Modified quasilinear equations with strongly singular and critical exponential nonlinearity
-
Reshmi Biswas
Abstract
In this article, we study global multiplicity result for a class of modified quasilinear singular equations involving the critical exponential growth:
where
1 Introduction and statement of main results
In this article, we study the existence, nonexistence, and multiplicity of the positive solutions for the following modified quasilinear equation:
where
where
Without the quasilinear term
Such kind of equations have significant applications in the physical modelling related to the boundary layer phenomenon for viscous fluids, non-Newtonian fluids, etc. If
One of the main features of problem (
Theorem 1
(Trudinger-Moser inequality) Let
Moreover,
if and only if
Here, the Sobolev space
where
Motivated by the seminal work of Ambrosetti et al. [3], many researchers studied the global multiplicity results for singular-convex problems. In [19], the author proved the global multiplicity result for (1.2), considering critical polynomial growth as
On the other hand, for modified quasilinear Schrödinger equations involving singular nonlinearity, there are a few results in the existing literature. Authors in [5,30,34] discussed existence of multiple solutions of such equations with singular nonlinearity
Inspired by all these aforementioned works, in this article, we investigate the existence, multiplicity, and non-existence results (i.e., the global multiplicity result) for problem (
The modified quasilinear term
The critical exponential nature of
The growth of the singular nonlinearity falls into the range
We now state all the hypotheses imposed on the continuous function
Critical growth assumption: For any
There exists a constant
There exists a constant
Example 2
Consider
Remark 3
From the condition
since
Moreover, the condition
Thus, for any
Now for any
equipped with the norm
That is, the set
We consider the following eigenvalue problem :
Let
Now we state the main results of this article:
Theorem 4
Let
Theorem 5
Let
2 Variational framework
For
equipped with the usual norm denoted by
endowed with the norm
Since
Observe that, the functional
Now we gather some properties of
Lemma 6
The function
the function
the function
After employing the change of variable
Remark 7
The functional
where
since
Next, we check for the Gateaux differentiability of the functional
Lemma 8
Let
Proof
We have to prove that
For the first term and third term in the right-hand side of the aforementioned expression, the proof follows in a standard way. So, we are left to show for the second term, i.e., for the singular term. Since
Now using mean value theorem, for any
for some
Now recalling (2.4), we obtain
Thus, by applying the Lebesgue dominated convergence theorem and passing to the limit
This completes the proof.□
Thus, using the properties of the functions
Moreover, by applying Lemma 6 and following the idea as in [36, Proposition 2.2], one can show that if
Definition 9
A function
In the next lemma, we discuss some comparison type result related to our problem (2.7).
Lemma 10
Let
where
Proof
The proof of this lemma follows in a similar fashion as in [35, Lemma 2.2].□
Notations. In the next subsequent sections, we make use of the following notations:
If
For any function
If
The arrows
The arrow
For any
3 Proof of Theorem 4: Existence and non-existence results
In this section, to prove the existence and non-existence results for the problem (2.7), we first need to study the existence and regularity result for the following purely singular problem:
Now, we have the following result for problem (3.1).
Theorem 11
Let
the problem (3.1) has a unique solution for each
the solution
the map
Proof
Let us set
Furthermore, for the case
This, in combination with standard elliptic regularity theory, implies that
Again, using [2, Proposition 4.1] (also see [9, Theorem 2.2]), we obtain
Finally, by using
Remark 12
One can check that (3.1) is the transformed form (with the transformation
where
From the assumption
For any
Let us define the set
and let
Lemma 13
Assume that the conditions in Theorem
4
hold and let h be defined as in (2.2). Then the set
Proof
First, we consider the case
Using (3.7), Lemma 6-
for any
Now we can choose
Set
which implies that
Now, without loss of generality, let us assume that
Next, by the mean value theorem, there exists
as
Next, using Lemma 6-
In the last relation, using Theorem 1, with sufficiently small
Now using Lemma 6-
The last relation, together with the Lebesgue dominated convergence theorem, implies that
Since by
Now from the weak lower semi-continuity of the norm, we have
which yields that
Furthermore, recalling (3.11) and (3.14), we obtain
Thus,
This yields that
Next, we claim that
It can be derived from the last expression that
Now for any
Letting the limit
Again, by taking
Next, we discuss the case
where
Then by the maximum principle,
where
So, using Lemma 6-
Plugging (3.19) in (3.18), we infer that
for some sufficiently small constant
We claim that
In virtue of Lemma 6-
for sufficiently small
Now for
Next, we will show that
On the other hand, by
Testing (3.17) against the test function
Thus,
For
Again for
Then, (3.23), (3.24), and (3.25) combining with the Lebesgue dominated convergence theorem yield that
Since for any
In the next lemma, we aim to discuss the regularity result for the solutions of (2.7).
