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Existence and multiplicity of radially symmetric kadmissible solutions for a kHessian equation
Boundary Value Problems volume 2022, Article number: 90 (2022)
Abstract
In this paper, we show that the radially symmetric kadmissible solutions set of a kHessian equation Dirichlet problem with homogeneous boundary condition contains a reversed Sshaped connected component. By determining the shape of unbounded continua of the solutions, we obtain the existence and multiplicity of radially symmetric kadmissible solutions with respect to the bifurcation parameter λ. The proof is based on the bifurcation technique.
1 Introduction
Consider the following kHessian equation Dirichlet problem with homogeneous boundary condition
where \(B_{1}=\{x\in \mathbb{R}^{n}:x<1\}\), λ is a positive parameter, \(S_{k}(D^{2}u)\) is the kHessian operator of \(u, k\in \{1,\ldots , n\}\), \(f:[0, \infty )\to [0, \infty )\) is a continuous function with \(f(s)>0\) for all \(s>0\). Let \(\lambda (D^{2}u):=(\lambda _{1}, \lambda _{2}, \ldots , \lambda _{n})\) be the eigenvalues of the Hessian matrix \(D^{2}u\) and \(\sigma _{k}(\lambda )\) be the kth elementary symmetric function of λ defined as follows:
Then, \(S_{k}(D^{2}u):=\sigma _{k}(\lambda (D^{2}u))\).
To work with elliptic operators, we always need u to be a kadmissible function. A function \(u \in C^{2}(B_{1})\cap C^{0}(\overline{B_{1}})\) is called kadmissible if \(\lambda (D^{2}u)\in \Gamma _{k}\), where
It is worth noting that the kHessian operator is elliptic for any kadmissible function u. Moreover, a kadmissible solution is subharmonic and, by the maximum principle, is negative in \(B_{1}\), see [1, p. 30]. The kHessian operator includes the following special examples:
• when \(k=1\), \(S_{1}(D^{2}u)= \Delta u=\sum \partial ^{2} u/\partial x_{i}^{2}\) and
• when \(k=n\), \(S_{n}(D^{2}u)=\det D^{2}u\).
Hence, the kHessian equation can be regarded as an extension of the semilinear elliptic equation and the Monge–Ampère equation. The study of the Monge–Ampère equation began with Monge [2] in 1784 and was continued by Ampère [3] in 1820. Since then, many scholars have studied this equation. For a full discussion, we recommend [4] and references therein.
The study of kadmissible solutions for a kHessian equation by the bifurcation technique can be traced back to Jacobsen [5]. In 1999, Jacobsen established the global bifurcation result for problem (1.1). Let \(\mu _{1}\) be the first eigenvalue of boundary value problem
which was first obtained in [1]. Jacobsen [5] proved that problem (1.1) with \(f(s)=s^{k}+g(s)\) and \(\lim_{s\to 0^{+}}g(s)/s^{k}=0\) possesses an unbounded continuum \(\mathcal{C}\) of nontrivial admissible solutions, which bifurcates from \((\mu _{1}, 0)\) and lies in the strip \(\{(\lambda , u):0\leq \lambda \leq \mu _{1}\}\). Dai and Luo [6] in 2018 pointed out that the conclusion of “\(\mathcal{C}\) lies in the strip \(\{(\lambda , u):0\leq \lambda \leq \mu _{1}\}\)” is not true. By using the bifurcation technique, they corrected this result and studied the global behavior of admissible solutions for problem (1.1). In another paper, Dai [7] established the existence, nonexistence, uniqueness, and multiplicity of radial symmetric kadmissible solutions of problem (1.1) by using the bifurcation technique according to the asymptotic behavior of f at 0 and ∞. However, the sublinear and superlinear conditions imposed on the nonlinearities only deduce a relatively simple shape of the component, and they provided no information on at least two direction turns of the connected component. In 2019, by the bifurcation technique, Ma, He and Yan [8] improved the result of [7], showing in their Theorem 1.1 that problem (1.1) has at least three radially symmetric kadmissible solutions under suitable conditions on the nonlinearity.
Very recently, He and Miao [9] showed that the kadmissible solutions of (1.1) are in general not convex, and they constructed a new cone and obtained the existence of three radially symmetric kadmissible solutions via the Leggett–Williams’ fixedpoint theorem. Zhang, Xu and Wu [10] studied the kadmissible solutions for the eigenvalue problem of a singular kHessian equation. By constructing the upper and lower solutions of the kHessian equation, the existence of a radially symmetric solution for the eigenvalue problem is established via Schauder’s fixedpoint theorem.
For other results concerning the existence, nonexistence, and multiplicity of kadmissible solutions for a kHessian equation, we refer the reader to [11–21] and the references therein.
Inspired by [5–10] and by a modified version of the global bifurcation result of problem (1.1) in [6], we show in this paper the existence of a reversed Sshaped connected component of radially symmetric kadmissible solutions of (1.1). As a byproduct, we assert further that (1.1) has one, two or three radially symmetric kadmissible solutions under the suitable conditions on the nonlinearity.
Throughout the paper, we make the following hypotheses on the nonlinearity f:

