Bifurcation and one-sign solutions for semilinear elliptic problems in $\mathbb{R}^{N}$

1.   Introduction

Consider the following semilinear elliptic problem

where $\lambda$ is a real parameter, $N\geq3$, and $a\in C^{\alpha}_{loc}(\mathbb{R}^{N}, \mathbb{R})$ for some $\alpha\in(0, 1)$ is a weighted function which can be sign-changing and $f\in C^{\alpha}(\mathbb{R}, \mathbb{R})$, and $f(s)s > 0$ for any $s\neq0$. Edelson and Rumbos ^[1] have shown that problem (1.1) with $f(u)\equiv u$ has a positive, simple and principal eigenvalue $\lambda_{1}$ and the positive principle eigenfunction $\phi$ satisfies the asymptotic decay law $\lim_{|x|\rightarrow +\infty}|x|^{N-2}\phi_{1}(x) = c$ for some constant $c$(where $a$ satisfied the following condition (A1)). Edelson and Rumbos ^[1,2] have also studied the existence of positive solution and the existence of global branches of minimal solutions of the problem (1.1) by the Schauder-Tychonoff fixed point theorem and the Dancer global bifurcation theorems ^[3]. By using the Rabinowitz global bifurcation method ^[4], Edelson and Furi ^[5] have shown the existence of positive minimal solution of the problem (1.1). In 2017, Dai ^[6] have established a global bifurcation result for the problem (1.1).

By ^[6], set

For any $u\in C^{\infty}_{c}(\Omega)$ with $\Omega\subseteq \mathbb{R}^{N}$, we define

Denote by $\mathcal{D}^{1, 2}(\Omega)$ the completion of $C^{\infty}_{c}(\Omega)$ with respect to the norm $\|u\|_{1}$. Denote by $S(\mathbb{R}^{N})$ the set of all measurable real functions defined on $\mathbb{R}^{N}$. Two functions in $S(\mathbb{R}^{N})$ are considered as the same element of $S(\mathbb{R}^{N})$ when they are equal almost everywhere. Let $L^{2}(\mathbb{R}^{N}; |a|) = :\{u\in S(\mathbb{R}^{N}): \int_{\mathbb{R}^{N}}|a| u^{2}dx < +\infty\}$. For $u\in \mathcal{D}^{1, 2}(\mathbb{R}^{N}), u\neq0$, define the Rayleigh quotient

Dai et al. ^[6] also assumed that $a$ satisfied the following condition:

(A1) Let $a\in M(\mathbb{R}^{N})$. Assume that $p, P\in C(\mathbb{R}^{N}, \mathbb{R})$ are positive, radially symmetric and satisfies

and

Furthermore, if $P$ satisfies the following more strong condition (with $r = |x|$)

Dai ^[5] established the following spectrum structure:

Lemma 1.1 (see [6, Theorem 1.1]). Let (A1) hold. Then there exists an orthonormal basis $\{\varphi_{k}\}^{+\infty}_{1}$ of $L^{2}(\mathbb{R}^{N}; |a|)$ and a sequence of positive real numbers $\{\lambda_{k}\}^{+\infty}_{1}$with $\lambda_{k}\rightarrow +\infty$ as $k\rightarrow +\infty$, such that

Moreover, one has that

and

For $k\geq2$. $\lambda_{1}$ is simple and principal eigenvalue. Furthermore, if $P$ satisfies (1.2), $\lambda_{1}$ is the unique positive principal eigenvalue and

for some constant $c$.

By the Rabinowitz global bifurcation theorem [2, Theorem 1.3] and the Dancer unilateral global bifurcation theorem [7, Theorem 2], Dai ^[6] obtained [6, Theorem 1.3]. The signum condition $f(s)s > 0$ for $s\neq0$ plays an important role in the [6, Theorem 1.3].

Of course, the natural question is that of what would happen without signum condition for the problem (1.1). Recently, Dai ^[8,9] studied the global behavior of the components of positive solutions for the Schr$\ddot{o}$dinger equation and one-sign solutions for the p-Laplacian without the signum condition, respectively.

Motivated by the above interesting and important studies, we shall show that the branches bifurcating from infinity and the trivial solution line for the problem (1.1) are disjoint and the existence results of radial nodal solutions to problem (1.1) without signum condition.

We now present the following assumptions on $f$ :

(A2) $f\in C(\mathbb{R}, \mathbb{R})$, and there exist two constants $s_{2} < 0 < s_{1},$ such that $f(s_{1}) = f(s_{2}) = f(0) = 0$ and $sf(s) > 0$ for $s\in\mathbb{R}\backslash\{s_{1}, 0, s_{2}\}.$

(A3) There exist two constants $\gamma_{1} > 0$ and $\gamma_{2} < 0$ such that

Let

The main results of this section are the following interesting results:

Theorem 1.1. Let (A1)–(A3) hold.

