Existence and multiplicity of positive solutions for one-dimensional $p$-Laplacian problem with sign-changing weight

1.   Introduction

Consider the following one-dimensional $p$-Laplacian problem

where $\varphi_p(s): = |s|^{p-2}s$, $p > 1$, $\lambda > 0$ is a parameter, $f\in C(\mathbb{R}, \mathbb{R})$ with $sf(s) > 0$ for $s\neq 0$ and $m\in C[0, 1]$ changes sign.

Notice that (1.1) is the one-dimensional version of the Dirichlet problem associated with the $p$-Laplacian equation

where $\lambda > 0$ is a parameter, $m\in C(\bar{\Omega})$, $f\in C(\mathbb{R}, \mathbb{R})$, $\Omega$ is a bounded domain in $\mathbb{R}^N, N\geq 1$.

Recently, the $p$-Laplacian problems with sign-definite weight have been studied by many authors. For example, Del Pino et al. ^[1] established the global bifurcation theorem for problem (1.1) in the case $m\equiv 1$. When $m(x)\geq 0$ and $m$ may be singular at $x = 0$ and/or $x = 1$, Lee et al. ^[2,3] obtained many different types of global existence results of positive solutions for problem (1.1). Dai et al. ^[4,5] established a Dancer-type unilateral global bifurcation result for one-dimensional $p$-Laplacian problem (1.1).

On the other hand, in high-dimensional case, Del Pino and Manásevich ^[6] studied the global behavior of continuum of positive solutions for problem (1.2) on the general domain of $\mathbb{R}^N$. In 2022, Ye and Han^[7] studied the global structure of problem (1.2) by using bifurcation technique.

To the authors' best knowledge, the study of solutions for $p$-Laplacian problems with sign-changing weight can be traced back to Drábek and Huang ^[8]. In 1997, the authors proved a Dancer-type bifurcation result for problem (1.2) with sign-changing weight. Since then, there have been many studies on problems (1.1) and (1.2) with sign-changing weight, see ^[9,10,11,12]. For example, Ma, Liu and Xu ^[9] in 2013 used bifurcation technique to show that (1.1) has a nodal solution, where the weight function $m$ changes sign. In 2014, Dai ^[10] also proved a Dancer-type unilateral global bifurcation result for problem (1.2) with sign-changing weight $m$ on the unit ball of $\mathbb{R}^N$. In 2015, Sim and Tanaka ^[11] proved in their Theorem 1.1 that the solution set of problem (1.1) with $m$ changes sign has an $S$-shaped continuum. Here $m$ satisfies

(F1) there exist $x_1, x_2\in [0, 1]$ such that $x_1 < x_2$, $m(x) > 0$ on $(x_1, x_2)$, and $m(x)\leq 0$ on $[0, 1]\setminus[x_1, x_2]$. As applications of this bifurcation result, they determined the intervals of the parameter $\lambda$ in which the problem (1.1) has one, two, or three positive solutions. In 2019, Chen and Ma ^[12] extended [11, Theorem 1.1] to the radial problem

where $\lambda > 0$ is a parameter, $f\in C([0, \infty), [0, \infty))$, $f(0) = 0, f(s) > 0$ for $s > 0$, and $m$ is a sign-changing function satisfying $H(B) = \{m\in C(\bar{B})\ \text{is radially symmetric}|m(x) > 0, x\in \Omega_1\ \text{and}\ m(x)\leq 0, x\in \bar{B}\backslash \Omega_1 \}$ with the annular domain $\Omega_1 = \{x\in \mathbb{R}^N: r_1 < |x| < r_2\}\subset B$ for some $0 < r_1 < r_2 < 1$.

Note that the solution of (1.3) is the radially symmetric solutions of the $N$-dimensional Dirichlet problem (1.2), where $\Omega = B$ is the unit ball of $\mathbb{R}^N, N\geq 2$.

It is worth remarking that ^[11,12] only studied the case of $f_0\in (0, \infty)$ and $f_\infty\in (0, \infty)$, which means that there is a constant $C > 0$ such that $f(s)\leq Cs^{p-1}$ for all $s\geq 0$, where $f_0: = \underset{s\to0}\lim\frac{f(s)}{s^{p-1}}, f_\infty: = \underset{s\to\infty}\lim\frac{f(s)}{s^{p-1}}$. However, if $f$ is superlinear near $\infty$ (i.e., $f_\infty = \underset{s\to\infty}\lim\frac{f(s)}{s^{p-1}} = \infty$), similar results have not been studied. One possible reason is that in this case we cannot use standard bifurcation techniques by linearization. Another reason is that we cannot prove the direction of bifurcation using the method in ^[11].

