Abstract
In this paper, we consider the singular Kirchhoff equation with two parameters
By using the sub-supersolution method together with the comparison principle for elliptic equations, we obtain several existence and nonexistence theorems. Our works improve the results in the previous literature.
Similar content being viewed by others
1 Introduction
In this paper, we study the existence and nonexistence of solutions to the following singular nonlocal elliptic problem with two parameters:
where Ω is a smooth bounded domain in \(\mathbb{R}^{N}\) (\(N\geq 2\)), \(a:[0,+\infty)\rightarrow(0,+\infty)\) is a continuous and nondecreasing function with
\(K,h\in C^{0,\gamma}(\bar{\varOmega})\) with \(K>0\) and \(h>0\) on Ω, and λ and μ are positive real numbers.
This problem is related to the general form of the stationary counterpart of the hyperbolic Kirchhoff equation
for free vibrations of elastic strings [7]. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations, in which L is the length of the string, ρ is the mass density, and \(P_{0}\) is the initial tension. Many researchers began an extensive research in equations of this type after Lions [10] put forward a basic framework. For example, Perera and Zhang [17, 30] considered the following Kirchhof-type problems with 4-sublinear case, asymptotically 4-linear case, and 4-superlinear case terms:
They obtained nontrivial solutions via the Yang index and the invariant sets of descent flow. Sun and Tang [22] obtained the existence and multiplicity of weak solutions of (1.2) by using the mountain pass theorem, the local linking theorem, the fountain theorem, and the symmetric mountain pass lemma in critical point theory. Utilizing the local linking theory, Yang and Zhang [29] obtained the nontrivial solutions of (1.2) with quasilinear terms. Yang and Han [28] obtained infinitely many solutions when \(f(x,u)\) is odd by using the fountain theorem. In recent years, many researchers studied the fractional Kirchhoff equation. In [9] and [24] the authors deal with fractional Schrödinger–Kirchhoff equations. In [23] the authors obtained the multiplicity of solutions of a fractional p-Laplacian Kirchhoff system. In [16] authors deal with nonlocal fractional problems. In addition, in [25] the authors obtained the local existence and blowup of solutions of nonlocal Kirchhoff diffusion problems. Some other results can be found in [3,4,5, 11,12,13,14,15, 20, 26], and their references.
Since the method of sub-supersolutions is an important tool to solve the existence of solutions for boundary value problems, naturally, some authors hope to use the theorem of sub-supersolutions to discuss the elliptic problems of Kirchhoff type. However, the nonlocal term brings some difficulties. To overcome the difficulty from the nonlocal term, Alves and Corrêa [1] considered following problem:
and obtained the existence of positive solutions by using the sub-supersolution method under the condition that \(f:\varOmega\times \mathbb{R}\rightarrow\mathbb{R}\) is an increasing function. Moreover, Alves and Corrêa [2] dealed with the quasilinear stationary Kirchhoff equation
and using the pseudomonotone operator theory, they established the existence of weak solutions by constructing a supersolution and a family of subsolutions. Recently, Yan, O’Regan, and Agarwal [27] discussed the following problem:
They presented some new definitions of a sub-supersolution to problem (1.5) and obtained the existence of classical solution to problem (1.5) when \(F(x,u)=\lambda u^{q}-u^{p+1}\) (\(0< q<1\), \(p>0\)).
Motivated by the ideas introduced in [6], we suppose that \(f:\overline{\varOmega}\times[0,\infty)\rightarrow[0,\infty)\) is a Hölder-continuous function that is positive on \(\overline{\varOmega}\times (0,\infty)\). We also assume that f is nondecreasing with respect to the second variable and is sublinear, that is,
-
(f1)
the mapping \((0,\infty)\ni s\mapsto\frac{f(x,s)}{s}\) is nonincreasing for all \(x\in\overline{\varOmega}\);
-
(f2)
\(\lim_{s\rightarrow0}\frac{f(x,s)}{s}=+\infty\) and \(\lim_{s\rightarrow+\infty}\frac{f(x,s)}{s}=0\) uniformly in \(x\in\overline {\varOmega}\);
-
(f3)
there exists \(n_{0}>0\) such that \(\lambda^{n_{0}}f(x,s)\geq f(x,\lambda^{n_{0}+1}s)\) for all \(\lambda\geq1\).
