-
Previous Article
Optimal regularity for parabolic Schrödinger operators
- CPAA Home
- This Issue
-
Next Article
Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior
Multiple solutions for a class of $(p_1, \ldots, p_n)$-biharmonic systems
1. | Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, United States |
2. | Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran |
References:
[1] |
G. A. Afrouzi and S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the $(p_1, \ldots, p_n)$-Laplacian, Nonlinear Anal., 70 (2009), 135-143.
doi: 10.1016/j.na.2007.11.038. |
[2] |
G. A. Afrouzi and S. Heidarkhani, Multiplicity theorems for a class of Dirichlet quasilinear elliptic systems involving the $(p_1, \ldots, p_n)$-Laplacian, Nonlinear Anal., 73 (2010), 2594-2602.
doi: 10.1016/j.na.2010.06.038. |
[3] |
G. A. Afrouzi, S. Heidarkhani and D. O'Regan, Existence of three solutions for a doubly eigenvalue fourth-order boundary value problem, Taiwanese J. Math., 15 (2011), 201-210. |
[4] |
G. A. Afrouzi, S. Heidarkhani and D. O'Regan, Three solutions to a class of Neumann doubly eigenvalue elliptic systems driven by a $(p_1, \ldots, p_n)$-Laplacian, Bull. Korean Math. Soc., 47 (2010), 1235-1250.
doi: 10.4134/BKMS.2010.47.6.1235. |
[5] |
C. Amrouche, Singular boundary conditions and regularity for the biharmonic problem in the half-space, Commun. Pure Appl. Anal. 6 (2007), 957-982.
doi: 10.3934/cpaa.2007.6.957. |
[6] |
D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. Methods Nonlinear Anal., 22 (2003), 93-103. |
[7] |
D. Averna and G. Bonanno, A mountain pass theorem for a suitable class of functions, Rocky Mountain J. Math., 39 (2009), 707-727.
doi: 10.1216/RMJ-2009-39-3-707. |
[8] |
M. B. Ayed and M. Hammami, On a fourth order elliptic equation with critical nonlinearity in dimension six, Nonlinear Anal., 64 (2006), 924-957.
doi: 10.1016/j.na.2005.05.050. |
[9] |
M. Ayed and A. Selmi, Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain, Commun. Pure Appl. Anal., 9 (2012), 1705-1722. |
[10] |
Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270 (2002), 357-368.
doi: 10.1016/S0022-247X(02)00071-9. |
[11] |
L. Boccardo and D. Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonlinear Differential Equations Appl., 9 (2002), 309-323.
doi: 10.1007/s00030-002-8130-0. |
[12] |
G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176.
doi: 10.1016/j.jmaa.2008.01.049. |
[13] |
Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190 (2003), 239-267.
doi: 10.1016/S0022-0396(02)00112-2. |
[14] |
A. Cabada, J. A. Cid and L. Sanchez, Positivity and lower and upper solutions for fourth-order boundary value problems, Nonlinear Anal., 67 (2007), 1599-1612.
doi: 10.1016/j.na.2006.08.002. |
[15] |
J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $R^N$, Nonlinear Anal., 49 (2002), 861-884.
doi: 10.1016/S0362-546X(01)00144-4. |
[16] |
C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, Discrete Contin. Dyn. Syst., 28 (2010), 1033-1050.
doi: 10.3934/dcds.2010.28.1033. |
[17] |
A. Djellit and S. Tas, On some nonlinear elliptic systems, Nonlinear Anal., 59 (2004), 695-706. |
[18] |
A. Djellit and S. Tas, Quasilinear elliptic systems with critical Sobolev exponents in $R^N$, Nonlinear Anal., 66 (2007), 1485-1497.
doi: 10.1016/j.na.2006.02.005. |
[19] |
P. Drábek, N. M. Stavrakakis and N. B. Zographopoulos, Multiple nonsemitrivial solutions for quasilinear elliptic systems, Differential Integral Equations, 16 (2003), 1519-1531. |
[20] |
S. Federica, A biharmonic equation in $R^4$ involving nonlinearities with critical exponential growth, Commun. Pure Appl. Anal., 12 (2013), 405-428. |
[21] |
M. R. Grossinho, L. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation, Appl. Math. Lett., 18 (2005), 439-444.
doi: 10.1016/j.aml.2004.03.011. |
[22] |
G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equation, Nonlinear Anal., 68 (2008), 3646-3656.
doi: 10.1016/j.na.2007.04.007. |
[23] |
S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters, Nonlinear Anal., 73 (2010), 547-554.