(a) If $f_{0}, f_{\infty}\in (0, \infty)$, then the problem (1.1) has at least two solutions $u_{\infty}^{+}$ and $u_{\infty}^{-}$ for $\lambda\in(\min\{\frac{\lambda_{1}}{f_{0}}, \frac{\lambda_{1}}{f_{\infty}}\}, \max\{\frac{\lambda_{1}}{f_{0}}, \frac{\lambda_{1}}{f_{\infty}}\}]$ such that $u_{\infty}^{+}$ is positive in $\mathbb{R}^{N}$ and $u_{\infty}^{-}$ is negative in $\mathbb{R}^{N}$; the problem (1.1) has at least four solutions $u_{\infty}^{+}$ and $u_{\infty}^{-}$, $u_{0}^{+}$ and $u_{0}^{-}$ for $\lambda\in(\max\{\frac{\lambda_{1}}{f_{0}}, \frac{\lambda_{1}}{f_{\infty}}\}, \infty)$ such that $u_{\infty}^{+}$, $u_{0}^{+}$ are positive in $\mathbb{R}^{N}$ and $u_{\infty}^{-}$, $u_{0}^{-}$ are negative in $\mathbb{R}^{N}$. Moreover, if $P$ satisfies (1.2), we have that

for all $x\in\mathbb{R}^{N}$ and some constants $c_{1}^{i}\neq0$, where $i = 0, \infty$. Do the same for $u_{i}^{-}$.

(b) If $f_{0}\in (0, \infty)$ and $f_{\infty} = \infty$, then the problem (1.1) has at least two solutions $u_{\infty}^{+}$ and $u_{\infty}^{-}$ for $\lambda\in(0, \frac{\lambda_{1}}{f_{0}}]$ such that $u_{\infty}^{+}$ is positive in $\mathbb{R}^{N}$ and $u_{\infty}^{-}$ is negative in $\mathbb{R}^{N}$; the problem (1.1) has at least four solutions $u_{\infty}^{+}$ and $u_{\infty}^{-}$, $u_{0}^{+}$ and $u_{0}^{-}$ for $\lambda\in(\frac{\lambda_{1}}{f_{0}}, \infty)$ such that $u_{\infty}^{+}$, $u_{0}^{+}$ are positive in $\mathbb{R}^{N}$ and $u_{\infty}^{-}$, $u_{0}^{-}$ are negative in $\mathbb{R}^{N}$. Moreover, if $P$ satisfies (1.2), we have that

for all $x\in\mathbb{R}^{N}$ and some constants $c_{1}^{i}\neq0$, where $i = 0, \infty$. Do the same for $u_{i}^{-}$.

(c) If $f_{0} = \infty$ and $f_{\infty}\in (0, \infty)$, then the problem (1.1) has at least two solutions $u_{\infty}^{+}$ and $u_{\infty}^{-}$ for $\lambda\in(0, \frac{\lambda_{1}}{f_{\infty}}]$ such that $u_{\infty}^{+}$ is positive in $\mathbb{R}^{N}$ and $u_{\infty}^{-}$ is negative in $\mathbb{R}^{N}$; the problem (1.1) has at least four solutions $u_{\infty}^{+}$ and $u_{\infty}^{-}$, $u_{0}^{+}$ and $u_{0}^{-}$ for $\lambda\in(\frac{\lambda_{1}}{f_{\infty}}, +\infty)$ such that $u_{\infty}^{+}$, $u_{0}^{+}$ are positive in $\mathbb{R}^{N}$ and $u_{\infty}^{-}$, $u_{0}^{-}$ are negative in $\mathbb{R}^{N}$. Moreover, if $P$ satisfies (1.2), we have

for all $x\in\mathbb{R}^{N}$ and some constants $c_{1}^{i}\neq0$, where $i = 0, \infty$. Do the same for $u_{i}^{-}$.

(d) If $f_{0} = \infty$ and $f_{\infty} = \infty$, then the problem (1.1) has at least four solutions $u_{\infty}^{+}$ and $u_{\infty}^{-}$, $u_{0}^{+}$ and $u_{0}^{-}$ for $\lambda\in(0, \infty)$ such that $u_{\infty}^{+}$, $u_{0}^{+}$ are positive in $\mathbb{R}^{N}$ and $u_{\infty}^{-}$, $u_{0}^{-}$ are negative in $\mathbb{R}^{N}$. Moreover, if $P$ satisfies (1.2), we have that

for all $x\in\mathbb{R}^{N}$ and some constants $c_{1}^{i}\neq0$, where $i = 0, \infty$. Do the same for $u_{i}^{-}$.

Furthermore, we can get the exact multiplicity of one-sign solutions for problem (1.1) under more strict hypotheses.

(A4) $f(s)\equiv0$ for any $s\not\in[s_{2}, s_{1}]$, $f(s)$ is $C^{1}$ with respect to $s\in[s_{2}, s_{1}]$, and such that $f(s)/s$ is decreasing in $[0, s_{1}]$ and is increasing in $[s_{2}, 0]$.

