Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
-
Shanbing Li
and
Jianhua Wu
Abstract
This article is concerned with the stationary problem for a prey-predator model with prey-taxis/predator-taxis under homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. We make a detailed description of the global bifurcation structure of coexistence states and find the ranges of parameters for which there exist coexistence states. At the same time, some sufficient conditions for the nonexistence of coexistence states are also established. Our method of analysis uses the idea developed by Cintra et al. (Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619–657). Our results indicate that the presence of prey-taxis/predator-taxis makes mathematical analysis more difficult, and the Beddington-DeAngelis functional response leads to some different phenomena.
1 Introduction
The present article is concerned with the following Dirichlet problem of quasilinear elliptic equations:
where
Such a prey-predator system with predator-taxis and prey-taxis was proposed by Tsyganov et al. [35]. It should be noted that the relevance of attractive prey-taxis (i.e., “predators move towards their prey”) was first biologically verified by Kareiva and Odell in their study of heterogeneous aggregative patterns [20], and the repulsive predator-taxis (i.e., “prey moves away from their predators”) has been detected for crayfish seeking shelter [14]. As far as we know, among prey-predator systems with predator-taxis or prey-taxis, the corresponding Neumann problem has been studied most extensively (see, for instance, [2,13,18,19,22,31,34,36–39] and references therein). While less extensively studied than those with Neumann boundary conditions, those with Dirichlet conditions have been mathematically examined by Cintra et al. [4–6].
When there are no prey-taxis and predator-taxis effects (i.e.,
Main results. To present our main results, we introduce some notations and basic facts. For any given
has an infinite sequence of eigenvalues that are bound from below. We denote the
admits a unique positive solution if and only if
The first purpose of this article is to study the positive solutions of (1.1) in the case
By regarding
Theorem 1.1
For any given
If
If
there exists a positive number
there exists a continuum
the continuum
therefore, (1.2) admits at least one positive solution for
If
there exists a positive number
the semi-trivial solution
In Figure 1, an indication of the behavior of the global bifurcation branches of the positive solutions to (1.2) is shown, where drastic changes can be observed between the cases
![Figure 1
Possible bifurcation diagram of positive solutions to (1.2). (a)
d
N
σ
1
[
−
Δ
;
1
]
<
λ
<
d
N
σ
1
[
−
Δ
;
1
]
+
1
/
k
{d}_{N}{\sigma }_{1}\left[-\Delta ;\hspace{0.33em}1]\lt \lambda \lt {d}_{N}{\sigma }_{1}\left[-\Delta ;\hspace{0.33em}1]+1\hspace{0.1em}\text{/}\hspace{0.1em}k
(b)
λ
≥
d
N
σ
1
[
−
Δ
;
1
]
+
1
/
k
\lambda \ge {d}_{N}{\sigma }_{1}\left[-\Delta ;\hspace{0.33em}1]+1\hspace{0.1em}\text{/}\hspace{0.1em}k
.](/document/doi/10.1515/ans-2022-0060/asset/graphic/j_ans-2022-0060_fig_001.jpg)
Possible bifurcation diagram of positive solutions to (1.2). (a)
The second purpose of this article is to study the positive solutions of (1.1) in the case
By regarding
Theorem 1.2
For any given
If
If
(1.3) has no positive solution for
there exists a continuum
If
(1.3) has no positive solution for
there exists a continuum
In Figure 2, an indication of the behavior of the global bifurcation branches of the positive solutions to (1.3) is shown, where the differences can be observed between the cases
![Figure 2
Possible bifurcation diagram of positive solutions to (1.2). (a)
d
P
σ
1
[
−
Δ
;
1
]
−
γ
/
m
<
μ
<
d
P
σ
1
[
−
Δ
;
1
]
{d}_{P}{\sigma }_{1}\left[-\Delta ;\hspace{0.33em}1]-\gamma \hspace{0.1em}\text{/}\hspace{0.1em}m\lt \mu \lt {d}_{P}{\sigma }_{1}\left[-\Delta ;\hspace{0.33em}1]
(b)
μ
>
d
P
σ
1
[
−
Δ
;
1
]
\mu \gt {d}_{P}{\sigma }_{1}\left[-\Delta ;\hspace{0.33em}1]
.](/document/doi/10.1515/ans-2022-0060/asset/graphic/j_ans-2022-0060_fig_002.jpg)
Possible bifurcation diagram of positive solutions to (1.2). (a)
Remark 1.1
The unilateral bifurcation result used in this article is developed by Cintra et al. [5] (see Theorems 1.1 and 1.2 in [5], which is based on the unilateral bifurcation result of [23]). Precisely, Cintra et al. have extended the unilateral bifurcation result for semilinear elliptic systems to quasilinear elliptic systems. It is worth noting that another interesting bifurcation result for quasilinear elliptic systems developed by Shi and Wang has been widely used in recent years (see Theorem 4.4 in [32], which is based on Degree theory for