Naturally, a new question is, can we study the existence and multiplicity of positive solutions to problem (1.1) when $f_0\in (0, \infty)$, $f_\infty = \infty$ and the weight function $m$ changes sign?

Motivated by the interesting studies of ^[11,12] and some earlier works in the literature, in present paper, we prove that an unbounded subcontinuum of positive solutions of (1.1) bifurcates from the trivial solution and grows to the left from the initial point, to the right at some point and to the left near $\lambda = 0$. Roughly speaking, we obtain that there exists a reversed $S$-shaped continuum in the positive solution set of problem (1.1).

Throughout this paper, we assume that

(H1) $f\in C(\mathbb{R}, \mathbb{R})$ with $sf(s) > 0$ for $s\neq 0$;

(H2) $m\in C[0, 1]$ changes sign and ${{\rm{meas}}}$$\{x\in [0, 1]|m(x) = 0\} = 0$;

(H3) there exist constants $\alpha > 0, f_0 > 0$, and $f_1 > 0$ such that

(H4) $f_\infty = \underset{s\to\infty}\lim\frac{f(s)}{s^{p-1}} = \infty;$

(H5) there exists $s_{0} > 0$ such that $0\leq s\leq s_0$ implies that

where $\mu_1$ is the first eigenvalue for the following linear eigenvalue problem

It is well-known that $\mu_1$ is simple, isolated and the associated eigenfunction $\phi_1$ has fixed sign in $[0, 1)$ (see for example ^[11,13]).

Arguing the shape of bifurcation, we have the following main result:

Theorem 1.1 (see Figure 1). Assume that (H1)–(H5) hold. Then there exist $\lambda_{\ast}\in(0, \frac{\mu_1}{f_0})$ and $\lambda^{\ast} > \frac{\mu_1}{f_0}$ such that

(ⅰ) (1.1) has at least one positive solution if $0 < \lambda < \lambda_{\ast}$;

(ⅱ) (1.1) has at least two positive solutions if $\lambda = \lambda_{\ast}$;

(ⅲ) (1.1) has at least three positive solutions if $\lambda_{\ast} < \lambda < \mu_1/f_0$;

(ⅳ) (1.1) has at least two positive solutions if $\mu_1/f_0 < \lambda\leq\lambda^{\ast}$;

(ⅴ) (1.1) has at least one positive solution if $\lambda = \lambda^{\ast}$;

(ⅵ) (1.1) has no positive solution if $\lambda > \lambda^{\ast}$.

Remark 1.1. Let us consider the function

Obviously, $f$ satisfies (H3) and (H4) with $\alpha = 1, f_0 = k, f_1 = 1$. It is easy to see that if $k > 0$ is sufficiently large, then the function $f$ satisfies (H5).

Remark 1.2. We note that, in ^[11], condition (F1) is the key condition to obtain the $S$-shaped continuum in the positive solutions set of problem (1.1). In other words, if the condition (F1) is replaced by (H1), the method in ^[11] cannot prove whether there is $S$-shaped continuum in the solution set of problem (1.1), even if $f_0, f_\infty \in (0, +\infty)$.

Remark 1.3. Recently, many scholars have studied other problems with $p$-Laplacian operator, such as about characterization of solutions, see ^{[14,15,16,17]}; Extensions of a $p$-Laplacian operator to higher order operator, see ^[18,19]; Aplication of $p$-Laplacian operator to a physical phenomena, see ^[20].

The rest of this paper is arranged as follows. In Section 2, we show global bifurcation phenomena from the trivial branch. Section 3 is devoted to showing that there are at least two direction turns of the continuum and completing the proof of Theorem 1.1.

2.   Existence of unbounded continuum

Let $X = \{u\in C[0, 1]: u(0) = u(1) = 0\}$ with the norm $||u||_\infty = \sup_{x\in [0, 1]} u(x)$. Let $E$ be the Banach space $C_0^1[0, 1]$ with the norm $||u|| = \max\{||u||_\infty, ||u'||_\infty\}$. Let $Y = L^1(0, 1)$ with its usual normal $||\cdot||_{L^1}$.

Consider the following boundary value problem

where $h\in L^1(0, 1)$. Problem (2.1) is equivalently

where $a:Y\to \mathbb{R}$ is a continuous function satisfying

It is known that $G_p:Y\to E$ is continuous and maps equi-integrable sets of $E$ into relatively compacts of $E$ (see ^[2]).