Lemma 14
Assume that
Proof
Let
with
Note that
Since
from (3.27), we deduce
In the last relation, letting
Now following the arguments as in [20, Lemma 10, Theorem C] combining with Theorem 1, from the last relation, we infer that
Now we claim that
Hence, the claim holds. Next, let
Then, arguing similarly as above, we obtain
Moreover, if
Finally, combining (3.29), (3.30), and (3.31) and recalling the standard elliptic regularity theory, it follows that
Remark 15
Using Lemma 14, and adapting the proof of Corollary 1.1 in [2] (also see [17]), one can show that if
The following lemma basically ensures the non-existence of solution to (2.7) for
Lemma 16
Let the conditions in Theorem
4
hold and let h be defined as in (2.2). Then,
Proof
From Lemma 13, we can infer that
We claim that there exist
where
Now we show that, up to some sub-sequence,
That is,
Arguing in a similar manner as in (3.33), it can be deduced that
Since
This implies that
Therefore,
In the next result, by using a sub-super solution technique, we show the existence of at least one solution to (2.7).
Proposition 17
Let the conditions in Theorem
4
be satisfied and let h be defined as in (2.2). Then for each
Proof
Let
Thus,
Hence, Lemma 10 yields that
Let
Next, for
For
since
Therefore, by the Lebesgue dominated convergence theorem,
Using the last three limits and the weak lower semicontinuity property of the norm, it follows that
Now we show that
For
where
for some
Moreover, using Lemma 6-
for all
where
We define the set
Arguing similarly, we have
Thus, from (3.41), we obtain
for all
Lemma 18
Assume that the conditions in Theorem
4
hold and let h be defined as in (2.2). Let
Proof
We prove this lemma for the case
Now suppose the statement of the lemma does not hold. So, let us assume that
Next, we define
Then,
where
From Proposition 17, we obtain
Let us prove (3.46). For any
Now recalling the proof of Proposition 17, we have
On the other hand, by
where
Given
This yields that
Similarly,
Now by the definition of the function
By using this combining with Lemma 6-
where in the last line, we used Theorem 1, (3.47), and (3.46). Similarly, from (3.45), we can show that
which is a contradiction to (3.42). Hence,
The next result is a consequence of Lemma 18, which yields that at the threshold level of
Proposition 19
Let the conditions in Theorem
4
be satisfied and let h be defined as in (2.2). Then for
Proof
From the definition of
For
Now Lemma 18 yields that
Plugging (3.50) in (3.51), we obtain
Thus, for all
Case-I:
Now (3.49) and (3.52) combining with Lemma 6-
Plugging (3.54) and (3.53) in (3.49) and using Lemma 6-
Case-II:
Again using (3.49), (3.52), Lemma 6-
This gives that
Using (3.57) and (3.53), from (3.49), we deduce
Case-III:
From (3.54), it follows that
Employing the last relation in (3.49) and using Theorem 11-
since
Thus, each of the expressions in (3.55), (3.59), (3.58) from Cases I, II, III, respectively, yields that
Therefore, up to a sub-sequence, there exists
Thus, by the Lebesgue dominated convergence theorem, for any
Next, we show that for any
To prove (3.62), first observe that for
Furthermore, for
Then repeating a similar argument as in (3.13), we obtain (3.62). Gathering (3.62) and (3.63), we finally infer that
Proof of Theorem 4
Combining Lemmas 13, 14, 16, and 18 along with Propositions 17 and 19, we infer that
4 Proof of Theorem 5: Multiplicity result
This section is dedicated toward establishing the existence of second solution of (2.7) using the mountain pass lemma in combination with Ekeland variational principle. Let us define the set
Since by Lemma 18,
(Zero Altitude):
(Mountain Pass): There exists a
Proposition 20
Let the conditions in Theorem
5
hold and let
Proof
Let us fix
Clearly
For
Letting
for all
Claim:
For
Clearly,
Note that
Furthermore, by using the fact that both
Hence, by employing the Lebesgue dominated convergence theorem, as
We define the sets
Then,
Therefore, as
Combining (4.3), (4.4), (4.5), (4.6), (4.7), and (4.8), letting
Since
Plugging (4.10) into (4.9), letting
Considering
for all
Now we show that
By applying the Brézis-Lieb lemma, we obtain
By putting
Let us denote
Now choosing
Since
In a similar way, we also have
Next, we show that
By the construction, we have
Case I:
Moreover, by Lemma 6-
due to the Sobolev inequality and the Hölder inequality. Now using (3.10),
Case II:
From this and the hypothesis
Case III:
since
Thus, (4.21) and (4.22), combining with the hypothesis
Therefore, taking into account (4.16) and (4.17), from (4.12), we finally obtain (4.11). This completes the proof of the proposition.□
Now we prove the existence of second solution in the case where
Now for
Then,
We choose
Lemma 21
Let the conditions in Theorem
5
and
Proof
We prove this lemma for the case
(i) By
for all
(ii). Suppose
Then,
From
Now (4.24) and (4.25) imply that
Using mean value theorem and Lemma 6-
where
Plugging the last relation in (4.26) and using Lemma 6-
On the other hand, again using the mean value theorem and Lemma 6-
where
In virtue of the relation
Now we estimate the right-hand side in (4.30).