The contents of this article are as follows: in Section 2, we state some preliminary results that will be used repeatedly in later discussions. In Section 3, we study (1.2), where for both cases
2 Preliminaries
In this section, we will introduce two important lemmas that will be used repeatedly throughout this article. The first lemma provides some important properties of the principal eigenvalue
Lemma 2.1
For any fixed
admits a principal eigenvalue, which is denoted by
Furthermore, the following monotonicity properties hold:
the monotonicity of
if
if
if
The second lemma provides some information on the diffusive logistic equation (see, for instance, Lemma 2.3 in [4]).
Lemma 2.2
For any fixed
admits a unique positive solution, denoted by
where
If
By virtue of Lemma 2.2, it is clear that (2.1) admits a unique positive solution
and (2.2) admits a unique positive solution
3 Structure of solutions for
β
N
=
0
and
β
P
>
0
This section is devoted to the understanding of the global bifurcation structure of the set of positive solutions to (1.2) by treating
3.1 A priori estimates
The main purpose of this subsection is to prove some a priori estimates of positive solutions. The first proposition of this subsection gives the
Proposition 3.1
If
for all
Proof
Observe that
When
Thus, we have
This implies that
Let us denote
The second equation with the boundary condition of (1.2) can be rewritten as follows:
Suppose that
By virtue of (3.1), we have
and so,
Thus,
for all
The next proposition of this subsection gives the
Proposition 3.2
Let
Proof
For simplicity, we denote the positive constants by
for any
As above, we apply the elliptic regularity [15] to (3.1) to conclude that there exists a positive constant
Since
the triangular inequality and Proposition 3.1 yield
Thus, there exists a positive constant
Note that the second equation of (1.2) can be expressed as follows:
Since
for any
3.2 Nonexistence of positive solutions
The main purpose of this subsection is to study the nonexistence of positive solutions. By virtue of Proposition 3.1, one has known that (1.2) has no positive solution for
Proposition 3.3
If
Proof
Assume that
where
By virtue of Lemma 2.1 and Proposition 3.1, we have
Moreover, Lemma 2.1 also shows that the monotonicity of
Consequently, there exists a constant
For any fixed
Proposition 3.4
If
Proof
To achieve the proof, we will adapt the proof of Lemma 5.5 of [5], which in turn came from Proposition 6.5 of [10]. Suppose the conclusion is false. Then (1.2) admits at least one positive solution
Step 1: It follows from Proposition 3.1 that
In view of (3.1), we deduce that
Thus,
By virtue of Lemma 2.2, (3.2) has a unique positive solution if
where
is a sub-solution of (3.2). Thus, the sub-supersolution method and the uniqueness of the positive solution of (3.2) show that
Furthermore, it follows from Lemma 2.1 and Proposition 3.1 that
This, together with
That is, if
Consequently, we prove that
Step 2: For writing convenience, we denote
According to Theorem 4.1 in [33], it is well known that
By virtue of the first equation of (1.2), we obtain from Lemma 2.1 that
We further denote
Consequently, we prove that
Step 3: We next want to show
If so, then we obtain a contradiction since
By virtue of the variational characterization of the principal eigenvalue (see Lemma 2.1), it follows that
Thereby, for all
Taking the infimum for all
On the other hand,
This implies that
Since
as
In order to take the limit in (3.3) with respect to
By virtue of Lemma 2.1 and the inequality
This means that the principal eigenvalue
by choosing a subsequence if necessary. On the other hand, we multiply both sides of (3.4) by
Since
as
where
We now begin to take the limit in (3.3) with respect to
In addition, Lebesgue’s dominated convergence theorem ensures that
Thus, we derive
In view of (3.3), it follows from Poincaré inequality
By summarizing the above analysis, we obtain
This completes the whole proof of this proposition.□
3.3 Bifurcation structure of positive solutions
The main purpose of this subsection is to investigate the bifurcation structure of positive solutions to (1.2) by regarding
For any
which bifurcates from
We now apply the result of Crandall and Rabinowitz [7] on bifurcation from a simple eigenvalue to obtain a local result on bifurcation from
given by
Clearly,
In order to find a bifurcation point on the semi-trivial solution branch
By setting
Let
Since
This ensures the invertibility of the operator
where
This shows that
We now show that
As above, we set
Since the operator
is self-adjoint, it follows from the Fredholm alternative theorem that (3.6) has a solution
Thus, the invertibility of the operator
Therefore,
The desired result is obtained.