Lemma 2.1 ([11, Lemma 2.1]). Assume that (H1)–(H3) hold. Then there exists an unbounded subcontinuum $\mathcal{C}$ of the set of solution of problem (1.1) in $\mathbb{R}\times X$ bifurcating from $(\mu_1/f_0, 0)$ such that

Here $P = \{u\in C[0, 1]:u(t)\geq 0\}$ is the positive cone in $X$.

Lemma 2.2. Assume that (H1) and (H2) hold. Let $u$ be a positive solution of (1.1). If there is a constant $f_*\in (0, \infty)$ such that $0\leq f(s)\leq f_*s^{p-1}$. Then

Proof. By Rolle's theorem, there exists $\tau\in (0, 1)$ such that $u'(\tau) = 0$. Integrating the equation of (1.1) over $[x, \tau]$, we have

By (H1) and (H2), we get

The proof is complete.

Lemma 2.3 ([9, Lemma 10]). Let (H2) holds. Let $I = [a, b]$ be such that $I\subset I^+: = \{x\in [0, 1]|m(x) > 0\}$ and ${\rm{meas}}\ I > 0$. Let $g_n: [0, 1]\to (0, \infty)$ be such that

Let $y_n\in E$ be a solution of the equation

Then the number of zeros of $y_n|_I$ goes to infinity as $n\to \infty$.

Lemma 2.4. Assume that (H1)–(H3) hold. If $f_0\in (0, \infty)$ and $f_\infty\in(0, \infty)$, then for any $\lambda\in (\mu_1/f_\infty, \mu_1/f_0)\cup (\mu_1/f_0, \mu_1/f_\infty)$, problem (1.1) has one solution $u$ such that $u$ is positive on $(0, 1)$.

Proof. The arguments are quite similar to those from the proof of Theorem 11 in ^[9]. However, for the sake of completeness, we give a sketch of the proof below.

Let $\zeta\in C(\mathbb{R})$ such that $f(s) = f_0\varphi_p(s)+\zeta(s)$ with $\lim_{s\to 0}\zeta(s)/\varphi_p(s) = 0$. By Lemma 2.1, there exists an unbounded subcontinuum $\mathcal{C}$, such that

Let $(\lambda_n, u_n)\in \mathcal{C}$ satisfy $\lambda_n+||u_n||\to \infty$. We note that $\lambda_n > 0$ for all $n\in \mathbb{N}$ since $(0, 0)$ is the only solution of problem (1.1) for $\lambda = 0$ and $\mathcal{C}\cap (\{0\}\times E) = \emptyset$.

We divide the rest proofs into two steps.

Step 1. We show that there exists a constant $M$ such that $\lambda_n\in (0, M]$ for $n\in \mathbb{N}$ large enough.

On the contrary, we suppose that $\lim_{n\to \infty}\lambda_n = \infty$. Note that

where

Conditions (H1), (H3) and the fact that $f_\infty\in (0, \infty)$ imply that there exists a positive constant $\rho$ such that $\tilde f_n(x)\geq \rho$ for any $x\in (0, 1)$. By Lemma 2.3, we get that $u_n$ must change its sign in $(0, 1)$ for $n$ large enough, and this contradicts the fact that $u_n\in \mathcal{C}$.

Step 2. We show that $\mathcal{C}$ joins $(\mu_1/f_0, 0)$ to $(\mu_1/f_\infty, \infty)$ It follows from Step 1 that $||u_n||\to \infty$. Let $\xi\in C(\mathbb{R})$ be such that $f(s) = f_\infty\varphi_p(s)+\xi(s)$. Then $\lim_{s\to \infty}\xi(s)/\varphi_p(s) = 0$. Let $\tilde \xi(u) = \max_{u\leq s\leq 2u}|\xi(s)|$. Then $\tilde \xi$ is nondecreasing and

We divide the equation

by $||u_n||$ and set $\bar u_n = u_n/||u_n||$. Since $\bar u_n$ is bounded in $E$, after taking a subsequence if necessary, we have $\bar u_n\rightharpoonup \bar u$ for some $\bar u\in E$ and $\bar u_n\to \bar u$ in Y with $||\bar u|| = 1$. Moreover, from (2.2) and the fact that $\tilde \xi$ is nondecreasing, we have

since

By the continuity and compactness of $G_p$, it follows that

where $\bar\mu = \lim_{n\to \infty} \lambda_n$, again choosing a subsequence and relabeling if necessary.

It is clear that $||\bar u|| = 1$ and $\bar u\in \mathcal{C}$ since $\mathcal{C}$ is closed in $\mathbb{R}\times E$. Thus, $\bar\mu f_\infty = \mu_1$, i.e., $\bar \mu = \mu_1/f_\infty$. Therefore, $\mathcal{C}$ joins $(\mu_1/f_0, 0)$ to $(\mu_1/f_\infty, \infty)$.