We assume that \(g\in C^{0,\gamma}(0,\infty)\) is a nonnegative and nonincreasing function satisfying:
-
(g1)
\(\lim_{s\rightarrow0}g(s)=+\infty\);
-
(g2)
there exist C, \(\delta_{0}>0\) and \(\tau\in(0,1)\) such that \(g(s)\leq Cs^{-\tau}\) for all \(s\in(0,\delta_{0})\).
Example
The function \(f(s)=s^{\alpha}\) (\(0<\alpha<1\)) fulfills (f1)–(f3), whereas \(g(s)=(s^{\alpha}+s^{\beta})^{-1}\) (\(0<\alpha <1\), \(\beta>1\)) satisfies assumptions (g1)–(g2).
Denote \(D=\{u\in C^{2}(\varOmega)\cap C(\overline{\varOmega});g(u)\in L^{1}(\varOmega)\}\).
We show in this paper that (1.1) has at least one solution in D for λ belonging to a certain range and any \(\mu>0\). We also prove that in some cases, (1.1) has no solutions in D, provided that λ and μ are sufficiently small.
Remark 1.1
-
(i)
If \(u\in D\), \(v\in C^{2}(\varOmega)\cap C(\overline {\varOmega})\), and \(0< u< v\) in Ω, then \(v\in D\).
-
(ii)
Let \(u\in C^{2}(\varOmega)\cap C(\overline{\varOmega})\) be a solution of (1.1). Then \(u\in D\) if and only if \(\triangle u\in L^{1}(\varOmega)\).
Our main results are the following.
Theorem 1.1
Assume that f satisfies (f1)–(f3) and g satisfies (g1)–(g2).
-
(1)
If \(\lim_{\lambda\rightarrow+\infty}\frac{\lambda}{a(\lambda ^{2+2n_{0}})}=+\infty\), then there exists λ̄ such that (1.1) has at least one solution in D for all \(\lambda>\bar{\lambda}\) and \(\mu>0\).
-
(2)
There exist \(\lambda^{\ast}\) and \(\mu_{\ast}\) small enough such that (1.1) has no solution in D for all \(\lambda<\lambda^{\ast}\) and \(\mu<\mu_{\ast}\).
Theorem 1.2
Assume that f satisfies (f1)–(f2). If \(\int _{0}^{1}g(s)\,ds=+\infty\), then (1.1) has no solution in D for any \(\lambda,\mu>0\).
This paper is organized as follows. Some preliminary lemmas are given in Sect. 2, and Sect. 3 is devoted to proofs of the results. Some ideas also come from [19] and [18].
2 Preliminaries
In this section, we consider the general problem
Let \(F(x,u):=F_{1}(x,u)+F_{2}(x,u)\).
Definition 2.1
The pair of functions \(\alpha,\beta\in C^{1}(\overline{\varOmega })\cap C^{2}(\varOmega)\) are a subsolution and supersolution of (2.1) if \(\alpha(x)\leq\beta(x)\) for \(x\in\varOmega\) and
where \(a_{0}=a(0)\) and \(b_{0}=a ( \int_{\varOmega}H(x)^{2}\,dx )\), \(E\in L^{p}(\varOmega)\) (\(p>N\)); here
\(G(x,y)\) is the Green function for \(-\triangle u(x)=h\), and \(u|_{\partial\varOmega}=0\).