By using the fact that
By
Therefore, for any
On the other hand, by Lemma 6-
Plugging (4.33) and (4.34) in (4.31) and using (4.29), for
Now incorporating (4.35) and (4.31) in (4.30) and using Hölder’s inequality, we obtain
Since
This is absurd since,
Next, we recall the following result due to Lions [24].
Theorem 22
Let
Proposition 23
Let the conditions in Theorem
5
hold and let
Proof
First, we define the complete metric space
with the metric
Using Lemma 21-
Then, in light of Lemma 21-
Set
Again, applying Ekeland’s variational principle to
Now choosing
In virtue of Lemma 6-
We claim that
For
Since
Next, for
Finally, for the case
which gives (4.39). Therefore, there exists a
Claim:
To establish this claim, it is sufficient to prove that
Now borrowing the similar arguments as in (4.17), we have
Next, we show that
Since
Now using (4.44) and Lemma 6-
The last relation, combining with the Lebesgue dominated convergence theorem, yields
as
Supposing the contrary, let us assume that
Set
Then,
Taking into account (4.47) and (4.48), we deduce
Hence, we choose
which together with the fact that
Now using Lemma 6-
thanks to the Lebesgue dominated convergence theorem. Thus, we obtain (4.41). Hence, taking into account (4.42) and (4.41) and arguing similarly as in the proof of Proposition 20, we can infer that
Proof of Theorem 5
From Proposition 23, it follows that
Acknowledgement
The authors are thankful to the editor and the reviewers for their valuable suggestions.
-
Funding information: Reshmi Biswas is funded by Fondecyt Postdocotal Project No. 3230657, Sarika Goyal is funded by SERB under the grant SPG/2022/002068, K. Sreenadh is funded by Janaki & K. A. Iyer Chair Professor grant.
-
Author contributions: Reshmi Biswas: writing – original draft preparation, Sarika Goyal: writing – original draft preparation, K. Sreenadh – writing, reviewing and editing.
-
Conflict of interest: The authors state no conflict of interest.
-
Data availability statement: No data were used for the research described in the article.