We further check the transversality condition:
By a simple calculation, we have
Suppose that (3.7) is not true. Then, there exists
As above, if this system admits a solution
This is a contradiction as
According to the local bifurcation theorem of Crandall and Rabinowitz (see Theorem 1.7 in [7]), positive solutions of (1.2) in a neighborhood of
for
We next calculate the signal of
satisfies
Let us multiply the above equation by
For the sake of convenience in writing, we denote
where
Then, (3.8) can be rewritten as follows:
Recall that
Let us multiply the above equation by
Let us divide both sides of this equation by
Setting
By virtue of the expressions of
Thereby, (3.11) becomes
Therefore, the bifurcation is supercritical if
Summarizing, we have the following result.
Theorem 3.1
Assume that
for
Although the above analysis provides some information on positive solutions of (1.2) in the neighborhood of
Proposition 3.5
Let
Then,
Proof
In view of the positivity of
for all
We next prove
The boundedness of
as
In addition, Lebesgue’s dominated convergence theorem ensures that
Thus, we derive
It follows from Poincaré inequality
On the other hand, by (3.13), we have
This completes the proof of Proposition 3.5.□
Proposition 3.6
Assume that
Proof
The linearization of
By the Riesz-Schauder theory, it is well known that the spectrum of
Since
Hence,
According to the local bifurcation theorem of Crandall and Rabinowitz [7], the necessary condition for bifurcation from
Since we hope to obtain positive solutions by bifurcating from
The following theorem gives the global structure of the local curve
Theorem 3.2
For any given
If
If
Proof
(1) Let
of positive solutions to (1.2) such that
there exists a positive solution
there exists another positive solution of (2.1), denoted by
We next claim that alternatives (a), (c), and (d) cannot occur. Since
(2) For any given
4 Structure of solutions for
β
N
>
0
and
β
P
=
0
In this section, we study (1.3) and investigate the global bifurcation structure of the set of positive solutions by treating
4.1 A priori estimates
Analogous to the case of (1.2), we first establish some a priori estimates of positive solutions to (1.3) so that we could make a detailed description for the global bifurcation structure of the set of positive solutions. Although the proofs of the next two propositions are similar to those of Propositions 3.1 and 3.2, we present them here for the reader’s convenience.
Proposition 4.1
If
for all
Proof
Suppose (1.3) has at least one positive solution
By virtue of the maximum principle (see, for instance, [24]), one sees that
When
Since
Thus,
We now consider the upper bound of
Then, it follows from the first equation of (1.3) that
Suppose that
By virtue of (4.1), we have
Since
With the help of Proposition 4.1, we establish the
Proposition 4.2
Let
Proof
For simplicity, we denote the positive constants by
for all
Note that
By virtue of the triangular inequality, we have
Consequently, there exists a positive constant
Observe that the first equation of (1.3) can be expressed as follows:
Since
for all
4.2 Nonexistence of positive solutions
The main purpose of this subsection is to give an appropriate nonexistence result of positive solutions to (1.3).
