complement of V in \(L^{p}(\varOmega )\). Then
Let P be a orthogonal projection in \(L^{p}(\varOmega )\) onto V and \(I-P\) be the orthogonal projection onto W. Then
Let \(u\in L^{p}(\varOmega )\). Then u can be written as
We note that P commutes with \(D=\frac{d}{\,dx}\). Thus (1.1) is equivalent to a pair of equations
We claim that for fixed \(v\in V\), (3.4) has a unique solution \(w(v)\) when \(2\le p\le \infty \). In fact, we suppose that (3.4) has two solutions \(w_{1}\), \(w_{2}\) for fixed \(v\in V\). Let us set \(\alpha _{v}(w)=b|v+w|^{p-2}(v+w)^{+} -a|v+w|^{p-2}(v+w)^{-}\). Then we have
Taking the inner product of (3.6) with \(w_{1}-w_{2}\), we have
The right-hand side of (3.7) is equal to
for \(0<\theta <1\). On the other hand, the left-hand side of (3.6) is equal to
by mean value theorem. On the other hand, by (3.8) and (3.9), we have
which is a contradiction because \(b<\lambda _{2}\). Thus \(w_{1}=w_{2}\). Thus for fixed \(v\in V\), every solution of (3.4) is unique. We note that \(w=0\) is a solution of (3.4) for every \(v\in V=PH\), \(v>0\) or \(v<0\) everywhere in Ω. If \(v>0\), then
If \(v>0\), then \((I-P) (b|v|^{p-2}v^{+} -a|v|^{p-2}v^{-} +s\phi _{1} ^{p-1} )=(I-P)b|v|^{p-2}v=0\). If \(v<0\), then \((I-P) (b|v|^{p-2}v ^{+} -a|v|^{p-2}v^{-} +s\phi _{1}^{p-1} )=(I-P)(a|v|^{p-2}v+s\phi _{1}^{p-1})=0\). Thus (1.1) is reduced to
where \(v=c\phi _{1}\), \(c\in R\).
If \(c>0\), then
If \(c<0\), then
Thus (1.1) has exactly two solutions.
4 Proof of Theorem 1.3
Lemma 4.1
(A priori bound)
Assume that \(1< p<\infty \), \(-\infty < a< \lambda _{1}\), \(\lambda _{2}< b<\lambda _{3}\), \(s\in R\). Then there exist \(s_{1}<0\), \(s_{2}>0\) and a constant \(C>0\) depending only on a and b such that for any s with \(s_{1}\le s\le s_{2}\), any solution u of (1.1) satisfies \(\|u\|_{W^{1,p}(\varOmega )}< C\).
Proof
Let u be any solution of (1.1). Suppose that any solution of (1.1) is not bounded. Then there exists a sequence \((u_{n})_{n}\) such that \(\|u_{n}\|_{W^{1,p}(\varOmega )}\to \infty \) so that
or equivalently,
Let us set \(w_{n}=\frac{u_{n}}{\|u_{n}\|_{W^{1,p}(\varOmega )}}\). Then \((w_{n})_{n}\) is bounded, so by passing to a subsequence if necessary, which we still denote by \((w_{n})_{n}\), we get that \((w_{n})_{n} \to w\) weakly for some w in \(W^{1,p}(\varOmega )\). Dividing (4.1) by \(\|u_{n}\|^{p-1}_{W^{1,p}(\varOmega )}\), we have
i.e.,
Since, by Lemma 2.1, the embedding \(W^{1,p}(\varOmega )\hookrightarrow L ^{p}(\varOmega )\) is compact, and \((-L_{p})^{-1}\) is compact operator, \((w_{n})_{n}\to w\) strongly in \(W^{1,p}(\varOmega )\). Taking the limit of (4.2) as \(n\to \infty \), we have
By Theorem 1.1(i), (4.3) has only the trivial solution, which is absurd, because \(\|w\|_{W^{1,p}(\varOmega )}=1\). Thus the lemma is proved. □
We shall consider the Leray–Schauder degree on a large ball.
