Global bifurcation of sign-changing radial solutions of elliptic equations of order 2m in annular domains

1.   Introduction

Let $\Omega(a, b)$ denote the annulus $\{x\in \mathbb{R}^n:\, a < |x| < b\}$, $0 < a < b < \infty$, $n \geq 2$, and consider the semilinear elliptic problems

where $\lambda > 0$ is a parameter, $\frac{\partial }{\partial\nu}$ is the outward normal derivative, $m$ is a positive integer and $f, g$ satisfy at least the following assumptions:

(H1) $f: \mathbb{R}\to \mathbb{R}$ is a continuous function;

(H2) $g: [a, b]\to [0, \infty)$ is a continuous function such that $g\not\equiv 0$ in $[a, b]$.

When $m = 1$ and $n = 1$, the existence of positive radial solutions of problem (1.1), (1.2) has been intensively studied in the case where $f$ is superlinear/sublinear at $0$ and $\infty$, see Coffman and Marcus ^[1], Erbe and Wang ^[2], Erbe, Hu and Wang ^[3], Lan and Webb ^[4] and references therein. When $m = 1$ and $n\geq 1$, the existence of radial positive solutions of (1.1), (1.2) was studied by Lin and Pai ^[5], Wang ^[6]. When $n\geq 2$ and $m > 1$, Dalmasso ^[7] investigated the existence of positive radial solutions for (1.1), (1.2) in an annulus with Dirichlet boundary conditions. He considered a nonlinearity which is either sublinear or the sum of a sublinear and a superlinear term. All of these work are based upon the fixed point theorem in cones. Ma and Thompson ^[8] used bifurcation theorem ^[9] to study the multiplicity of nodal solutions of (1.1), (1.2) when $m = 1$ and $n = 1$ under the conditions: either

or

Of course, the natural question is whether or not similar result can be established for (1.1), (1.2) in the case $m > 1$ and $n > 1$?

The purpose of this paper is to generalize the main results in Ma and Thompson ^[8] to the case $m > 1$ and $n > 1$.

Since we are interested in radial nodal solutions, problems (1.1), (1.2) reduce to the one-dimensional boundary value problems

where $\mathcal{L}$ denotes the polar form of the Laplacian, i.e.:

Thus

In particular,

The rest of the paper is organized as follows. In Section 2 we state some properties of Green function $G_m(t, s)$ due to Dalmasso ^[7] and the radial eigenvalues theory of the corresponding linear eigenvalue problem of (1.3), (1.4) due to Elias ^[10] and Rynne ^[11]. Section 3 we introduce Dancer's unilateral global bifurcation theorem of Dancer ^[9]. Finally in Section 4, we state and prove our main result via global bifurcation technique.

2.   Preliminaries

The homogeneous Dirichlet problem

has only the trivial solution. Then it is well-known(see e.g. M. A. Naimark [12,P.29]) that the operator $(-1)^m \mathcal{L}^m$ with Dirichlet boundary conditions has one and only one Green's function $G_m(t, s)$.

Lemma 2.1.(Dalmasso [7,Theorem 2.1]) $G_m(t, s) > 0$ for $a < t, s < b$.

Proof. From (1.5) and (1.6), it follows that $(-1)^m \mathcal{L}^m$ has a Pólya's factorization in $[a, b]$. So, it is a disconjugate operator on $[a, b]$. Therefore, $G_m(t, s) > 0$ for $a < t, s < b$ is an immediate consequence of Coppel [13,Theorem 11 on p.108], or Elias [14,Theorem 0.13].

Lemma 2.2.(Dalmasso [7,Theorem 2.2]) (ⅰ) There exists a positive constant $C_m$, such that

(ⅱ) For any $\delta \in (0, (b - a)/2)$ there exists $\eta\in (0, 1)$ such that

We consider the Banach spaces

and

Denote

with the norm $||\cdot||_{2m-1}$ which, for convenience, we will write as $||\cdot||$. Define an operator $A:X\to Y$

Then $Au = 0$ is disconjugate on $[a, b]$, and the boundary conditions (1.4) are such that $A$ is formally self-adjoint, that is

where $\langle \cdot, \cdot\rangle$ denotes the standard $L^2(a, b)$ inner product. Moreover, Then $A^{-1}: Y\rightarrow E$ is compact.

For each integer $k\geq 1$ and $\nu\in \{+, -\}$, let $\mathcal{S}_{k, \nu}$ denote the set of functions $u\in E$ such that:

(1) $u$ has only simple zeros in $(a, b)$ and no quasi-derivative of $u$ is zero at $a$ or $b$, other than those specified in (1.4);

(2) $u$ has exactly $k-1$ zeros in $(a, b)$;

(3) $\nu u > 0$ in a deleted neighborhood of $t = 0$.