Lemma 2.5. (^[21]) Let $X$ be a Banach space and let $C_n$ be a family of closed connected subsets of $X$. Assume that:

(ⅰ) there exist $z_n \in C_n$, $n = 1, 2, \cdots,$ and $z^*\in X$, such that $z_n\to z^\ast;$

(ⅱ) $r_n = \sup \{||x|||x\in C_n\} = \infty;$

(ⅲ) for every $R > 0$, $(\bigcup_{n = 1}^{\infty}C_n)\cap \bar B_R(0)$ is a relatively compact set of $X$.

Then $D: = \limsup_{n\to \infty}C_n$ is unbounded, closed and connected.

Lemma 2.6. If $f_0\in (0, \infty)$ and $f_\infty = \infty$, then the unbounded subcontinuum $\mathcal{C}$ of positive solutions for (1.1) joins $(\frac{\mu_1}{f_0}, 0)$ to $(0, \infty)$.

Proof. Note that Lemma 2.6 cannot be proved using standard bifurcation techniques by linearization. To overcome this difficulty we shall employ a limiting procedure. Let us define a function $f_n$ as the following

Next, we consider the following problem

Clearly, $\lim_{n\to \infty} f_n(s) = f(s), \ (f_n)_0 = f_0$ and $(f_n)_\infty = n$. Lemma 2.1 implies that there exists a sequence of unbounded continua $\mathcal{C}_n$ of solutions to problem (2.3) emanating from $(\frac{\mu_1}{f_0}, 0)$ and joining to $(\frac{\mu_1}{n}, \infty)$.

By Lemma 2.5, there exists an unbounded component $\mathcal{C}$ of $\limsup_{n\to \infty} \mathcal{C}_n$ such that $(\frac{\mu_1}{f_0}, 0) \in \mathcal{C}$ and $(0, \infty)\in \mathcal{C}$. This completes the proof.

3.   Direction turn of bifurcation

In this section, we show that there are at least two direction turns of the continuum under conditions (H3) and (H5), and accordingly we finish the proof of Theorem 1.1.

Lemma 3.1. Assume that (H1)–(H3) hold. Let $\{(\lambda_n, u_n)\}$ be a sequence of positive solutions of (1.1) which satisfies $||u_n||\to 0$ and $\lambda_n\to \frac{\mu_1}{f_0}$. Let $\phi_1(x)$ be the eigenfunction of (1.4) which satisfies $||\phi_1|| = 1$. Then there exists a subsequence of $\{u_n\}$, again denoted by $\{u_n\}$, such that $\frac{u_n}{||u_n||}$ converges uniformly to $\phi_1$ on $[0, 1]$.

Proof. As the proof is very similar to that in [11, Lemma 2.3], we omit it.

Lemma 3.2. Assume that (H2) holds. Let $\alpha > 0$ and let $\phi_1 > 0$ be a first eigenfunction of (1.4). Then

Proof. Multiplying the equation of (1.4) by $\phi_1^{\alpha+1}$ and integrating it over $[0, 1]$, we obtain

Lemma 3.3. Let the hypotheses of Lemma 2.1 hold. Then there exists $\delta > 0$ such that $(\lambda, u)\in \mathcal{C}$ and $|\lambda-\mu_1/f_0|+||u||\leq \delta$ imply $\lambda < \mu_1/f_0$.

Proof. For contradiction we assume that there exists a sequence $\{(\beta_n, u_n)\}$ such that $(\beta_n, u_n)\in \mathcal{C}$ satisfying

By Lemma 3.1, there exists a subsequence of $\{u_n\}$, again denoted by $\{u_n\}$, such that $u_n/||u_n||$ converges uniformly to $\phi_1$ on $[0, 1]$, where $\phi_1$ is the eigenfunction of (1.4) which satisfies $||\phi_1|| = 1$. Multiplying the equation of (1.1) with $(\lambda, u) = (\beta_n, u_n)$ by $u_n$ and integrating it over $[0, 1]$, we obtain

that is,

From Lemma 3.1, after taking a subsequence and relabeling if necessary, $u_n/||u_n||$ converges to $\phi_1$ in $E$.

it follows that

and accordingly,

with $\hat\zeta:\mathbb{N}\to \mathbb{N}$ satisfying $\lim_{n\to \infty}\hat\zeta(n) = 0$.

That is

Lebesgue's dominated convergence theorem, condition (H3) imply that

and

This contradicts $\beta_n\geq \mu_1/f_0.$

Lemma 3.4. Assume that (H1), (H2) and (H5) hold. Let $\mathcal{C}$ be as in Lemma 2.6. If $(\lambda, u)\in \mathcal{C}$ such that $||u||_\infty = s_0$, we have $\lambda > \frac{\lambda_1}{f_0}$.