Lemma 4.2
Assume that \(1< p<\infty \), \(-\infty < a<\lambda _{1}\), \(\lambda _{2}< b<\lambda _{3}\). Then there exist a constant \(R>0\) depending on a, b, s, \(s_{1}<0\) and \(s_{2}>0\) such that for any s with \(s_{1}\le s\le s_{2}\), the Leray–Schauder degree
where \(-L_{p} u=-(|u^{\prime }|^{p-2}u^{\prime })^{\prime }\).
Proof
Let us consider the homotopy
By Theorem 1.3(ii), for any \(s>0\), (1.1) has no solution. Thus there exist \(s_{2}>0\) and a large \(R>0\) such that (4.3) has no zero in \(B_{R}(0)\) for any \(s\ge s_{2}\), and by the a priori bound in Lemma 4.1, there exists \(s_{1}<0\) such that for any s with \(s_{1}\le s\le s _{2}\), all solutions of
satisfy \(\|u\|_{W^{1,p}(\varOmega )}\le R\) and (4.3) has no zero on \(\partial B_{R}\) for any s with \(s_{1}\le s\le s_{2}\). Since
by homotopy arguments, for any any s with \(s_{1}\le s\le s_{2}\), we have
for any \(0\le \lambda \le 1\). Thus the lemma is proved. □
Lemma 4.3
Let K be a compact set in \(L^{p}(\varOmega )\). Let \(\xi >0\) a.e. Then there exists a modulus of continuity \(\alpha :R \to R\) depending only on K and ξ such that
It follows that
and
Proof
For any \(\tau \in K\), let \(\tau _{n}=(|\tau |-\frac{\xi }{\eta })^{+}\). Since \(0\le \tau _{n} \le |\tau |\) and \(\tau _{n}(x)\to 0\) as \(\eta \to 0\) a.e., it follows that \(\|\tau _{n}\|_{L^{p}(\varOmega )}\to 0\) for all \(\tau \in K\). We claim that, for a given \(\epsilon >0\), there exists \(\delta >0\) such that if \(\tau \in K\), then \(\|\tau _{n}\|_{L^{p}(\varOmega )}\le 2\epsilon \) for all \(\eta \in [0,\delta ]\). Choose \(\{\tau _{i}, i=1,\dots ,N\}\) as an ϵ net for K. Choose δ so that \(\|(\tau _{i})_{ \delta }\|_{L^{p}(\varOmega )}<\epsilon \) for \(i=1,\dots , N\). Then for any \(\tau \in K\), there exists \(\tau _{k}\), α, \(\|\alpha \|_{L^{P}( \varOmega )}<\epsilon \) that \(\tau =\tau _{K}+\alpha \). Since \((u+v)^{+} \le u^{+}+v^{+}\), we have \(\|\tau _{\delta }\|_{L^{P}(\varOmega )}\le ( \tau _{K})_{\delta }+|\alpha |\) and therefore \(\|\tau _{\eta }\|_{L^{P}( \varOmega )}\le \|\tau _{\delta }\|_{L^{P}(\varOmega )}+\|\alpha \|_{L^{p}( \varOmega )}\le 2\epsilon \). □
Lemma 4.4
Assume that \(1< p<\infty \), \(-\infty < a<\lambda _{1}, \lambda _{2}<b<\lambda _{3}\). Then there exist a constant \(R>0\) depending on a, b, s and \(s_{1}<0\) such that for \(s_{1}\le s<0\), the Leray–Schauder degree
where \(u_{0}=(\frac{s}{\lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1}>0\) is a positive solution of (1.1).