Proposition 4.3
If
Proof
If
In view of Poincaré’s inequality, one sees that
This implies that
4.3 Bifurcation structure of positive solutions
The main purpose of this subsection is to investigate the bifurcation structure of positive solutions to (1.3) by regarding
Suppose
We now use bifurcation techniques to make a direct investigation of this semi-trivial solution branch
Define
Clearly,
By setting
where
and
Since
Therefore, we apply the local bifurcation theorem of Crandall and Rabinowitz (see Theorem 1.7 in [7]) to conclude that positive solutions of (1.3) in the neighborhood of
for
Consequently, we have the following result.
Theorem 4.1
Assume that
for
The following theorem gives the global structure of the local curve
Theorem 4.2
If
Proof
By Definition 1.1 in [5],
of positive solutions to (1.3), where
there exists another positive solution of (2.2), denoted by
there exists a positive solution
We next claim that alternatives (b), (c), and (d) cannot occur. Since
This means that for any given
According to Proposition 4.3, one has known that if
which bifurcates from
Since bifurcation from
we first discuss the properties of this principal eigenvalue.
Proposition 4.4
Let
Then,
Proof
The proof can be completed by the similar arguments to those of Proposition 3.5.□
It can be proved as before that there is the local curve
Theorem 4.3
Assume that
There exists a neighborhood
for
Positive solutions contained in
The local curve
5 Summary and discussion
In this article, we study the stationary problem for a prey-predator model with prey-taxis/predator-taxis under the homogeneous Dirichlet boundary conditions, where the interaction is governed by a Beddington-DeAngelis functional response. By applying the local and global bifurcation theory, eigenvalue theory of the second-order linear elliptic operators, and various elliptic estimates, we establish the sufficient conditions for the existence/nonexistence of coexistence states. These results provide an easy way to predict the coexistence of two species and manage to explain the occurrence of stationary patterns. In the following, we compare the findings of this study with those of the previous articles as follows:
When the Lotka-Volterra type functional response is adopted (i.e.,
When the prey-taxis is ignored (i.e.,
This means that
Based on a priori estimate of solutions and standard elliptic regularity theory, it is not difficult to show that as
(5.1)Hence, it is reasonable to think that the existing results of this article have been perturbed from the existing results of [3] (see also [27]) as
There are various interesting questions that deserve further exploration. For the semilinear elliptic system (5.1), Casal et al. analyzed theoretically the multiplicity of positive solutions and found numerically the existence of a Hopf bifurcation in [3]. Some time later, some of these pioneering findings were sharpened by Du and Lou in [11,12]. However, it is unclear whether or not these results hold for the quasilinear elliptic systems (1.2) and (1.3). Moreover, when the domain of habitation is one-dimensional, the uniqueness of positive solutions of (5.1) has been established in [3] (see also the result of Dancer et al. [8] and the result of López-Gómez and Pardo [28]). Whether the method developed in [3] can successfully solve the uniqueness of positive solutions of (1.2) and (1.3) is unknown. In addition, a more interesting question is how to study the positive solutions of the complete system (1.1). All these questions are very interesting and worthwhile to pursue in the future.
Acknowledgements
The authors would like to thank three reviewers for their very careful readings and helpful suggestions, which significantly improved the initial draft.
-
Funding information: The work was partially supported by the National Science Foundation of China (11901446, 12171296), the Postdoctoral Science Foundation of China (2021T140530), the National Science Foundation in Shaanxi Province of China (2023-JC-YB-038), and the Fundamental Research Funds for the Central Universities (ZYTS23051).
-
Conflict of interest: The authors state that there is no conflict of interest.