The following are the main results of this section.

Theorem 1.2. Let (A1), (A2) and (A4) hold. Assume that $f_{0}\in (0, \infty)$. Then,

(ⅰ) the problem (1.1) has exactly two solutions $u^{+}(\lambda, \cdot)$ and $u^{-}(\lambda, \cdot)$ for $\lambda\in(\frac{\lambda_{1}}{f_{0}}, +\infty)$, such that $0 < u^{+}(\lambda, \cdot)\leq s_{1}$ and $s_{2}\leq u^{-}(\lambda, \cdot) < 0$ in $\mathbb{R}^{N}$;

(ⅱ) all one-sign solutions of problem (1.1) lie on two smooth curves

$\mathscr{C}^{+}$ and $\mathscr{C}^{-}$ join at $(\lambda_{1}/f_{0}, 0)$;

(ⅲ) $u^{+}(\lambda, \cdot)$($u^{-}(\lambda, \cdot)$) is increasing (decreasing) with respect to $\lambda$.

(ⅳ) If $P$ satisfies (1.2), we have that

for all $x\in\mathbb{R}^{N}$ and some constants $c_{1}, c_{2}\neq0$.

Theorem 1.3. Let (A1), (A2) and (A4) hold. Assume that $f_{0} = \infty$. Then,

(ⅰ) the problem (1.1) has exactly two solutions $u^{+}(\lambda, \cdot)$ and $u^{-}(\lambda, \cdot)$ for $\lambda \in(0, +\infty)$, such that $0 < u^{+}(\lambda, \cdot)\leq s_{1}$ and $s_{2}\leq u^{-}(\lambda, \cdot) < 0$ in $\mathbb{R}^{N}$;

(ⅱ) all one-sign solutions of problem (1.1) lie on two smooth curves

$\mathscr{C}^{+}$ and $\mathscr{C}^{-}$ join at $(\lambda_{1}/f_{0}, 0)$;

(ⅲ) $u^{+}(\lambda, \cdot)$($u^{-}(\lambda, \cdot)$) is increasing (decreasing) with respect to $\lambda$.

(ⅳ) If $P$ satisfies (1.2), we have that

for all $x\in\mathbb{R}^{N}$ and some constants $c_{1}, c_{2}\neq0$.

The rest of this paper is arranged as follows: Section 2, provides some preliminaries. In Section 3, we prove Theorem 1.1, which considers the existence of one-sign solutions for the problem (1.1) without signum condition. In Section 4, we consider exact multiplicity of one-sign solutions for the problem (1.1) and give the proof of Theorems 1.2 and 1.3.

2.   Preliminaries

Let

with the norm

Clearly, $E$ is a Banach space. Let $P^{+} = \{u\in E|u > 0, \, \, {\texttt{for}}\, \, {\texttt{all}}\, \, x\in \mathbb{R}^{N}\}$ and set $P^{-} = -P^{+}$ and $P = P^{+}\cup P^{-}$.

By ^[6], we can show that $u$ is a one-sign $C^{2+\alpha}$ solution of problem (1.1) if and only if u is a solution of the operator equation

where $\Gamma_{N}(x-y) = \frac{1}{N(N-2)\omega_{N}}|x-y|^{2-N}$, $\omega_{N}$ being the volume of the unit ball in $\mathbb{R}^{N}$. Dai ^[6] also can show that $L:E\rightarrow E$ is completely continuous.

Consider the following problem

Suppose $g\in C (\mathbb{R}^{N}\times E\times\mathbb{R}, E)$ satisfies

uniformly for $x\in\mathbb{R}^{N}$ and $\lambda$ on bounded sets.

Similar the proof of [6, Theorem 1.3], we can obtain that the following result:

Theorem 2.1. Assume that (A1) and (2.3) hold. The pair $(\lambda_{1}, 0)$ is a bifurcation point of the problem (2.2) and there are two distinct unbounded continuum $\mathscr{C}^{+}$ and $\mathscr{C}^{-}$ in $\mathbb{R }\times E$ of solutions of the problem (2.2) emanating from $(\lambda_{1}, 0)$. Moreover, we have

where $\nu\in\{+, -\}$.

By (2.1), Eq (2.2) is equivalent to

where $H(\lambda, u) = o(\|u\|)$ at $u = 0$ uniformly on bounded $\lambda$ intervals, $H(\lambda, u) = L[g(x, u, \lambda)]$. $G(\lambda, u): \mathbb{R}\times E\rightarrow E$ is completely continuous.

In order to prove Theorem 1.1, we also need to establish the unilateral global bifurcation result of the problem (2.2) from infinity under assumption

uniformly for $x\in \mathbb{R}^{N}$ and $\lambda$ on bounded sets.

By Rabinowitz ^[10], we have the following theorem.