Proof
Let us set \(M=(-L_{p}-b g_{p})^{-1}\). Then (1.1) can be rewritten as
or equivalently,
where \(M=(-L_{p}-b g_{p})^{-1}\). The operator M is compact on \(L^{p}(\varOmega )\), and the set \(K=M(\bar{B})\), where B̄ is the closed unit ball in \(L^{p}(\varOmega )\). Then K is a compact set. Let us set \(\gamma =\min \{b-\lambda _{2},\lambda _{3}-b\}\). We can observe that \(\|M(u)\|_{L^{p}(\varOmega )}\le \frac{1}{\gamma }\|g_{p}^{-1}(u)\|_{L ^{p}(\varOmega )}\). Let α be the modulus continuity given in Lemma 4.3 corresponding to K and \(\xi =M(s\phi _{1}^{p-1})=(\frac{s\phi _{1}}{\lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1}\), and choose \(\epsilon >0\) so that
We have
For \(u\in (\frac{s}{\lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1}+(|s| \epsilon v)^{\frac{1}{p-1}}\) with \(v\in \bar{B}\),
since \((\frac{s}{\lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1}>0\). Then \(T(u)=M(b|u|^{p-2}u^{+} -a|u|^{p-2}u^{-} -b|u|^{p-2}u+s\frac{\phi _{1}}{ \lambda _{1}-b})\) can be rewritten as
If u is a solution of (4.4), then \(u=Tu\) and, by Lemma 4.3,
It follows that
Thus we have shown that any solution \(u\in (\frac{s}{\lambda _{1}-b})^{ \frac{1}{p-1}}\phi _{1}+|s|\epsilon \bar{B}\) of (4.4) belongs to \((\frac{s}{\lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1}+\frac{1}{4}|s| \epsilon \bar{B.}\) This estimate holds if we replace \(b|u|^{p-2}u^{+} -a|u|^{p-2}u^{-} -b|u|^{p-2}u\) by \(\lambda (b|u|^{p-2}u^{+} -a|u|^{p-2}u ^{-} -b|u|^{p-2}u)\) with \(0\le \lambda \le 1\). Thus the equation
has no solution on the boundary of the ball \(B_{\epsilon |s|}((\frac{s}{ \lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1})\) for \(0\le \lambda \le 1\). By the homotopy invariance degree,
is defined for \(0\le \lambda \le 1\) and is independent of λ. For \(\lambda =0\),
since \(u=(\frac{s}{\lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1}\) is the unique solution of the equation, and, since there are 2 eigenvalues \(\lambda _{1}\), \(\lambda _{2}\) of \(-L_{p}\) to the left of b, the operator \(I-b(-L_{p})^{-1}\) has two negative eigenvalues, while all the rest are positive. When \(\lambda =1\), we have
Thus by the homotopy invariance of degree, we have
Thus the lemma is proved. □
Lemma 4.5
Assume that \(1< p<\infty \), \(-\infty < a<\lambda _{1}\), \(\lambda _{2}< b<\lambda _{3}\) and \(s_{1}<0\). Then there exist a constant \(\epsilon >0\) depending on a, b, s such that for \(s_{1}\le s<0\), the Leray–Schauder degree
where \(u_{1}=-(\frac{s}{a-\lambda _{1}})^{\frac{1}{p-1}}\phi _{1}<0\) is a negative solution of (1.1).
Proof
We can prove this lemma by an almost identical argument as that of Lemma 4.4. □
Proof of Theorem 1.3
The proofs of Theorem 1.3(i)–(ii) are the same as those of Theorem 1.2(i)–(ii).