References
[1] J. R. Beddington, Mutual interference between parasites or predators and its effect on searching efficiency, J. Animal Ecol. 44 (1975), 331–340. 10.2307/3866Search in Google Scholar
[2] Y. L. Cai, Q. Cao, and Z. A. Wang, Asymptotic dynamics and spatial patterns of a ratio-dependent predator-prey system with prey-taxis, Appl. Anal. 101 (2022), 81–99. 10.1080/00036811.2020.1728259Search in Google Scholar
[3] A. Casal, J. C. Eilbeck, and J. López-Gómez, Existence and uniqueness of coexistence states for a predator-prey model with diffusion, Diff. Int. Eqns. 7 (1994)411–439. 10.57262/die/1369330437Search in Google Scholar
[4] W. Cintra, C. Morales-Rodrigo, and A. Suárez, Coexistence states in a cross-diffusion system of a predator-prey model with predator satiation term, Math. Models Methods Appl. Sci. 28 (2018), 2131–2159. 10.1142/S0218202518400109Search in Google Scholar
[5] W. Cintra, C. Morales-Rodrigo, and A. Suárez, Unilateral global bifurcation for a class of quasilinear elliptic systems and applications, J. Differential Equations 267 (2019), 619–657. 10.1016/j.jde.2019.01.021Search in Google Scholar
[6] W. Cintra, C. A. dos Santos, and J. Z. Zhou, Coexistence states of a Holling type II predator-prey system with self and cross-diffusion terms, Discret. Contin. Dyn. Syst. Ser. B 27 (2022), 3913–3931. 10.3934/dcdsb.2021211Search in Google Scholar
[7] M. G. Crandall and P. H. Rabinowitz, Bifurcation from simple eigenvalues, J. Funct. Anal. 8 (1971), 321–340. 10.1016/0022-1236(71)90015-2Search in Google Scholar
[8] E. N. Dancer, J. López-Gómez, and R. Ortega, Ortega, On the spectrum of some linear noncooperative elliptic systems with radial symmetry, Diff. Int. Eqns. 8 (1995), 515–523. 10.57262/die/1369316502Search in Google Scholar
[9] D. L. DeAngelis, R. A. Goldstein, and R. V. O’Neill, A model for tropic interaction, Ecology 56 (1975), 881–892. 10.2307/1936298Search in Google Scholar
[10] M. Delgado and A. Suárez, Study of an elliptic system arising from angiogenesis with chemotaxis and flux at the boundary, J. Differential Equations 244 (2008), 3119–3150. 10.1016/j.jde.2007.12.007Search in Google Scholar
[11] Y. H. Du and Y. Lou, Some uniqueness and exact multiplicity results for a predator-prey model, Trans. Amer. Math. Soc. 349 (1997), 2443–2475. 10.1090/S0002-9947-97-01842-4Search in Google Scholar
[12] Y. H. Du and Y. Lou, S-shaped global bifurcation curve and Hopf bifurcation of positive solutions to a predator-prey model, J. Differential Equations 144 (1998), no. 2, 390–440. 10.1006/jdeq.1997.3394Search in Google Scholar
[13] M. Fuest, Global solutions near homogeneous steady states in a multidimensional population model with both predator- and prey-taxis, SIAM J. Math. Anal. 52 (2020), 5865–5891. 10.1137/20M1344536Search in Google Scholar
[14] J. E. Garvey, R. A. Stein, and H. M. Thomas, Assessing how fish predation and interspecific prey competition influence a crayfish assemblage, Ecology 75 (1994), 532–547. 10.2307/1939556Search in Google Scholar
[15] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second edition, Springer-Verlag, Berlin, 1983. Search in Google Scholar
[16] G. H. Guo and J. H. Wu, Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response, Nonlinear Anal. 72 (2010), 1632–1646. 10.1016/j.na.2009.09.003Search in Google Scholar
[17] G. H. Guo and J. H. Wu, The effect of mutual interference between predators on a predator-prey model with diffusion, J. Math. Anal. Appl. 389 (2012), 179–194. 10.1016/j.jmaa.2011.11.044Search in Google Scholar
[18] H. Y. Jin and Z. A. Wang, Global stability of prey-taxis systems, J. Differential Equations 262 (2017), 1257–1290. 10.1016/j.jde.2016.10.010Search in Google Scholar
[19] H. Y. Jin and Z. A. Wang, Global dynamics and spatio-temporal patterns of predator-prey systems with density-dependent motion, European J. Appl. Math. 32 (2021), 652–682. 10.1017/S0956792520000248Search in Google Scholar