Theorem 2.2. Let (A1) and (2.4) hold. There exists a connected component $\mathscr{D}^{\nu}$ of of solutions of the problem (2.2), containing $\lambda_{1}\times\{\infty\}$. Moreover, if $\Lambda\subset\mathbb{R}$ is an interval such that $\Lambda\backslash\{\lambda_{1}\}$ doesn't contain any other eigenvalue of problem (1.3), and $\mathscr{M}$ is a neighborhood of $\lambda_{1}\times\{\infty\}$ whose projection on $\mathbb{R}$ lies in $\Lambda$ and whose projection on $E$ is bounded away from $0$, then either

$1^{o}$ $\mathscr{D}^{\nu}-\mathscr{M}$ is bounded in $\mathbb{R}\times E$ in which case $\mathscr{D}^{\nu}-\mathscr{M}$ meets $\mathscr{R} = \{(\lambda, 0)|\lambda\in\mathbb{R}\}$ or

$2^{o}$ $\mathscr{D}^{\nu}-\mathscr{M}$ is unbounded.

If $2^{o}$ occurs and $\mathscr{D}^{\nu}-\mathscr{M}$ has a bounded projection on $\mathbb{R}$, then $\mathscr{D}^{\nu}-\mathscr{M}$ meets $\lambda_{j}\times\{\infty\}$ for some $j\neq 1$, and $\nu\in\{+, -\}$.

To prove our main results, we need the following results:

Lemma 2.1. (see ^[9]) Let $X$ be a normal space and let $\{C_{n}|n = 1, 2, ...\}$ be a sequence of unbounded connected subsets of $X$. Assume that:

$(i)$ there exists $z^{\ast}\in \liminf_{n\rightarrow +\infty}C_{n}$ with $\|z^{\ast}\| < +\infty$;

$(ii)$ for every $R > 0$, $\left(\cup_{n = 1}^{+\infty}C_{n}\right)\cap \overline{B}_{R}$ is a relatively compact set of $X$, where

Then, $\mathbb{D}: = \limsup_{n\rightarrow \infty}C_{n}$ is unbounded, closed and connected.

In order to treat the problems with non-asymptotic nonlinearity at $\infty$, we shall need the following lemmas.

Lemma 2.2. (see ^[9]) Let $(X, \rho)$ be a metric space. If $\{C_{i}\}_{i\in \mathbb{N}}$ is a sequence of sets whose limit superior is $L$ and there exists a homeomorphism $T : X\rightarrow X$ such that for every $R > 0$, $\left(\cup_{i = 1}^{+\infty}T(C_{i})\right)\cap \overline{B}_{R}$ is a relatively compact set, then for each $\epsilon > 0$ there exists an $m$ such that for every $n > m, C_{n}\subset V_{\epsilon}(L),$ where $V_{\epsilon}(L)$ denotes the set of all points $p$ with $\rho(p, x) < \epsilon$ for any $x\in L$.

Now, in order to study the exact multiplicity of one-sign solutions for (1.1), let $\mathbb{E} = \mathbb{ R }\times E,$ $\Phi(\lambda, u) = u-G(\lambda, u)$ and

For $\lambda\in\mathbb{R}$ and $0 < s < +\infty$, define an open neighborhood of $(\lambda_{1}, 0)$ in $\mathbb{E}$ as follows:

Let $E_{0}$ be a closed subset of $E$ satisfying $E = {\texttt{span}}\{\psi_{1}\}\oplus E_{0}$, where $\psi_{1}$ is an eigenfunction corresponding to $\lambda_{1}$ with $\|\psi_{1}\| = 1$. According to the Hahn-Banach theorem, we have $l\in E^{\ast}$ satisfying

where $E^{\ast}$ denotes the dual space of $E$. For any $0 < \varepsilon < +\infty$ and $0 < \eta < 1$, define

Obviously, $K_{\varepsilon, \eta}$ is an open subset of $E$, $K_{\varepsilon, \eta} = K_{\varepsilon, \eta}^{+}\cup K_{\varepsilon, \eta}^{-}$, with $K_{\varepsilon, \eta}^{+} = \{(\lambda, u)\in\mathbb{E}:|\lambda-\lambda_{1}| < \varepsilon, l(u) > \eta\|u\|\},$ $K_{\varepsilon, \eta}^{-} = -K_{\varepsilon, \eta}^{+}$, which are disjoint and open in $E$.

Similar to that of [11, Lemma 6.4.1], we can show the following lemma.

Lemma 2.3. Let $\eta\in(0, 1)$, there is $\delta_{0} > 0$ such that for each $\delta: 0 < \delta < \delta_{0}$, it holds that

And there exist $s\in\mathbb{R}$ and a unique $y\in E_{0}$ such that

for each $(\lambda, v)\in((\mathcal {S}\backslash \{(\lambda_{1}, 0)\})\cap\mathbb{B}_{\delta}(\lambda_{1}, 0)).$ Further, $\lambda = \lambda_{1}+o(1)$ and $y = o(s)$ as $s\rightarrow0$ for these solutions $(\lambda, v)$.