Lemma 2.2
(see [27])
Let \(\varOmega\subseteq\mathbb{R}^{N}\) (\(N\geq 1\)) be a smooth bounded domain. Let \(F: \varOmega\times\mathbb {R}\rightarrow\mathbb{R}\) be a continuous function. Let α and β be the subsolution and supersolution of (2.1), respectively. If
then problem (2.1) has at least one solution u such that
Let \(\varphi_{1}\) be the normalized positive eigenfunction corresponding to the first eigenvalue \(\lambda_{1}\) of the problem
Lemma 2.3
(see [8])
\(\int_{\varOmega}\varphi_{1}^{-s}<\infty\) if and only if \(s<1\).
Lemma 2.4
(see [21])
Let \(F:\bar{\varOmega}\times(0,\infty )\rightarrow\mathbb{R}\) be a Hölder-continuous function with exponent \(\gamma\in(0,1)\) on each compact subset of \(\overline{\varOmega }\times(0,\infty)\) satisfying:
-
(F1)
\(\lim_{s\rightarrow\infty}\sup(s^{-1} \max_{x\in\bar {\varOmega}}F(x,s)) <\lambda_{1}\);
-
(F2)
for each \(t>0\), there exists a constant \(D(t)>0\) such that
$$F(x,r)-F(x,s)\geq-D(t) (r-s)\quad \textit{for } x\in\overline{\varOmega} \textit{ and } r\geq s\geq t; $$ -
(F3)
there exist \(\eta_{0}>0\) and an open subset \(\varOmega _{0}\subset\varOmega\) such that
$$\min_{x\in\bar{\varOmega}}F(x,s)\geq0 \quad \textit{for } x\in(0, \eta_{0}) $$and
$$\lim_{s\rightarrow0}\frac{F(x,s)}{s}=+\infty\quad \textit{uniformly in } x\in \varOmega_{0}. $$
Then for any nonnegative function \(\varphi_{0}\in C^{2,\gamma}(\partial \varOmega)\), the problem
has at least one positive solution \(u\in C^{2,\gamma}(G)\cap C(\overline {\varOmega})\) for any compact set \(G\subset\varOmega\cup\{x\in\partial\varOmega; \varphi_{0}(x)> 0\}\).
Lemma 2.5
(see [21])
Let \(F:\overline{\varOmega}\times(0,\infty )\rightarrow\mathbb{R}\) be a continuous function such that the mapping \((0,\infty)\ni s\mapsto\frac{F(x,s)}{s}\) is strictly decreasing at each \(x\in\varOmega\). Assume that there exists \(\nu,\omega\in C^{2}(\varOmega )\cap C(\overline{\varOmega})\) such that:
-
(a)
\(\triangle\omega+F(x,\omega)\leq0\leq\triangle\nu+F(x,\nu)\) in Ω;
-
(b)
\(\nu,\omega>0\) in Ω and \(\nu\leq\omega\) on ∂Ω;
-
(c)
\(\triangle\nu\in L^{1}(\varOmega)\).
Then \(\nu\leq\omega\) in Ω.
We observe that the hypotheses of Lemmas 2.4 and 2.5 are fulfilled for
Lemma 2.6
Let f satisfy (f1)–(f2). Then for any \(\lambda,\mu>0\), according to Lemmas 2.4 and 2.5, the boundary value problem
has a unique solution \(U_{\lambda,\mu}\in C^{2,\gamma}(\varOmega)\cap C(\overline{\varOmega})\).
3 Proofs of main theorems
Denote \(K^{\ast}= \max_{x\in\overline{\varOmega}}K(x)\), \(K_{\ast}= \min_{x\in\overline{\varOmega}}K(x)\).
Proof of Theorem 1.1
According to Lemma 2.6, the boundary value problem
has a unique solution \(U\in C^{2,\gamma}(\varOmega)\cap C(\overline{\varOmega })\). Let \(H: [0,\infty)\rightarrow[0,\infty)\) be such that
Obviously, \(H\in C^{2}(0,+\infty)\cap C^{1}[0,+\infty)\) exists by our assumption (g2). From (3.2) it follows that \(H''\) is nonincreasing, whereas H and \(H'\) are nondecreasing on \((0,\infty)\). Using this fact and applying the mean value theorem, we deduce that, for all \(t>0\), there exist \(\xi_{t}^{1},\xi_{t}^{2}\in(0,t)\) such that
and
These inequalities imply
Hence
On the other hand, by (g2) and (3.2) there exists \(\eta>0\) such that
which yields
where \(c>0\) is a constant.