References
[1] A. Adimurthi, Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the n-Laplacian, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 17 (1990), 393–413. Search in Google Scholar
[2] A. Adimurthi and J. Giacomoni, Multiplicity of positive solutions for singular and critical elliptic problem in R2, Commun. Contemp. Math. 8 (2006), 621–656. 10.1142/S0219199706002222Search in Google Scholar
[3] A. Ambrosetti, H. Brézis, and G. Cerami, Combined effect of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519–543. 10.1006/jfan.1994.1078Search in Google Scholar
[4] M. Badiale and G. Tarantello, Existence and multiplicity for elliptic problems with critical growths and discontinuous nonlinearities, Nonlinear Anal. 29 (1997), 639–677. 10.1016/S0362-546X(96)00071-5Search in Google Scholar
[5] K. Bal, P. Garain, I. Mandal, and K. Sreenadh, Multiplicity result to a singular quasilinear Schrödinger equation, J. Math. Anal. Appl. 497 (2021), 124904. 10.1016/j.jmaa.2020.124904Search in Google Scholar
[6] F. Bass and N. N. Nasanov, Nonlinear electromagnetic-spin waves, Phys. Rep. 189 (1990), 165–223. 10.1016/0370-1573(90)90093-HSearch in Google Scholar
[7] H. Brézis and L. Nirenberg, H1 versus C1 local minimizers, C. R. Acad. Sci. Paris Sér. I Math. 317 (1983), no. 5, 465–472. Search in Google Scholar
[8] M. Colin and L. Jeanjean, Solutions for a quasilinear Schrödinger equation: A dual approach, Nonlinear Anal. 56 (2004), 213–226. 10.1016/j.na.2003.09.008Search in Google Scholar
[9] M. G. Crandall, P. H. Rabinowitz, and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations 2 (1977), 193–222. 10.1080/03605307708820029Search in Google Scholar
[10] D. G. de Figueiredo, O. H. Miyagaki, and B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations 3 (1995), no. 2, 139–153. 10.1007/BF01205003Search in Google Scholar
[11] J. M. do Ó, O. H. Miyagaki, and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations: The critical exponential case, Nonlinear Anal. 67 (2007), 3357–3372. 10.1016/j.na.2006.10.018Search in Google Scholar
[12] J. M. do Ó and U. Severo, Solitary waves for a class of quasilinear Schrödinger equations in dimension two, Calc. Var. Partial Differential Equations 38 (2010), 275–315. 10.1007/s00526-009-0286-6Search in Google Scholar
[13] J. M. do Ó, O. H. Miyagaki, and S. H. M. Soares, Soliton solutions for quasilinear Schrödinger equations with critical growth, J. Differential Equations 248 (2010), 722–744. 10.1016/j.jde.2009.11.030Search in Google Scholar
[14] R. Dhanya, S Prashanth, K. Sreenadh, and S. Tiwari, Critical growth elliptic problem in R2 with singular discontinuous nonlinearities, Adv. Differ. Equ. 19 (2014), 409–440. 10.57262/ade/1396558057Search in Google Scholar
[15] R. Dhanya, S. Prashanth, S. Tiwari, and K. Sreenadh, Elliptic problems in RN with critical and singular discontinuous nonlinearities, Complex Var. Elliptic Equ. 61 (2016), 1668–1688. 10.1080/17476933.2016.1198787Search in Google Scholar
[16] P. Drabek, A. Kufner, and F. Nicolosi, Quasilinear Elliptic Equations with Degenerations and Singularities. Walter de Gruyter, Berlin, 2011. Search in Google Scholar
[17] J. Giacomoni and K. Saoudi, Multiplicity of positive solutions for a singular and critical problem, Nonlinear Anal. 71 (2009), 4060–4077. 10.1016/j.na.2009.02.087Search in Google Scholar
[18] R. W. Hasse, A general method for the solution of nonlinear soliton and kink Schrödinger equation, Z. Phys. B. 37 (1980), 83–87. 10.1007/BF01325508Search in Google Scholar
[19] Y. Haitao, Multiplicity and asymptotic behaviour of positive solutions for a singular semilinear elliptic problem, J. Differential Equations 189 (2003), 487–512. 10.1016/S0022-0396(02)00098-0Search in Google Scholar
[20] N. Hirano, C. Saccon, and N. Shioji, Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations 9 (2004), 197–220. 10.57262/ade/1355867973Search in Google Scholar
[21] Y. He and G. B. Li, Concentrating soliton solutions for quasilinear Schrödinger equations involving critical Soblev exponents, Discrete Contin. Dyn. Syst. 36 (2016), 73–762. 10.3934/dcds.2016.36.731Search in Google Scholar
[22] L. Jeanjean, T. J. Luo, and Z. Q. Wang, Multiple normalized solutions for quasi-linear Schrödinger equations, J. Differential Equations 259 (2015), 3894–3928. 10.1016/j.jde.2015.05.008Search in Google Scholar
[23] S. Kurihara, Large-amplitude quasi-solitons in superfluids films, J. Phys. Soc. Japan 50 (1981), 3262–3267. 10.1143/JPSJ.50.3262Search in Google Scholar
[24] P. L. Lions, The concentration compactness principle in the calculus of variations part-I, Rev. Mat. Iberoamericana 1 (1985), 185–201. 10.4171/rmi/6Search in Google Scholar
[25] Q. Liu, X. Q. Liu, and Z. Q. Wang, Multiple sign-changing solutions for quasilinear elliptic equations via perturbation method, Comm. Partial Differential Equations 39 (2014), 2216–2239. 10.1080/03605302.2014.942738Search in Google Scholar