(F1)
there exist \(\alpha > 0\), \(f_{0} > 0\) and \(f_{1} > 0\) such that
$$ \lim_{s\to 0^{+}} \frac{f(s)f_{0}^{k}s^{k}}{s^{k+\alpha}}=f_{1}^{k}; $$(1.3) 
(F2)
$$ f_{\infty}:=\lim_{s\to \infty} \frac{f(s)}{s^{k}}=\infty ; $$(1.4)

(F3)
there exists \(s_{0} > 0\) such that if \(0\leq s\leq s_{0}\) we have
$$ 0\leq f(s)\leq \frac{C_{n}^{k} f_{0}^{k}}{\mu _{1}^{k} n}s_{0}^{k}, $$where \(\mu _{1} > 0\) is the first eigenvalue of (1.2). Moreover, \(\mu _{1}\) is simple (see [1]).
Let \(X=C[0,1]\) with the normal \(\u\ = \max_{r\in [0,1]} u(r)\). Let \(E=\{u\in C^{1}[0, 1]: u'(0)=u(1)=0\}\) with the norm \(\u\_{1}=\max_{r\in [0,1]}u(r)+\max_{r\in [0,1]}u'(r)\). Let \(X^{+} = \{u\in X : u \geq 0\}\) and \(P^{+}\) be the set of functions in \(X^{+}\) that are positive in \([0, 1)\). Also, set \(K^{+} = \mathbb{R} \times P^{+}\) under the product topology.
Our main result is the following theorem.
Theorem 1.1
(see Fig. 1)
Assume that (F1)–(F3) 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 radially symmetric kadmissible solution if \(0<\lambda <\lambda _{\ast}\);

(ii)
(1.1) has at least two radially symmetric kadmissible solutions if \(\lambda =\lambda _{\ast}\);

(iii)
(1.1) has at least three radially symmetric kadmissible solutions if \(\lambda _{\ast}<\lambda <\mu _{1}/f_{0}\);

(iv)
(1.1) has at least two radially symmetric kadmissible solutions if \(\mu _{1}/f_{0}<\lambda \leq \lambda ^{\ast}\);

(v)
(1.1) has at least one radially symmetric kadmissible solution if \(\lambda =\lambda ^{\ast}\);