Set
where M is a positive constant. Then we can get
Fix \(M\geq1\). The monotonicity of g leads to
and by (3.6)
By Hopf’s maximum principle there exist \(\delta_{0}\) and \(\varSigma\subset \varOmega\) such that
On \(\varOmega\setminus\varSigma\), we choose \(M\geq M_{1}= \max\{1,\frac {K^{\ast}}{a_{0}\delta^{2}}\}\). Then we have
Choosing \(M\geq\max\{M_{1},\frac{K^{\ast}g(H(\delta_{0}))}{a_{0}\lambda _{1}H'(\delta_{0})\delta_{0}}\}\), we have
It follows from (3.7)–(3.9) that
Since \(\varphi_{1}>0\) in Ω, from (3.3) we have
Take \(A_{0}=4a_{0}\lambda_{1}c^{-1}|\alpha(x)|_{\infty}\), where \(c = \inf_{x\in\bar{\varOmega}}f(x,|\alpha_{\lambda,\mu}|_{\infty})>0\). If \(\lambda>A_{0}\), then assumption (f1) produces
Combined with (3.12), this gives
Denote \(\varOmega_{0}=\{x\in\varOmega;\varphi_{1}(x)<\eta\}\). By (3.4) and (3.5) it follows that
These estimates combined with Lemma 2.3 yield \(g(\alpha_{\lambda,\mu })\in L^{1}(\varOmega)\), and so \(\triangle\alpha_{\lambda,\mu}\in L^{1}(\varOmega)\).
Set
where \(\lambda\geq\max\{1,\mu\}\).
We now prove that there exists \(A_{0}\) such that \(\alpha_{\lambda,\mu }\leq\beta_{\lambda,\mu}\) for all \(\lambda>A_{0}\).
Since
we can get
By Lemma 2.5 it follows that \(\alpha_{\lambda,\mu}\leq\beta_{\lambda,\mu }\) on Ω̅ for all \(\lambda>A_{0}\).
Define
that is,
We have
Let
that is,
where C is a positive constant. Since \(\lim_{\lambda\rightarrow +\infty}\frac{\lambda}{a(\lambda^{2+2n_{0}})}=+\infty\), there exists \(B_{0}\) such that
Choosing \(\bar{\lambda}=\max\{1,\mu, A_{0},B_{0}\}\), we easily to see that
Since
we get
Hence
By Lemma 2.2 (1.1) has at least one solution \(u_{\lambda,\mu}\) such that \(\alpha_{\lambda,\mu}\leq u_{\lambda,\mu}\leq\beta_{\lambda,\mu}\) in Ω̅. Since \(g(\alpha_{\lambda,\mu})\in L^{1}(\varOmega )\), we have \(\alpha_{\lambda,\mu}\in D\). By Remark 1.1 we deduce that \(u_{\lambda,\mu}\in D\). Hence, for all \(\lambda\geq\bar{\lambda}\) and \(\mu>0\), problem (1.1) has at least one solution \(u_{\lambda,\mu}\in D\).