Remark 2.1. From Lemma 2.3, we can see that $\mathscr{D} = \mathscr{D}^{+}\cup\mathscr{D}^{-}$ near $(\lambda_{1}, 0)$ is given by a curve $(\lambda(s), u(s)) = (\lambda_{1} + o(1), s\psi_{1} + o(s))$ for s near $0$. Moreover, we can distinguish between two portions of this curve by $s\geq0$ and $s\leq0$.

3.   One-sign solutions without the signum condition

When $f_{0}\in (0, \infty)$, let $\zeta(u)\in C(\mathbb{R}, \mathbb{R})$ be such that

with

Let us consider

as a bifurcation problem from the trivial solution $u\equiv0$.

Applying Theorem 2.1 to (3.1), we have the following result.

Remark 3.1. There is an unbounded continuum $\mathscr{C}^{\nu}$ of solutions of the problem (1.1) emanating from $(\frac{\lambda_{1}}{f_{0}}, 0)$, such that $\mathscr{C}^{\nu}\subset((\mathbb{R}\times P^{\nu})\cup\{(\frac{\lambda_{1}}{f_{0}}, 0)\})$, where $\nu\in\{+, -\}$.

We now analyze the global behavior of $\mathscr{C}^{+}$+ and $\mathscr{C}^{-}$.

Lemma 3.1. Let (A1)–(A3) hold. Then

(ⅰ) for $(\lambda, u)\in(\mathscr{C}^{+}\cup(\mathscr{C}^{-})$, we have that $s_{2} < u(x) < s_{1}$ for all $x\in\mathbb{R}^{N}$;

(ⅱ) for $(\lambda, u)\in(\mathscr{D}^{+}\cup\mathscr{D}^{-})$, we have that either $\sup_{x\in\mathbb{R}^{N}}u(x) > s_{1}$ or $\inf_{x\in\mathbb{R}^{N}}u(x) < s_{2}$.

Proof. Suppose on the contrary that there exists $(\lambda, u)\in(\mathscr{C}^{+}\cup\mathscr{C}^{-}\cup\mathscr{D}^{+}\cup\mathscr{D}^{-})$ such that either $\sup_{x\in\mathbb{R}^{N}}u(x) = s_{1}$ or $\inf_{x\in\mathbb{R}^{N}}u(x) = s_{2}$.

We only treat the case of $\sup_{x\in\mathbb{R}^{N}}u(x) = s_{1}$ because the proof for the case of $\inf_{x\in\mathbb{R}^{N}}u(x) = s_{2}$ can be given similarly.

We claim that there exists $0 < m < +\infty$ such that $f (s)\leq m(s_{1}-s)$ for any $s\in[0, s_{1}]$. Clearly, the claim is true for the case of $s = 0$ or $s = s_{1}$ by virtue of (A2).

For any $\epsilon\in(0, \gamma_{1})$, it follows from (A3) that there exists $\delta > 0$, such that

for any $s\in(s_{1}-\delta, s_{1})$. From (A2), it arrives

So, the claim is verified by choosing $m = \max\{\rho, \gamma_{1}+\epsilon\}$.

Now, let us consider the following problem

It is obvious that $f (s)\leq m(s_{1}-s)$ for any $s\in[0, s_{1}]$ implies

The strong maximum principle of ^[12] implies that $s_{1} > u$ in $\mathbb{R}^{N}$. This is a contradiction.

Lemma 3.2. Let (A1)–(A3) hold. Then

Proof. We show that the projection of $\mathscr{C}^{+}$ on $\mathbb{R}$ is unbounded. It is sufficient to show that the set $\{(\lambda, u)\in\mathcal {\mathscr{C}}^{+}|\lambda\in[0, d]\}$ is bounded for any fixed $d\in(0, +\infty)$. Suppose on the contrary that there exists $(\lambda_{n}, u_{n})\in\mathscr{C}^{+}, n\in\mathbb{N}$, such that $\lambda_{n}\rightarrow \mu\leq d$, $\|u_{n}\|\rightarrow +\infty$ as $n\rightarrow +\infty$. Let $v_{n} = u_{n}/\|u_{n}\|$. Then, $v_{n}$ should be the solutions of problem

By Lemma 3.1 (ⅰ), we have that

By (A2), one can obtain that $f(s)\in C([0, s_{1}])$.

Thus, we can show

By the compactness of $L(.)$, it follows from (3.2) that $v_{n}\rightarrow v_{0}\equiv0$ as $n\rightarrow +\infty$. This contradicts $\|v_{0}\| = 1$.

This together with the fact that $\mathscr{C}^{+}$ joins $(\frac{\lambda_{1}}{f_{0}}, 0)$ to infinity yields that

Similarly, we can show that

In the following we will investigate the other one-sign solutions of problem (1.1).

When $f_{\infty}\in (0, \infty)$, let $\xi(u)\in C(\mathbb{R}, \mathbb{R})$ be such that

with

Let us consider

as a bifurcation problem from infinity. We add the points $\{(\lambda, \infty)|\lambda\in\mathbb{R}\}$ to space $\mathbb{R}\times E$.