[20] P. Kareiva and G. Odell, Swarms of predators exhibit “preytaxis” if individual predators use area-restricted search, Am. Nat. 130 (1987), 233–270. 10.1086/284707Search in Google Scholar
[21] S. Kesavan, Homogenization of elliptic eigenvalue problems. I, Appl. Math. Optim. 5 (1979), 153–167. 10.1007/BF01442551Search in Google Scholar
[22] J. M. Lee, T. Hillen, and M. A. Lewis, Pattern formation in prey-taxis systems, J. Biol. Dyn. 3 (2009), 551–573. 10.1080/17513750802716112Search in Google Scholar PubMed
[23] J. López-Gómez, Spectral Theory and Nonlinear Function Analysis, Chapman & Hall/CRC, New York, 2001. Search in Google Scholar
[24] J. López-Gómez, Linear Second Order Elliptic Operators, World Scientific Publishing Company, Singapore, 2013. 10.1142/8664Search in Google Scholar
[25] J. López-Gómez, Global bifurcation for Fredholm operators, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 539–564. Search in Google Scholar
[26] J. López-Gómez and C. Mora-Corral, Counting zeros of C1 Fredholm maps of index 1, Bull. Lond. Math. Soc. 37 (2005), 778–792. 10.1112/S0024609305004716Search in Google Scholar
[27] J. López-Gómez and E. Munnnoz-Hernández, A spatially heterogeneous predator-prey model, Discret. Contin. Dyn. Syst. Ser. B 26 (2021), 2085–2113. 10.3934/dcdsb.2020081Search in Google Scholar
[28] J. López-Gómez and R. Pardo, Existence and uniqueness of coexistence states for the predator-prey model with diffusion: The scalar case, Diff. Int. Eqns. 6 (1993), 1025–1031. 10.57262/die/1370021908Search in Google Scholar
[29] J. Pejsachowicz and P. J. Rabier, Degree theory for C1 Fredholm mappings of index 0, J. Anal. Math. 76 (1998), 289–319. 10.1007/BF02786939Search in Google Scholar
[30] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7 (1971), 487–513. 10.1016/0022-1236(71)90030-9Search in Google Scholar
[31] G. Q. Ren and B. Liu, Global existence and convergence to steady states for a predator-prey model with both predator- and prey-taxis, Discret. Contin. Dyn. Syst. 42 (2022), 759–779. 10.3934/dcds.2021136Search in Google Scholar
[32] J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations 246 (2009), 2788–2812. 10.1016/j.jde.2008.09.009Search in Google Scholar
[33] G. Stampacchia, Le problème de Dirichlet pour les équations elliptiques du second ordre coefficients discontinus, Ann. Inst. Fourier (Grenoble) 15 (1965), 189–257. 10.5802/aif.204Search in Google Scholar
[34] Y. S. Tao and M. Winkler, A fully cross-diffusive two-component evolution system: existence and qualitative analysis via entropy-consistent thin-film-type approximation, J. Funct. Anal. 281 (2021), 109069. 10.1016/j.jfa.2021.109069Search in Google Scholar
[35] M. A. Tsyganov, J. Brindley, A. V. Holden, and V. N. Biktashev, Quasisoliton interaction of pursuit-evasion waves in a predator-prey system, Phys. Rev. Lett. 91 (2003), 218102. 10.1103/PhysRevLett.91.218102Search in Google Scholar PubMed
[36] M. Winkler, Asymptotic homogenization in a three-dimensional nutrient taxis system involving food-supported proliferation, J. Differential Equations 263 (2017), 4826–4869. 10.1016/j.jde.2017.06.002Search in Google Scholar
[37] S. N. Wu, J. P. Shi, and B. Y. Wu, Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis, J. Differential Equations 260 (2016), 5847–5874. 10.1016/j.jde.2015.12.024Search in Google Scholar
[38] S. N. Wu, J. F. Wang, and J. P. Shi, Dynamics and pattern formation of a diffusive predator-prey model with predator-taxis, Math. Models Methods Appl. Sci. 28 (2018), 2275–2312. 10.1142/S0218202518400158Search in Google Scholar
[39] T. Xiang, Global dynamics for a diffusive predator-prey model with prey-taxis and classical Lotka-Volterra kinetics, Nonlinear Anal. Real. World Appl. 39 (2018), 278–299. 10.1016/j.nonrwa.2017.07.001Search in Google Scholar
© 2023 the author(s), published by De Gruyter
This work is licensed under the Creative Commons Attribution 4.0 International License.