Proof. Let $(\lambda, u)$ be a solution of (1.1) with $||u||_\infty = u(\tau) = s_0$. Let $f_* = \frac{f_0}{\mu_1\int_0^1|m(x)|dx}$ be from Lemma 2.2. By condition (H5) and Lemma 2.2, we have

that is

Proof of Theorem 1.1. Let $\mathcal{C}$ be as in Lemma 2.6. By Lemma 2.6, $\mathcal{C}$ is bifurcating from $(\frac{\mu_1}{f_0}, 0)$ and joins $(\mu_1/f_0, 0)$ to $(0, \infty)$. Since $\mathcal{C}$ is unbounded, there exists $\{(\lambda_n, u_n)\}$ such that $(\lambda_n, u_n)\in \mathcal{C}$ and $\lambda_n+||u_n||_\infty\to \infty$. By Lemma 2.6, we have that $||u_n||_\infty\to \infty$ and $\lambda_n\to 0$, then there exists $(\lambda_0, u_0)\in \mathcal{C}$ such that $||u_0||_\infty = s_0$ and Lemma 3.4 shows that $\lambda_0 > \frac{\mu_1}{f_0}$.

By Lemmas 2.6, 3.3, 3.4, $\mathcal{C}$ passes through some points $(\frac{\mu_1}{f_0}, v_{1})$ and $(\frac{\mu_1}{f_0}, v_{2})$ with $\|v_{1}\|_\infty < s_{0} < \|v_{2}\|_\infty$, and there exist $\underline{\lambda}$ and $\overline{\lambda}$ which satisfy $0 < \underline{\lambda} < \frac{\mu_1}{f_0} < \overline{\lambda}$ and both (ⅰ) and (ⅱ):

(ⅰ) if $\lambda\in(\frac{\mu_1}{f_0}, \overline{\lambda}]$, then there exist $u$ and $v$ such that $(\lambda, u), (\lambda, v)\in\mathcal{C}$ and $\|u\|_\infty < s_{0} < \|v\|_\infty$;

(ⅱ) if $\lambda\in(\underline{\lambda}, \frac{\mu_1}{f_0}]$, then there exist $u$ and $v$ such that $(\lambda, u), (\lambda, v)\in\mathcal{C}$ and $\|u\|_\infty < \|v\|_\infty < s_{0}$.

Define $\lambda^{\ast} = \text{sup}\{\overline{\lambda}:\ \overline{\lambda}\ \text{satisfies}\ (\text{i}) \}$ and $\lambda_{\ast} = \text{inf}\{\underline{\lambda}:\ \underline{\lambda}\ \text{satisfies} \ (\text{ii}) \}$. Then by the standard arguments, (1.1) has a positive solution at $\lambda = \lambda_{\ast}$ and $\lambda = \lambda^{\ast}$, respectively. Clearly, $\mathcal{C}$ is bifurcating from $(\frac{\mu_1}{f_0}, 0)$ and goes leftward. Moreover, $\mathcal{C}$ turns to the right at $(\lambda_{\ast}, u_{\lambda_{\ast}})$ and to the left at $(\lambda^{\ast}, u_{\lambda^{\ast}})$, finally to the left near $\lambda = 0$ (see Figure 1). That is, $\mathcal{C}$ is a reversed $S$-shaped continuum. By figuring the shape of unbounded continuum $\mathcal{C}$ of positive solutions, the statements (i)–(vi) hold. This complete the proof of Theorem 1.1.

4.   Conclusions

The $p$-Laplacian operator in one and multi-dimensions is a current vigorous area of research. In this paper, we extend the seminal work by Sim and Tanaka ^[11] on "Three positive solutions for the one-dimensional $p$-Laplacian problem'', where the authors studied $f(s)s^{1-p} = f_\infty$ with $f_\infty\in (0, \infty)$. The current work considers the case where $f_\infty = \infty$ and aims to prove the existence of a reversed $S$-shaped continuum. As a by-product, we assert further that (1.1) has one, or two or three positive solutions under the suitable conditions on the weight function and nonlinearity. More interesting and complex behavior of such problem will further be explored.

Acknowledgments

L. Miao was supported by the NSF of China (No. 12161071) and the NSF of Qinghai Province (No.2023-ZJ-949Q). Z. He was supported by the Education and Teaching Research Project of Qinghai University (JY202138). M. Xu was supported by the NSF of Gansu Province(No. 21JR1RA230).

Conflict of interest

The authors declare that they have no competing interests.