(vi)
(1.1) has no radially symmetric kadmissible solution if \(\lambda >\lambda ^{\ast}\).
Remark 1.1
Condition (F1) implies
Remark 1.2
Let us consider the function
Obviously, f satisfies (F1) and (F2) with
It is easy to see that if \(m>\frac{m C_{n}^{k}}{\mu _{1}^{k} n}\) is sufficiently large, then the function f also satisfies (F3).
Remark 1.3
Note that Condition (F1) has never been used before, as far as the authors know. Indeed, under (F1), we have an unbounded subcontinuum that is bifurcating from \((\mu _{1}/f_{0}, 0)\) and goes leftward. Conditions (F2) and (F3) lead the unbounded subcontinuum to the right at some point, and finally to the left near \(\lambda =0\).
The paper is organized as follows. In Sect. 2, we show global bifurcation phenomena from the trivial branch with the leftward direction near the initial point. Section 3 is devoted to showing that there are at least two direction turns of the component and completing the proof of Theorem 1.1.
2 Preliminaries
In this section, we give some lemmas and show a global bifurcation phenomenon from the trivial branch.
Lemma 2.1
([11])
Assume \(z(r)\in C^{2}[0, R]\) is radially symmetric and \(z'(0)=0\). Then, the function \(u(x)=z(r)\) with \(r=x< R\) belongs to \(C^{2}(B_{R})\), and
where \(B_{R}=\{x\in \mathbb{R}^{n}: x< R\}\) and \(C^{m}_{n}=\frac{n!}{(nm)! m!}\).
Lemma 2.2
The function \(u\in C^{2}(B_{1})\) is a radially symmetric kadmissible solution of the problem (1.1) if and only if \(z(r)\) is a solution of the boundary value problem
Let \(v = z\). The problem (2.1) can be written as
Next, we will establish the global bifurcation result for the problem (2.2) with \(f(s)=s^{k}+g(s)\), i.e.,
Here, \(g: \mathbb{R}^{+}\to \mathbb{R}\) satisfies \(\lim_{s\to 0^{+}} g(s)/s^{k} = 0\) and the following subcritical growth restriction
for some \(q\in (0, k^{*})\), where
is the critical exponent for the kHessian operator [12]. In particular, in [12], the author proved that the boundary value problem
has no solution in \(C^{1}(\bar{B}_{R})\cap C^{4}(B_{R})\) for the supercritical (or critical) case \(p\geq k^{*}\) and it admits a solution that is radially symmetric and is in \(C^{2}(\bar{B}_{R})\) in the subcritical case \(0< p< k^{*}\), with \(p\neq k\). Moreover, if \(k = 1\), the 1Hessian operator is the Laplacian and \(k^{*}=(n+2)/(n2)\) is the critical Sobolev exponent.
Lemma 2.3
([7, Theorem 1.1])
The pair \((\mu _{1}, 0)\) is a bifurcation point of problem (2.3) and the associated bifurcation branch \(\mathcal{C} \subseteq (K^{+} \cup \{(\mu _{1}, 0)\})\) is unbounded in \([0, \infty ) \times X\).
Lemma 2.4
([7, Theorem 6.5])
If \(f_{0}\in (0, \infty )\), \(f_{\infty}=\infty \) and (2.4) holds, there is an unbounded component \(\mathcal{C}\) of the set of positive solutions of problem (2.2) bifurcating from \((\mu _{1}/f_{0}, 0)\) such that \(\mathcal{C} \subseteq (K^{+} \cup \{(\mu _{1}/f_{0}, 0)\})\). Moreover, \(\mathcal{C}\) joins \((\mu _{1}/f_{0}, 0)\) to \((0, \infty )\).
Lemma 2.5
Let the hypotheses of Lemma 2.4hold. Suppose \((\lambda _{j}, u_{j})\subset \mathcal{C}\) is a sequence of positive solutions of (2.2) that satisfies
Then, there exists a subsequence of \(\{u_{j}\}\), again denoted by \(\{u_{j}\}\), such that \(u_{j}/\u_{j}\\) converges uniformly to \(\phi _{1}\) on \([0, 1]\). Moreover, \(u_{j}/\u_{j}\_{1}\) converges to \(\phi _{1}\) in \(C^{1}[0, 1]\). Here, \(\phi _{1}(r)\) is the eigenfunction of (1.2) corresponding to \(\mu _{1}\).
Proof
As the proof is very similar to that in [8, Lemma 2.3] we omit it here. □
3 Direction turn of bifurcation
Lemma 3.1
Let the hypotheses of Lemma 2.4hold. Then, there exists \(\delta >0\) such that \((\lambda , u) \in \mathcal{C}\) and \(\lambda  \mu _{1}/f_{0} + \u\ \leq \delta \) implies \(\lambda < \mu _{1}/f_{0}\).
Proof
For contradiction, we assume that there exists a sequence \(\{(\lambda _{j}, u_{j})\}\subset \mathcal{C}\) satisfying
By Lemma 2.5, there exists a subsequence of \(\{u_{j}\}\), again denoted by \(\{u_{j}\}\), such that \(u_{j}/\u_{j}\\) converges uniformly to \(\phi _{1}\) on \([0, 1]\), where \(\phi _{1}(r) > 0\) is the first eigenfunction of (1.2) that satisfies \(\\phi _{1}\= 1\). Multiplying Eq. (2.2) with \((\lambda , v) = (\lambda _{j}, u_{j})\) by \(u_{j}\) and integrating it on \([0, 1]\), we obtain
that is
From Lemma 2.5, after taking a subsequence and relabeling if necessary, \(u_{j}/\u_{j}\\) converges to \(\phi _{1}\) in \(C^{1}[0, 1]\). Since
it follows that
and accordingly,
with \(\hat{\zeta}:\mathbb{N}\to \mathbb{R}\) satisfying \(\lim_{j\to \infty}\hat{\zeta}(j)=0\).
That is,
Lebesgue’s dominated convergence theorem, condition (F1) implies that
and
This contradicts \(\lambda _{j} \geq \mu _{1}/f_{0}\). □
Remark 3.1
Lemma 3.1 implies that the bifurcation branch \(\mathcal{C}\) has the leftward direction from the bifurcation point \((\mu _{1}/f_{0}, 0)\).
Lemma 3.2
Assume that (F1)–(F3) hold. Let u be a positive solution of (2.2) with \(0< f(s)\leq f_{*}s^{k}\) for some \(f_{*}>0\). Then, there exists a constant \(C>0\) independently of u such that
Proof
Integrating Eq. (2.2) on \([0, r]\) and recalling that \(f(s)\leq f_{*}s^{k}\), we have
that is,
□
Lemma 3.3
Assume that (F3) holds. Suppose u is a positive solution of (2.2) with \(\u\ = s_{0}\). Then,
Proof
Let \((\lambda , u)\) be a positive solution of (2.2). By Lemma 3.2 and condition (F3), it follows that
That is, \(\lambda >\frac{\mu _{1}}{f_{0}}\). □
Remark 3.2
By Lemma 3.3, we know that there exists a direction turn of the bifurcation branch \(\mathcal{C}\) that grows to the right at some point \((\lambda ^{*}, u_{\lambda ^{*}})\in \mathcal{C}\), where \(\u_{\lambda ^{*}}\=s_{0}\).
Proof of Theorem 1.1
Let \(\mathcal{C}\) be as in Lemma 2.4. By Lemma 2.4, \(\mathcal{C}\) 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 \(\lambda _{n}+\u_{n}\\to \infty \). By Lemma 2.4, we have that \(\u_{n}\\to \infty \) and \(\lambda _{n}\to 0\), then there exists \((\lambda _{0}, u_{0})\in \mathcal{C}\) such that \(\u_{0}\=s_{0}\) and Lemma 3.3 shows that \(\lambda _{0}>\frac{\mu _{1}}{f_{0}}\).
By Lemmas 3.1 and 3.3, \(\mathcal{C}\) passes through some points \((\frac{\mu _{1}}{f_{0}}, v_{1})\) and \((\frac{\mu _{1}}{f_{0}}, v_{2})\) with \(\v_{1}\< s_{0}<\v_{2}\\), and there exist \(\underline{\lambda}\) and λ̅ that 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\< s_{0}<\v\\);