Nonexistence for λ, μ small. Let \(\lambda,\mu>0\). Set
Since \(K_{\ast}>0\), assumption (g1) implies \(\lim_{s\rightarrow0}\varTheta _{\lambda,\mu}(x.s)=-\infty\) uniformly in \(x\in\overline{\varOmega}\). Then there exists \(c>0\) such that
Let \(s\geq c\). From (f1) we deduce
for all \(x\in\bar{\varOmega}\). Fix \(\mu\leq\frac{c\lambda _{1}a_{0}}{2|h|_{\infty}}\) and let \(M= \sup_{x\in\overline{\varOmega}}\frac {f(x,c)}{a_{0}c}>0\). From the above inequality we have
Set \(A(\lambda)=\lambda M\). Inequalities (3.13) and (3.14) yield
Moreover, \(A(\lambda)\rightarrow0\) as \(\lambda\rightarrow0\). If (1.1) has a solution \(u_{\lambda,u}\), then
and using (3.15), we get
Since \(A(\lambda)\rightarrow0\) as \(\lambda\rightarrow0\), this relation leads to a contradiction for \(\lambda,\mu>0\) sufficiently small. The proof of Theorem 1.1 is now complete. □
Proof of Theorem 1.2
Suppose to the contrary that exist λ and μ such that (1.1) has a solution \(u_{\lambda,\mu}\in D\), and by Lemma 2.6 the boundary value problem
has a unique solution \(V_{\lambda,\mu}\in C^{2,\gamma}(\varOmega)\cap C(\overline{\varOmega})\).
Set
Since
by Lemma 2.5 we get \(u_{\lambda,\mu}\leq V_{\lambda,\mu}\) in Ω.
Consider the perturbed problem
Since \(K_{\ast}>0\), it follows that \(u_{\lambda,\mu}\) and \(V_{\lambda ,\mu}\) are subsolution and supersolution for (3.17), respectively. Then there exists a solution \(u_{\epsilon}\in C^{2,\gamma}(\overline{\varOmega})\) of (3.17) such that
Integrating in (3.17), we deduce
Hence
where \(M>0\) is a constant. Since \(\frac{\partial u_{\epsilon }}{\partial n}\leq0\) on ∂Ω, relation (3.18) yields \(K_{\ast}\int_{\varOmega}g(u_{\epsilon}+\epsilon)\,dx\leq M\), and so \(K_{\ast }\int_{\varOmega}g(V_{\lambda,\mu}+\epsilon)\,dx\leq M\). Thus, for any compact subset \(\omega\subset\varOmega\), we have
Letting \(\epsilon\rightarrow0\), this relation leads to \(K_{\ast}\int _{\omega}g(V_{\lambda,\mu})\,dx\leq M\). Therefore
Choose \(\delta>0\) sufficiently small and define \(\varOmega_{\delta}:=\{x\in \varOmega; \operatorname{dist}(x,\partial\varOmega)\leq\delta\}\). Taking into account the regularity of the domain, we get that there exists \(k>0\) such that
Then
which contradicts (3.19). It follows that problem (1.1) has no solution in D, and the proof of Theorem 1.2 is now complete. □
References
Alves, C.O., Corrêa, F.J.S.A.: On existence of solutions for a class of problem involving a nonlinear operator. Commun. Appl. Nonlinear Anal. 8, 43–56 (2014)
Alves, C.O., Corrêa, F.J.S.A.: A sub-supersolution approach for a quasilinear Kirchhoff equation. J. Math. Phys. 56, 591–608 (2015)
Alves, C.O., Corrêa, F.J.S.A., Ma, T.M.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl. 49, 85–93 (2005)
Bensedik, A., Bouchekif, M.: On an elliptic equation of Kirchhoff type with a potential asymptotically linear at infinity. Math. Comput. Model. 49, 1089–1096 (2009)
Cheng, B., Wu, X.: Existence results of positive solutions of Kirchhoff type problems. Nonlinear Anal. 71, 4883–4892 (2009)
Ghergu, M., Rădulescu, V.: Sublinear singular elliptic problems with two parameters. J. Differ. Equ. 195, 520–536 (2003)
Kirchhoff, G.: Mechanik. Teubner, Leipzig (1883)
Lazer, A.C., McKenna, P.J.: On a singular nonlinear elliptic boundary value problem. Proc. Am. Math. Soc. 111, 721–730 (1991)
Liang, S., Repovš, D., Zhang, B.: On the fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity. Comput. Math. Appl. 75, 1778–1794 (2018)
Lions, J.L.: On some questions in boundary value problems of mathematical physics. In: Contemporary Developments in Continuum Mechanics and Partial Differential Equations. North-Holland Mathematics Studies, vol. 36, pp. 284–346 (1977)