Articles in the same Issue
- Research Articles
- Asymptotic properties of critical points for subcritical Trudinger-Moser functional
- The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
- On some dense sets in the space of dynamical systems
- Sharp profiles for diffusive logistic equation with spatial heterogeneity
- Generic properties of the Rabinowitz unbounded continuum
- Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
- Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
- Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
- The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
- A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
- Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
- Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
- Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
- Ground states of Schrödinger systems with the Chern-Simons gauge fields
- Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
- A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
- A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
- On a version of hybrid existence result for a system of nonlinear equations
- Special Issue: Geometric PDEs and applications
- Preface for the special issue on “Geometric Partial Differential Equations and Applications”
- Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
- Total mean curvatures of Riemannian hypersurfaces
- On degenerate case of prescribed curvature measure problems
- A curvature flow to the Lp Minkowski-type problem of q-capacity
- Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
- A note on second derivative estimates for Monge-Ampère-type equations
- The Lp chord Minkowski problem
- Widths of balls and free boundary minimal submanifolds
- Smooth approximation of twisted Kähler-Einstein metrics
- The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
- A Carleman inequality on product manifolds and applications to rigidity problems
- Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
- Pinched hypersurfaces are compact
- The spinorial energy for asymptotically Euclidean Ricci flow
- Geometry of CMC surfaces of finite index
- Capillary Schwarz symmetrization in the half-space
- Regularity of optimal mapping between hypercubes
- Special Issue: In honor of David Jerison
- Preface for the special issue in honor of David Jerison
- Homogenization of oblique boundary value problems
- A proof of a trace formula by Richard Melrose
- Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
- Regularity properties of monotone measure-preserving maps
- Examples of non-Dini domains with large singular sets
- Sharp inequalities for coherent states and their optimizers
- Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
- Propagation of symmetries for Ricci shrinkers
- Linear extension operators for Sobolev spaces on radially symmetric binary trees
- The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
- On an effective equation of the reduced Hartree-Fock theory
- Polynomial sequences in discrete nilpotent groups of step 2
- Integral inequalities with an extended Poisson kernel and the existence of the extremals
- On singular solutions of Lane-Emden equation on the Heisenberg group
Articles in the same Issue
- Research Articles
- Asymptotic properties of critical points for subcritical Trudinger-Moser functional
- The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
- On some dense sets in the space of dynamical systems
- Sharp profiles for diffusive logistic equation with spatial heterogeneity
- Generic properties of the Rabinowitz unbounded continuum
- Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
- Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
- Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
- The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
- A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
- Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
- Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
- Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
- Ground states of Schrödinger systems with the Chern-Simons gauge fields
- Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
- A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
- A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
- On a version of hybrid existence result for a system of nonlinear equations
- Special Issue: Geometric PDEs and applications
- Preface for the special issue on “Geometric Partial Differential Equations and Applications”
- Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
- Total mean curvatures of Riemannian hypersurfaces
- On degenerate case of prescribed curvature measure problems
- A curvature flow to the Lp Minkowski-type problem of q-capacity
- Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
- A note on second derivative estimates for Monge-Ampère-type equations
- The Lp chord Minkowski problem
- Widths of balls and free boundary minimal submanifolds
- Smooth approximation of twisted Kähler-Einstein metrics
- The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
- A Carleman inequality on product manifolds and applications to rigidity problems
- Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
- Pinched hypersurfaces are compact
- The spinorial energy for asymptotically Euclidean Ricci flow
- Geometry of CMC surfaces of finite index
- Capillary Schwarz symmetrization in the half-space
- Regularity of optimal mapping between hypercubes
- Special Issue: In honor of David Jerison
- Preface for the special issue in honor of David Jerison
- Homogenization of oblique boundary value problems
- A proof of a trace formula by Richard Melrose
- Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
- Regularity properties of monotone measure-preserving maps
- Examples of non-Dini domains with large singular sets
- Sharp inequalities for coherent states and their optimizers
- Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
- Propagation of symmetries for Ricci shrinkers
- Linear extension operators for Sobolev spaces on radially symmetric binary trees
- The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
- On an effective equation of the reduced Hartree-Fock theory
- Polynomial sequences in discrete nilpotent groups of step 2
- Integral inequalities with an extended Poisson kernel and the existence of the extremals
- On singular solutions of Lane-Emden equation on the Heisenberg group