Applying Theorem 2.2 to (3.4), we have the following result.

Remark 3.2. There exists an unbounded continua $\mathscr{D}^{\nu}$ of solutions of (1.1), emanating from $(\frac{\lambda_{1}}{f_{\infty}}, \infty)$, such that $\mathscr{D}^{\nu}\subset((\mathbb{R}\times P^{\nu})\cup\{(\frac{\lambda_{1}}{f_{\infty}}, \infty)\})$, where $\nu\in\{+, -\}$.

We now analyze the global behavior of $\mathscr{D}^{+}$+ and $\mathscr{D}^{-}$.

Lemma 3.3. Let (A1)–(A3) hold. Then

Proof. We show that $\mathscr{D}^{\nu}-\mathscr{M}$ has an unbounded projection on $\mathbb{R}$.

Applying Theorem 2.2 to (3.4), one can obtain that $(1^{0})$ of Theorem 2.2 does not occur by Lemma 3.1 (ii). So $\mathscr{D}^{\nu}-\mathscr {M}$ is unbounded.

Now, we show that the case of $\mathscr{D}^{\nu}-\mathscr{M}$ meeting $\lambda_{j}\times\{\infty\}$ for some $j > 1$ is impossible, where $\lambda_{j}$ denotes the $j$th eigenvalue of the problem (1.3). Assume on the contrary that $\mathscr{D}^{\nu}-\mathscr{M}$ meets $\lambda_{j}\times\{\infty\}$ for some $j > 1$. So there exists a neighborhood $\widetilde{\mathscr{N}}\subset \widetilde{\mathscr{M}}$ of $\lambda_{j}\times\{\infty\}$ such that $u$ must change sign for any $(\lambda, u)\in(\mathscr{D}^{\nu}-\mathscr{M})\cap(\widetilde{\mathscr{N}}\setminus(\lambda_{j}\times\{\infty\}))$, where $\widetilde{\mathscr{M}}$ is a neighborhood of $\lambda_{j}\times\{\infty\}$ which satisfies the assumptions of Theorem 2.2, which contradicts $\mathscr{D}^{\nu}\subset((\mathbb{R}\times P^{\nu})\cup\{(\frac{\lambda_{1}}{f_{\infty}}, \infty)\})$.

Thus,

Similarly, we have

Now, we give Proof of Theorem 1.1

Proof of Theorem 1.1.

(a) Since problem (1.1) has a unique solution $u\equiv0$ for $\lambda\equiv0$, we get

By Lemmas 3.1–3.3, we conclude the desired results.

We only derive the rate of decay of $u_{\infty}^{+}$ since the proof for the other case is completely analogous.

By (2.1), we have

where $\Gamma_{N}(x-y) = \frac{1}{N(N-2)\omega_{N}}|x-y|^{2-N}$, $\omega_{N}$ being the volume of the unit ball in $\mathbb{R}^{N}$.

By $f_{0}, f_{\infty}\in (0, \infty)$, there exist some constant $\varrho > 0$ such that $|f(s)|\leq \varrho|s|$ for any $s\in\mathbb{R}$. Then, we have that

Since $u_{\infty}^{+}$ is bounded, one can get $u_{\infty}^{+}\leq c_{3}$ for some constants $c_{3} > 0$. By condition (1.2), it follows that

By (3.5), for any $\varepsilon > 0$, there exists a $R > 0$ such that for all $x\in \mathbb{R}^{N}$

and

where $\Omega_{R} = \{y\in\mathbb{R}^{N}:|y| > R\},$ $B_{R} = \{y\in\mathbb{R}^{N}:|y| < R\}.$

Furthermore, by (3.6), (3.7), and proof of [6, Theorem 1.1: p. 5948-5949], one can obtain

where $c_{N} = \frac{1}{N(N-2)\omega_{N}}$.

By (3.5), it follows that

Therefore, we have that

for all $x\in\mathbb{R}^{N}$ and some constants $c_{1}^{i}\neq0$, where $i = 0, \infty$. Do the same for $u_{i}^{-}$.

(b) Inspired by the idea of ^[13], we define the cut-off function of $f$ as the following

We consider the following problem

Clearly, we can see that $\lim_{n\rightarrow +\infty}f^{[n]}(s) = f(s)$, $(f^{[n]})_{0} = f_{0}$ and $(f^{[n]})_{\infty} = n$.

By Remarks 3.1 and 3.2, there are two distinct unbounded continuum $\mathscr{C}^{\nu}$ and $\mathscr{D}^{\nu[n]}$ of solutions of the problem (3.8) emanating from $(\frac{\lambda_{1}}{f_{0}}, 0)$, and $(\frac{\lambda_{1}}{n}, \infty)$) respectively, such that they are disjoint, unbounded in the direction of $\lambda$ and

where $\nu = +, -$.