Lu, S.S.: Multiple solutions for a Kirchhoff-type equation with general nonlinearity. Adv. Nonlinear Anal. 7, 293–306 (2016)
Ma, T.: Remarks on an elliptic equation of Kirchhoff type. Nonlinear Anal. 63, 1967–1977 (2005)
Mao, A., Luan, S.: Sign-changing of a class nonlocal quasilinear elliptic boundary value problems. J. Math. Anal. Appl. 383, 239–243 (2011)
Mao, A., Zhang, Z.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal., Theory Methods Appl. 70, 1275–1287 (2009)
Mao, A., Zhu, X.: Existence and multiplicity results for Kirchhoff problems. Mediterr. J. Math. 14, 58 (2017). https://doi.org/10.1007/s00009-017-0875-0
Molica Bisci, G., Rădulescu, V., Servadei, R.: Variational Methods for Nonlocal Fractional Problems. Encyclopedia of Mathematics and Its Applications, vol. 162. Cambridge University Press, Cambridge (2016)
Perera, K., Zhang, Z.: Nontrivial solutions of Kirchhoff-type problems via the Yang index. J. Differ. Equ. 221, 246–255 (2006)
Rădulescu, V.: Singular phenomena in nonlinear elliptic problems: from blow-up boundary solutions to equations with singular nonlinearities. In: Handbook of Differential Equations: Stationary Partial Differential Equations, vol. 4, Chap. 7, pp. 485–593 (2007)
Rădulescu, V.: Combined effects in nonlinear singular elliptic problems with convection. Rev. Roum. Math. Pures Appl. 53, 543–553 (2008)
Ricceri, B.: On an elliptic Kirchhoff-type problem depending on two parameters. J. Glob. Optim. 46, 543–549 (2010)
Shi, J., Yao, M.: On a singular nonlinear semilinear elliptic problem. Proc. R. Soc. Edinb. 128, 1389–1401 (1998)
Sun, J., Tang, C.: Existence and multiplicity of solutions for Kirchhoff type equations. Nonlinear Anal., Theory Methods Appl. 74, 1212–1222 (2011)
Xiang, M., Rădulescu, V., Zhang, B.: Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional p-Laplacian. Nonlinearity 29, 3186–3205 (2016)
Xiang, M., Rădulescu, V., Zhang, B.: Combined effects for fractional Schrödinger–Kirchhoff systems with critical nonlinearities. ESAIM Control Optim. Calc. Var. 24, 1249–1273 (2018)
Xiang, M., Rădulescu, V., Zhang, B.: Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions. Nonlinearity 31, 3228–3250 (2018)
Xu, L., Chen, H.: Ground state solutions for Kirchhoff-type equations with a general nonlinearity in the critical growth. Adv. Nonlinear Anal. 7, 535–546 (2018)
Yan, B., O’Regan, D., Agarwal, R.P.: The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method. An. Ştiinţ. Univ. ‘Ovidius’ Constanţa 26, 5–41 (2018)
Yang, M., Han, Z.: Existence and multiplicity results for Kirchhoff type problems with four-superlinear potentials. Appl. Anal. 91, 2045–2055 (2012)
Yang, Y., Zhang, J.: Nontrivial solutions of a class of nonlocal problems via local linking theory. Appl. Math. Lett. 23, 377–380 (2010)
Zhang, Z., Perera, K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317, 456–463 (2006)
Acknowledgements
The authors thank the referees for their comments.
Availability of data and materials
Not applicable.
Authors’ information
Not applicable.
Funding
This work is supported by the National Natural Science Foundation of China (61603226) and the Fund of Natural Science of Shandong Province (ZR2018MA022).
Author information
Authors and Affiliations
Contributions
Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
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
Di, K., Yan, B. The existence of positive solution for singular Kirchhoff equation with two parameters. Bound Value Probl 2019, 40 (2019). https://doi.org/10.1186/s13661-019-1154-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-019-1154-8