(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 \(\u\<\v\<s_{0}\).
Define \(\lambda ^{\ast}=\mathrm{sup}\{\overline{\lambda}: \overline{\lambda} \text{ satisfies } ({\text{i}}) \}\) and \(\lambda _{\ast}=\mathrm{inf}\{\underline{\lambda}: \underline{\lambda} \text{ satisfies } ({\text{ii}}) \}\). Then, by the standard arguments, (2.2) has a positive solution at \(\lambda =\lambda _{\ast}\) and \(\lambda =\lambda ^{\ast}\), respectively.
Clearly, \(\mathcal{C}\) turns to the right at \((\lambda _{\ast}, \u_{\lambda _{\ast}}\)\) and to the left at \((\lambda ^{\ast}, \u_{\lambda ^{\ast}}\)\), and finally to the left near \(\lambda =0\). This complete the proof of Theorem 1.1. □
Now, we strengthen the assumptions on f. Assume, in addition to (F1)–(F3), that f satisfies:

(F4)
there exists \(s_{1}>2s_{0} > 0\) such that \(0\leq s\leq s_{1}\) implies that
$$ \min_{s_{1}\leq s\leq 2s_{1}}\frac{f(s)}{s^{k}}> \frac{f_{0}^{k}}{\mu _{1}^{k}}\eta _{1}^{k}, $$where \(\eta _{1}\) is the first positive eigenvalue of the following problem
$$ \textstyle\begin{cases} (r^{nk}(v')^{k} )'=\eta ^{k}\frac {n}{C^{k}_{n}}r^{n1}v^{k}, & r\in (0,1), \\ v'(0)= v(\frac{1}{2})=0. \end{cases} $$
By an argument similar to proving [8, Lemma 3.5] with small modification, we may obtain the following result.
Lemma 3.4
Assume (F4) holds. Let \((\lambda , u)\) be a positive solution of (2.2) with \(\u\=s_{1}\). Then, \(\lambda <\frac{\mu _{1}}{f_{0}}\).
Remark 3.3
Lemma 3.4 means that there exists a direction turn of the bifurcation continuum \(\mathcal{C}\) that grows to the left at some point \((\lambda ^{**}, u_{\lambda ^{**}})\in \mathcal{C}\), where \(\u_{\lambda ^{**}}\=s_{1}\).
Using the method similar to that used in proving Lemmas 3.3 and 3.4 infinite time, we have
Theorem 3.1
Assume that (F1)–(F4) hold. Then, the continuum \(\mathcal{C}\) is unbounded, joins \((\frac{\mu _{1}}{f_{0}}, 0)\) to \((0, \infty )\), and oscillates around the axis \(\{\lambda =\frac{\mu _{1}}{f_{0}}\}\) an infinite number of times.
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This work is supported by the Natural Science Foundation of Qinghai Province (2021ZJ957Q).
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LYM completed the main study and wrote the manuscript, ZQH checked the proofs process and verified the calculation. Moreover, all the authors read and approved the last version of the manuscript.
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Miao, L., He, Z. Existence and multiplicity of radially symmetric kadmissible solutions for a kHessian equation. Bound Value Probl 2022, 90 (2022). https://doi.org/10.1186/s13661022016760
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DOI: https://doi.org/10.1186/s13661022016760