By Lemma 2.2, one derives that for each $\epsilon > 0$ there exists an $N$, such that for $n > N$, $\mathscr{D}^{\nu[n]}\subset V_{\epsilon} (\mathscr{D}^{\nu})$, where $\mathscr{D}^{\nu} = \limsup_{n\rightarrow +\infty}\mathscr{D}^{\nu[n]}$. So it achieves

It follows $(\frac{\lambda_{1}}{n}+\epsilon, +\infty)\subseteq{\texttt{Proj}}_{\mathbb{R}}(\mathscr{D}^{\nu}).$ The arbitrariness of $\epsilon > 0$ and $n$ imply

Similar the proof of Lemma 3.2, we have that

Similar the proof of (a), we have that

for all $x\in\mathbb{R}^{N}$ and some constants $c_{1}^{i}\neq0$, where $i = 0, \infty$. Do the same for $u_{i}^{-}$.

(c) Define

We consider the following problem

Clearly, we can see that $\lim_{n\rightarrow +\infty}f^{[n]}(s) = f(s)$, $(f^{[n]})_{0} = n$ and $(f^{[n]})_{\infty} = f_{\infty}$.

By Remarks 3.1 and 3.2, there are two distinct unbounded continuum $\mathscr{C}^{\nu}$ and $\mathscr{D}^{\nu[n]}$ of solutions of the problem (3.9) emanating from $(\frac{\lambda_{1}}{n}, 0)$, and $(\frac{\lambda_{k}}{f_{\infty}}, \infty)$) respectively, such that they are disjoint, unbounded in the direction of $\lambda$ and

where $\nu = +, -$.

Taking $z^{\ast} = (0, 0),$ and applying Lemma 2.1 again, one derives that $\mathscr{C}^{\nu} = \limsup_{n\rightarrow +\infty}\mathscr{C}^{\nu[n]}$ is unbounded and connected, moreover, $z^{\ast}\in\mathscr{C}^{\nu}.$

We claim that $\mathscr{C}^{\nu}\cap(\mathbb{R}\times P^{\nu}) = \{(0, 0)\})$.

Suppose on the contrary that there exists a sequence $(\lambda_{n}, u_{n})\in\mathscr{C}^{\nu}\setminus\{(0, 0)\} = \limsup_{n\rightarrow \infty}\mathscr{C}^{\nu[n]}\setminus\{(0, 0)\}$ such that $\lim_{n\rightarrow \infty}\lambda_{n} = \mu\neq0$ and $\lim_{n\rightarrow \infty}\|u_{n}\| = 0.$ Hence, for any $N_{0}\in\mathbb{N}$, there exists $n_{0}\geq N_{0}$ such that $(\lambda_{n}, u_{n})\in\mathscr{C}^{\nu[n_{0}]}$. By (3.9), it follows that $\lambda_{n_{0}} = \frac{\lambda_{1}}{n}$ for $n_{0}\geq N_{0}$. From the arbitrary of $N_{0}$, it implies that $n_{0}\rightarrow \infty$, i.e., $\mu = 0$, which contradicts the assumption of $\mu\neq0$.

Lemma 3.1(ⅰ) implies that the projection of $\mathscr{C}_{k, 0}^{+}$ on $\mathbb{R}$ is unbounded.

Furthermore, we have

Similar the proof of Lemma 3.3, we have that

(d) Similar the proof of (b) and (c), respectively, we have that

and

4.   Exact multiplicity of one-sign solutions

To prove Theorems 1.2 and 1.3, by Dai and Han ^[14], Afrouzi and Rasouli ^[15], we first give the definition of linearly stable solution for the problem (1.1). For any $\phi\in E$ and positive solution $u$ of problem (1.1), we can calculate that the linearized eigenvalue problem of (1.1) at the direction $\phi$ is

where $f'(u)\phi$ denotes the Fr$^{'}$echet derivative of $f$ about $u$ at the direction $\phi$. A solution $u$ of problem (1.1) is stable if all eigenvalues of problem (4.1) are positive, otherwise it is unstable. We define the Morse index $M(u)$ of $u$ to problem (1.1) to be the number of negative eigenvalues of problem (4.1). A solution $u$ of problem (1.1) is degenerate if $0$ is an eigenvalue of problem (4.1), otherwise it is non-degenerate. The following lemma is our main stability result for the one-sign solution.

Lemma 4.1. Under the assumptions of Theorem 1.2 (a), then any positive or negative solution $u$ of problem (1.1) is stable and non-degenerate, and Morse index $M(u) = 0$.