The following result is an immediate consequence of Rynne[11,Theorem 2.4].

Lemma 2.3. Assume that

(H0) $p\in C^0[a, b]$, and $p\geq 0$ on $[a, b]$, while $p\not\equiv 0$ on any interval of $[a, b]$.

Then for each $k\geq 1$ and each $\nu\in \{+, -\}$, problem

has a unique solution $(\lambda_k, \psi_k)\in \mathbb{R}_+\times \mathcal{S}_{k, \nu}$ with $||\psi_k|| = 1$. In addition:

(1) $\sigma(\mathcal{L}, p) = \{\lambda_k:\; k\geq 1\}$;

(2) If $k' > k\geq 1$ then $\lambda_{k'} > \lambda_k > 0$;

(3) $\lim_{k\to\infty}\lambda_k = \infty$.

3.   Functional setting and preliminary properties

The main point to prove our main result (see Theorem 4.1 below) consists in using the unilateral global bifurcation theorem of Dancer ^[9].

Let $E_1$ be a real Banach space with norm $||\cdot||_{E_1}$. $\mathcal{E}$ will denote $\mathbb{R}\times E_1$. Let the mapping $\mathcal{G}: \mathcal{E}\to E_1$ satisfy

Assumption $\mathfrak{A}$: if $\mathcal{G}(\lambda, 0) = 0$ for $\lambda\in \mathbb{R}$, $\mathcal{G}$ is completely continuous and

where $L$ is a completely continuous linear operator on $E_1$ and $||H(\lambda, x)||_{E_1}/||x||_{E_1}\to 0$ uniformly on bounded subsets of $\mathbb{R}$ as $||x||_{E_1}\to 0$.

Define $\Phi(\lambda):E_1\to E_1$ by $\Phi(\lambda)(x) = x-\mathcal{G}(\lambda, x)$ and define $\mathfrak{L}$ to be the closure of $\{(\lambda, x) \in \mathcal{E}:\, x = \mathcal{G}(\lambda, x), \, x\neq 0\}$ in $\mathcal{E}$. Then (cp. Rabinowitz ^[15]) $\mathfrak{L}\cap(\mathbb{R}\times\{0\})\subseteq r(L) \times\{0\}$, where $r(L)$ denotes the set of real characteristic value of $L$. If $\mu\in r(L)$, define $C_\mu$ to be the component of $\mathfrak{L}$ containing $(\mu, 0)$.

Assume now that $\mu\in r(L)$ such that $\mu$ has multiplicity $1$. Suppose that $v\in E_1\setminus\{0\}$ and $l\in E_1^*$ such that

(where $L^*$ is the adjoint of $L$) and $l(v) = 1$. If $y\in (0, 1)$, define

By [15,Lemma 1.24], there exists an $S > 0$ such that

where ${\mathcal{E}}_S(\mu) = \{(\lambda, u)\in \mathcal{E}\, |\, |\lambda-\mu|+\|u\|_{E_1} < S\}$ and $\bar {\mathcal{E}}_S(\mu)$ denotes closure of $\mathcal{E}_S(\mu)$. For $0 < \epsilon\leq S$ and $\nu = \pm$, define $\mathcal{D}^\nu_{\mu, \epsilon}$ to be the component of $\{(\mu, 0)\}\cup (\mathfrak{L}\cap\bar {\mathcal{E}}_\epsilon(\mu)\cap K_y^\nu)$ containing $(\mu, 0)$, $C^\nu_{\mu, \epsilon}$ to be the component of $\overline{C_\mu \setminus \mathcal{D}^{-\nu}_{\mu, \epsilon}}$ containing $(\mu, 0)$ (where $-\nu$ is interpreted in the natural way), and $C_{\mu, \nu}$ to be the closure of $\bigcup_{S\geq \epsilon > 0} C^\nu_{\mu, \epsilon}$. Then $C_{\mu, \nu}$ is connected and, by ^[9], $C_\mu = C_{\mu, +}\cup C_{\mu, -}$. By ^[15], Lemma 1.24], the definition of $C_{\mu, \nu}$ is independent of $y$.

Theorem A. [9,Theorem 2] Either $C_{\mu, +}$ and $C_{\mu, -}$ are both unbounded or

4.   Proof of the main result

We shall make use of the following assumptions

(A1) $f\in C(\mathbb{R}, \mathbb{R})$ with $sf(s) > 0$ for $s\neq 0$;

(A2) there exist $f_0, \ f_\infty\in (0, \infty)$ such that

Theorem 4.1. Let (A1), (A2) and (H2) hold. Assume that for some $k\in \mathbb{N}$, either

or

Then (1.3), (1.4) has two solutions $u_k^+$ and $u_k^-$ such that $u_k^+$ has exactly $k-1$ zeros in $(a, b)$ and is positive near $0$, and $u_k^-$ has exactly $k-1$ zeros in $(a, b)$ and is negative near $0$.