[26] J. Q. Liu, Y. Q. Wang, and Z. Q. Wang, Solutions for quasilinear Schrödinger equations via the Nehari method, Comm. Partial Differential Equations 29 (2004), 879–901. 10.1081/PDE-120037335Search in Google Scholar
[27] J. Q. Liu and Z. Q. Wang, Soliton solutions for quasilinear Schrödinger equations, I, Proc. Amer. Math. Soc. 131 (2003), 441–448. 10.1090/S0002-9939-02-06783-7Search in Google Scholar
[28] X. Q. Liu, J. Q. Liu, and Z. Q. Wang, Quasilinear elliptic equations with critical growth via perturbation method, J. Differential Equations 254 (2013), 102–124. 10.1016/j.jde.2012.09.006Search in Google Scholar
[29] J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077–1092. 10.1512/iumj.1971.20.20101Search in Google Scholar
[30] J. M. B. do Ó and A. Moameni, Solutions for singular quasilinear Schrödinger equations with one parameter, Commun. Pure Appl. Anal. 9 (2010), no. 4, 1011–1023. 10.3934/cpaa.2010.9.1011Search in Google Scholar
[31] M. Poppenberg, K. Schmitt, and Z. Q. Wang, On the existence of soliton solutions to quasilinear Schrödinger equations, Calc. Var. Partial Differential Equations 14 (2002), 329–344. 10.1007/s005260100105Search in Google Scholar
[32] B. Ritchie, Relativistic self-focusing and channel formation in laser-plasma interactions, Phys. Rev. E. 50 (1994), 687–689. 10.1103/PhysRevE.50.R687Search in Google Scholar
[33] D. Ruiz and G. Siciliano, Existence of ground states for a modified nonlinear Schrödinger equation, Nonlinearity 23 (2010), 1221–1233. 10.1088/0951-7715/23/5/011Search in Google Scholar
[34] G. Santos, G. M. Figueiredo, and U. B. Severo, Multiple solution for a class of singular quasilinear problems, J. Math. Anal. Appl. 480 (2019), 123405. 10.1016/j.jmaa.2019.123405Search in Google Scholar
[35] C. A. Santos, M. Yang, and J. Zhou, Global multiplicity of solutions for a modified elliptic problem with singular terms, Nonlinearity 34 (2021), 7842–7871. 10.1088/1361-6544/ac2a50Search in Google Scholar
[36] U. Severo, Existence of weak solutions for quasilinear elliptic equatiuons involving the p-Laplacian, Electron. J. Differential Equations 2008 (2008), no. 56, 1–16. Search in Google Scholar
© 2024 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
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
Articles in the same Issue
- Regular Articles
- Existence of normalized peak solutions for a coupled nonlinear Schrödinger system
- Orbital stability of the trains of peaked solitary waves for the modified Camassa-Holm-Novikov equation
- Global boundedness in a two-dimensional chemotaxis system with nonlinear diffusion and singular sensitivity
- Existence and uniqueness of solution for a singular elliptic differential equation
- The bounded variation capacity and Sobolev-type inequalities on Dirichlet spaces
- Vanishing and blow-up solutions to a class of nonlinear complex differential equations near the singular point
- Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions
- Concentration phenomena for a fractional relativistic Schrödinger equation with critical growth
- Beyond the classical strong maximum principle: Sign-changing forcing term and flat solutions
- Monotonicity of solutions for parabolic equations involving nonlocal Monge-Ampère operator
- On a nonlinear Robin problem with an absorption term on the boundary and L1 data
- Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities
- Gradient estimates for a class of higher-order elliptic equations of p-growth over a nonsmooth domain
- Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ3
- Global existence and finite-time blowup for a mixed pseudo-parabolic r(x)-Laplacian equation
- Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent
- A minimization problem with free boundary for p-Laplacian weakly coupled system
- Multiplicity of semiclassical solutions for a class of nonlinear Hamiltonian elliptic system
- k-convex solutions for multiparameter Dirichlet systems with k-Hessian operator and Lane-Emden type nonlinearities
- Infinitely many solutions for Hamiltonian system with critical growth
- On optimal control in a nonlinear interface problem described by hemivariational inequalities
- Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid
- Potential and monotone homeomorphisms in Banach spaces
- Critical fractional Schrödinger-Poisson systems with lower perturbations: the existence and concentration behavior of ground state solutions
- Supercritical Hénon-type equation with a forcing term
- Concentration of blow-up solutions for the Gross-Pitaveskii equation
- Mountain-pass-type solutions for Schrödinger equations in R2 with unbounded or vanishing potentials and critical exponential growth nonlinearities
- Regularity for critical fractional Choquard equation with singular potential and its applications