Proof. Without loss of generality, let $u$ be a positive solution of problem (1.1), and let $(\mu_{1}, \phi_{1})$ be the corresponding principal eigenpair of problem (4.1) with $\phi_{1} > 0$ in $\mathbb{R}^{N}$. Notice that $u$ and $\phi_{1}$ satisfy

and

Multiplying the first equation of problem (4.3) by $u$ and the first equation of problem (4.2) by $\phi_{1}$, subtracting and integrating, we obtain

By some simple computations, we can show that it follows from (A4) that $f(s)-f'(s)s\geq0$ for any $s\geq0$. Since $u > 0$ and $\phi_{1} > 0$ in $\mathbb{R}^{N}$, we have $\mu_{1} > 0$ and the positive solution $u$ must be stable.

Proof of Theorem 1.2

(a) By Theorem 2.1, we can obtain the problem (1.1) possesses at least two one-sign solutions $u^{+}$ and $u^{-}$ such that $u^{+} > 0$ and $u^{-} < 0$ in $\mathbb{R}^{N}$. In order to prove exact multiplicity of one-sign solutions for (1.1). Define $F : \mathbb{R}\times E\rightarrow \mathbb{R}$ by

From Lemma 4.1, we know that any one-sign solution $(\lambda, u)$ of problem (1.1) is stable. Therefore, at any one-sign solution $(\lambda^{\ast}, u^{\ast})$ for the problem (1.1), we can apply Implicit Function Theorem to $F(\lambda, u) = 0$, and all the solutions of $F(\lambda, u) = 0$ near $(\lambda^{\ast}, u^{\ast})$ are on a curve $(\lambda, u(\lambda))$ with $|\lambda-\lambda^{\ast}|\leq\varepsilon$ for some small $\varepsilon > 0$. Furthermore, by virtue of Remark 2.2, the unbounded continua $\mathscr{D}^{+}$ and $\mathscr{D}^{-}$ are all curves.

To complete the proof, it suffices to show that $u^{+}(\lambda, \cdot)$($u^{-}(\lambda, \cdot)$) is increasing (decreasing) with respect to $\lambda$. We only prove the case of $u^{+}(\lambda, \cdot)$. The proof of $u^{-}(\lambda, \cdot)$ can be given similarly. Since $u^{+}(\lambda, \cdot)$ is differentiable with respect to $\lambda$ (as a consequence of Implicit Function Theorem), taking the derivative of the first equation of problem (4.2) by $\lambda$, one can obtain that

Multiplying the first equation of problem (4.4) by $u$ and the first equation of problem (4.2) by $\frac{du^{+}}{d\lambda}$, subtracting and integrating, we obtain

(A2) implies $f(s)s\geq0$ for any $s\in\mathbb{R}$. So we get $(f'(u^{+})u^{+}-f(u^{+}))\frac{du^{+}}{d\lambda}\leq0$ by (A1). While (A4) shows that $f'(u^{+})u^{+}-f(u^{+})\leq0$. Therefore, we have $\frac{du^{+}}{d\lambda}\geq0$.

Next, we only prove the case of the uniqueness of positive solution of problem (1.1) since the proof of the uniqueness of negative solution of problem (1.1) is similar. Suppose on the contrary that there exist two solutions $v_{1}^{+}$ and $v_{2}^{+}$ corresponding to $\lambda$ with $v_{1}^{+}\in\mathscr{D}^{+}$ of the problem (1.1) for $\lambda\in(\lambda_{1}/f_{0}, +\infty)$. For $\varepsilon > 0$, take $(\lambda-\varepsilon, v_{\lambda-\varepsilon}^{+}), (\lambda+\varepsilon, v_{\lambda+\varepsilon}^{+})\in\mathscr{D}^{+}$, then $v_{\lambda\pm\varepsilon}\rightarrow v_{1}^{+}$ as $\varepsilon\rightarrow 0$. By the monotonicity of $v_{2}^{+}$ with respect to $\lambda$, we get $v_{\lambda-\varepsilon}^{+}\leq v_{2}^{+} \leq v_{\lambda+\varepsilon}^{+}$. Then $v_{2}^{+} = v_{1}^{+}$.

Similar the proof of (a) of Theorem 1.1, we have that

for all $x\in\mathbb{R}^{N}$ and some constants $c_{1}, c_{2}\neq0$.

Proof of Theorem 1.3. By Theorem 4.1 and Lemma 2.1, there is a distinct unbounded continuum $\mathscr{D}^{\nu}(\nu\in\{+, -\})$ of solutions of the problem (1.2) emanating from $(0, 0)$. Furthermore, we have that the problem (1.1) possesses at least two one-sign solutions $u^{+}$ and $u^{-}$ such that $u^{+} > 0$ and $u^{-} < 0$ in $(0, +\infty)$. In view of the argument of Theorem 1.2, the desired conclusion can be obtained immediately.

5.   Conclusions

In this study, we have proved the existence of one-sign solutions without signum condition for the semilinear elliptic problems (1.1) by the bifurcation techniques and the approximation of connected components. We also obtained the exact multiplicity of one-sign solutions for the problem (1.1) under more strict hypotheses. Our fifindings can be applied to further other differential equations with various other boundary conditions, high dimensional case, and so on as future work.

Conflict of interest

The author declares no conflicts of interest in this paper.