(iii) By Lemmas 4.4 and 4.5, there is a solution \((\frac{s}{\lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1}>0\) in \(B_{|s|\epsilon }((\frac{s}{ \lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1})\) and a solution \(-(\frac{s}{a- \lambda _{1}})^{\frac{1}{p-1}}\phi _{1}<0\) in \(B_{|s|\epsilon }(-(\frac{s}{a- \lambda _{1}})^{\frac{1}{p-1}}\phi _{1})\). We may assume that \(\epsilon <(\frac{1}{b-\lambda _{1}})^{\frac{1}{p-1}}\) and \(\epsilon <(\frac{1}{ \lambda _{1}-a})^{\frac{1}{p-1}}\). Then these two balls are disjoint. This gives two solutions for any n. There is a large ball \(B_{R}\) centered at the origin and containing \(B_{|s|\epsilon }((\frac{s}{ \lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1})\) and \(B_{|s|\epsilon }(-(\frac{s}{a- \lambda _{1}})^{\frac{1}{p-1}}\phi _{1})\). Since
we have
Thus there exists a third solution in \(B_{R}(0)\backslash (B_{|s| \epsilon }((\frac{s}{\lambda _{1}-b})^{\frac{1}{p-1}}\phi _{1})\cup B _{|s|\epsilon }(-(\frac{s}{a-\lambda _{1}})^{\frac{1}{p-1}}\phi _{1})\). Thus we have proved Theorem 1.3(iii). □
References
Choi, Q.H., Jung, T.: An application of a variational reduction method to a nonlinear wave equation. J. Differ. Equ. 117, 390–410 (1995)
Choi, Q.H., Jung, T.: A nonlinear suspension bridge equation with nonconstant load. Nonlinear Anal. TMA 35, 649–668 (1999)
Choi, Q.H., Jung, T.: Multiplicity results for the nonlinear suspension bridge equation. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 9, 29–38 (2002)
Choi, Q.H., Jung, T., McKenna, P.J.: The study of a nonlinear suspension bridge equation by a variational reduction method. Appl. Anal. 50, 73–92 (1993)
del Pino, M., Elgueta, M., Manasevich, R.: A homotopic deformation along p of a Leray–Schauder degree result and existence for \((|u^{\prime }|^{p-2}u ^{\prime })^{\prime }+f(x,u)=0\), \(u(0)=u(T)=0\), \(p>1\). J. Differ. Equ. 80, 1–13 (1998)
Ghergu, M., Rádulescu, V.: Existence and nonexistence of entire solutions to the logistic differential equations. Abstr. Appl. Anal. 2003(17), 995–1003 (2003)
Ghergu, M., Rádulescu, V.: Singular Elliptic Problems, Bifurcation and Asymptotic Analysis. Oxford Lecture Series in Mathematics and Its Applications. Oxford University Press, London (2008)
Kim, Y.-H., Wang, L., Zhang, C.: Global bifurcation for a class of degenerate elliptic equations with variable exponents. J. Math. Anal. Appl. 371, 624–637 (2010)
Lê, A.: Eigenvalue problems for the p-Laplacian. Nonlinear Anal. 64, 1057–1099 (2006)
Manásevich, R., Mawhin, J.: Periodic solutions for nonlinear systems with p-Laplacian-like operators. J. Differ. Equ. 145, 367–393 (1998)
Manásevich, R., Mawhin, J.: Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators. J. Korean Math. Soc. 37, 665–685 (2000)
McKenna, P.J., Walter, W.: Nonlinear oscillations in a suspension bridge. Arch. Ration. Mech. Anal. 98, 167–177 (1987)
O’Regan, D.: Some general existence principles and results for \((\phi (y^{\prime }))^{\prime }=qf(t,y,y ^{\prime })\), \(0< t<1\). SIAM J. Math. Anal. 24, 648–668 (1993)
Acknowledgements
Not applicable
Availability of data and materials
Not applicable
Funding
Tacksun Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (NRF-2017R1A2B4005883). Q-Heung Choi was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2017R1D1A1B03030024).
Author information
Authors and Affiliations
Contributions
TJ introduced the main ideas of multiplicity study for this problem. Q-HC participated in applying the method for solving this problem and drafted the manuscript. All authors contributed equally to reading and approving the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Additional information
Abbreviations
Not applicable
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Jung, T., Choi, QH. Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities. Bound Value Probl 2019, 56 (2019). https://doi.org/10.1186/s13661-019-1165-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-019-1165-5