- Double phase anisotropic variational problems involving critical growth
- Normalized solutions of NLS equations with mixed nonlocal nonlinearities
- Nontrivial solutions for resonance quasilinear elliptic systems
- Standing waves for Choquard equation with noncritical rotation
- Low regularity conservation laws for Fokas-Lenells equation and Camassa-Holm equation
- Modified quasilinear equations with strongly singular and critical exponential nonlinearity
- Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent
- Nests of limit cycles in quadratic systems
- Regularity of minimizers for double phase functionals of borderline case with variable exponents
- Boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems with nonlinear boundary condition
- Ground state solutions for magnetic Schrödinger equations with polynomial growth
- Nonoccurrence of Lavrentiev gap for a class of functionals with nonstandard growth
- Dynamics for wave equations connected in parallel with nonlinear localized damping
- Boussinesq's equation for water waves: Asymptotics in Sector I
- Optimal global second-order regularity and improved integrability for parabolic equations with variable growth
- Normalized solutions of Schrödinger equations involving Moser-Trudinger critical growth
- Normalized solutions for the double-phase problem with nonlocal reaction
- Normalized solutions for Sobolev critical fractional Schrödinger equation
- Well-posedness of Cauchy problem of fractional drift diffusion system in non-critical spaces with power-law nonlinearity
- Improved results on planar Klein-Gordon-Maxwell system with critical exponential growth
- Existence and multiplicity of solutions for a new p(x)-Kirchhoff equation
- Quasiconvex bulk and surface energies: C1,α regularity
- Time decay estimates of solutions to a two-phase flow model in the whole space
- Bounds for the sum of the first k-eigenvalues of Dirichlet problem with logarithmic order of Klein-Gordon operators
- The properties of a new fractional g-Laplacian Monge-Ampère operator and its applications
- A fractional profile decomposition and its application to Kirchhoff-type fractional problems with prescribed mass
- Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content
- Multiplicity of normalized semi-classical states for a class of nonlinear Choquard equations
- The second Yamabe invariant with boundary
- Peaked solitary waves and shock waves of the Degasperis-Procesi-Kadomtsev-Petviashvili equation
- Normalized solutions for the Choquard equations with critical nonlinearities
- Boundedness and long-time behavior in a parabolic-elliptic system arising from biological transport networks
- Initial boundary value problem and exponential stability for the planar magnetohydrodynamics equations with temperature-dependent viscosity
- Nonexistence of mean curvature flow solitons with polynomial volume growth immersed in certain semi-Riemannian warped products
- Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment
- Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4
- Nonexistence of the compressible Euler equations with space-dependent damping in high dimensions
- Multiplicity of solutions for a nonhomogeneous quasilinear elliptic equation with concave-convex nonlinearities
- On semilinear inequalities involving the Dunkl Laplacian and an inverse-square potential outside a ball
- Besov regularity for the elliptic p-harmonic equations in the non-quadratic case
-
Uniqueness and nondegeneracy of ground states for
- Existence of solutions for a class of quasilinear Schrödinger equations with Choquard-type nonlinearity
- Weighted Hardy-Adams inequality on unit ball of any even dimension
- Existence and regularity for a p-Laplacian problem in ℝN with singular, convective, and critical reaction
- Temporal periodic solutions of non-isentropic compressible Euler equations with geometric effects
- Nodal solutions for a zero-mass Chern-Simons-Schrödinger equation
- The modified quasi-boundary-value method for an ill-posed generalized elliptic problem
- Nonlocal heat equations with generalized fractional Laplacian
- Choquard equations with recurrent potentials
- Local and global well-posedness of the Maxwell-Bloch system of equations with inhomogeneous broadening
- The Riemann problem for two-layer shallow water equations with bottom topography
- Global stability and asymptotic profiles of a partially degenerate reaction diffusion Cholera model with asymptomatic individuals
- On Kirchhoff-Schrödinger-Poisson-type systems with singular and critical nonlinearity
- Energy-variational solutions for viscoelastic fluid models
- Stability on 3D Boussinesq system with mixed partial dissipation
- Existence and multiplicity results for non-autonomous second-order Hamiltonian systems
- Erratum
- Erratum to “Normalized solutions for the Kirchhoff equation with combined nonlinearities in ℝ4”