Remark 4.1. Theorem 4.1 generalizes Dalmasso [7,Theorem 1.1] where only the existence of radial positive solution was studied under the conditions $k = 1, \nu = +$ and

Obviously, (4.3) is a special case of (4.2).

Remark 4.2. Theorem 4.1 generalizes Ma and Thompson [8,Theorem 1.1] where $n = 1$ and $m = 1$, i.e., the second order ODE version of Theorem 4.1 was considered.

Remark 4.3. Conditions (4.1) and (4.2) in Theorem 4.1 are optimal. Let us consider the following counterexample.

where

It is easy to check that

i.e.

We shall see that (4.4) has no positive solution when $\lambda = 1$. In fact, suppose on the contrary that $(1, u)$ is a positive solution of (4.4). Then, multiplying both sides of (4.4) by $\sin \pi t$ and integrating from $0$ to $1$, we get

This is a contradiction. Therefore, the conditions (4.1) and (4.2) are optimal.

Remark 4.4. For other related results on the fourth order problems, see Drábek and Holubová ^[16], Cabada and Enguiça ^[17] and Ma and Lu ^[18].

Let $\zeta, \ \xi \in C(\mathbb{R})$ be such that

Obviously

Let

then $\tilde \xi$ is nondecreasing and

Let us consider

as a bifurcation problem from the trivial solution $u\equiv 0$.

Equation (4.6) can be converted to the equivalent equation

where $\sigma(t): = t^{n-1}$. Further we note that $||A^{-1}[\sigma(\cdot)g(\cdot) \zeta(u(\cdot)]|| = o(||u||)$ for $u$ near $0$ in $E$, since

The results of Dancer ^[9] for (4.6) can be stated as follows: For each integer $k\geq 1$, $\nu\in \{+, \ -\}$, Either $C_k^+$ and $C_k^-$ are both unbounded or

We shall show that

Assume on the contrary that $(\bar\eta, y)\in \big((C_k^+ \cap C_k^-)\setminus \{(\lambda_k/f_0, 0)\}\big)$ for some $\bar\eta\neq \lambda_k/f_0$.

If $y\neq 0$ in $[a, b]$, then $y\in (\mathcal{S}_{k, +}\cap \mathcal{S}_{k, -})$ implies that $y(t)\equiv 0$ in a left neighborhood of $t = a$. This contradicts the fact $y$ has exactly $k-1$ simple zeros in $(a, b).$

If $y\equiv 0$ in $[a, b]$, then there exists a sequence $\{(\eta_n, y_n)\}\subset C^+_k$ or $\{(\eta_n, y_n)\}\subset C^-_k$, such that

we only deal with the first case.

Set

Then

Using the fact that any zeros on $v_n$ in $(a, b)$ are simple and the standard argument, we may assume that $v_n\to v_*$ for some $v_*\in \mathcal{S}_{k, +}$ with $||v_*|| = 1$, and

This implies $\bar\eta = \lambda_k/f_0$. However, it is a contradiction. Therefore, (4.8) is valid, and accordingly, both $C_k^+$ and $C_k^-$ are unbounded.

Proof of Theorem 4.1. It is clear that any solution of (4.6) of the form $(1, u)$ yields a solution $u$ of (1.3)-(1.4). We will show $C^\nu_k$ crosses the hyperplane $\{1\}\times E$ in $\mathbb{R}\times E$. To do this, it is enough to show that $C^\nu_k$ joins $\bigl(\frac{\lambda_k}{f_0}, 0\bigr)$ to $\bigl(\frac{\lambda_k}{f_\infty}, \infty\bigr)$. Let $(\mu_n, y_n)\in C^\nu_k$ satisfy

We note that $\mu_n > 0$ for all $n\in \mathbb{N}$ since $(0, 0)$ is the only solution of (4.6) for $\lambda = 0$ and $C^\nu_k\cap (\{0\}\times E) = \emptyset$.

Using the same method to prove Ma and Thompson [8,Theorem 1.1] with obvious change, and replacing 'from Lemma 2.1' in [8,Page 714] with 'from the proof of Elias [10,Lemma 4]' we may deduce the desired result.

5.   Conclusions

We are concerned with determining values of $\lambda$, for which there exist nodal solutions of elliptic equations of order $2m$ in annular domains with Dirichlet boundary conditions. Furthmore, we have given an example to show that the interval of $\lambda$ is optimal.

Acknowledgments

The authors are very grateful to the anonymous referees for their very valuable suggestions. This work was supported by National Natural Science Foundation of China (No.11671322).

Conflict of interest

All of the authors of this article claims that together they have no any competing interests each other.