doi: 10.1016/j.na.2010.03.051. |
[24] |
A. Kristály, Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains, Proc. Edinb. Math. Soc., 48 (2005), 465-477.
doi: 10.1017/S0013091504000112. |
[25] |
A. C. Lazer and P. J. McKenna, Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.
doi: 10.1137/1032120. |
[26] |
C. Li and C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian, Nonlinear Anal., 69 (2008), 3322-3329.
doi: 10.1016/j.na.2007.09.021. |
[27] |
C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal., 72 (2010), 1339-1347.
doi: 10.1016/j.na.2009.08.011. |
[28] |
L. Li and C.-L. Tang, Existence of three solutions for (p,q)-biharmonic systems, Nonlinear Anal., 73 (2010), 796-805.
doi: 10.1016/j.na.2010.04.018. |
[29] |
X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters, J. Math. Anal. Appl., 327 (2007), 362-375.
doi: 10.1016/j.jmaa.2006.04.021. |
[30] |
S. Liu and M. Squassina, On the existence of solutions to a fourth-order quasilinear resonant problem, Abstr. Appl. Anal., 7 (2002), 125-133.
doi: 10.1155/S1085337502000805. |
[31] |
A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal., 31 (1998), 895-908.
doi: 10.1016/S0362-546X(97)00446-X. |
[32] |
S. I. Pokhozhaev, On a constructive method of the calculus of variations, (Russian) Dokl. Akad. Nauk SSSR, 298 (1988), 1330-1333; translation in Soviet Math. Dokl., 37 (1988), 274-277. |
[33] |
B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226. |
[34] |
B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (2009), 3084-3089.
doi: 10.1016/j.na.2008.04.010. |
[35] |
J. Simon, Regularitè de la solution d'une equation non lineaire dans $R^N$, in "Journes d'Analyse Non Linaire" (P. Bénilan, J. Robert, eds.), Proc. Conf., Besanon, 1977, Lecture Notes in Math., 665, pp. 205-227, Springer, Berlin-Heidelberg-New York, 1978. |
[36] |
J. Su and Z. Liu, A bounded resonance problem for semilinear elliptic equations, Discrete Contin. Dyn. Syst., 19 (2007), 431-445.
doi: 10.3934/dcds.2007.19.431. |
[37] |
T. Teramoto, On Positive radial entire solutions of second-order quasilinear elliptic systems, J. Math. Anal. Appl., 282 (2003), 531-552.
doi: 10.1016/S0022-247X(03)00153-7. |
[38] |
W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of p-biharmonic type, J. Math. Anal. Appl., 348 (2008), 730-738.
doi: 10.1016/j.jmaa.2008.07.068. |
[39] |
Z. Wang, Nonradial positive solutions for a biharmonic critical growth problem, Commun. Pure Appl. Anal., 11 (2012), 517-545.
doi: 10.3934/cpaa.2012.11.517. |
[40] |
G. Warnault, Regularity of the extremal solution for a biharmonic problem with general nonlinearity, Commun. Pure Appl. Anal., 8 (2009), 1709-1723.
doi: 10.3934/cpaa.2009.8.1709. |
[41] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications," Vol. II, Berlin-Heidelberg-New York, 1985. |
[42] |
G. Q. Zhang, X. P. Liu and S. Y. Liu, Remarks on a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian, Electron. J. Differential Equations, 20 (2005), 1-10. |
[43] |
J. Zhang and S. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal., 60 (2005), 221-230. |
show all references
References:
[1] |
G. A. Afrouzi and S. Heidarkhani, Existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the $(p_1, \ldots, p_n)$-Laplacian, Nonlinear Anal., 70 (2009), 135-143.
doi: 10.1016/j.na.2007.11.038. |
[2] |
G. A. Afrouzi and S. Heidarkhani, Multiplicity theorems for a class of Dirichlet quasilinear elliptic systems involving the $(p_1, \ldots, p_n)$-Laplacian, Nonlinear Anal., 73 (2010), 2594-2602.
doi: 10.1016/j.na.2010.06.038. |
[3] |
G. A. Afrouzi, S. Heidarkhani and D. O'Regan, Existence of three solutions for a doubly eigenvalue fourth-order boundary value problem, Taiwanese J. Math., 15 (2011), 201-210. |
[4] |
G. A. Afrouzi, S. Heidarkhani and D. O'Regan, Three solutions to a class of Neumann doubly eigenvalue elliptic systems driven by a $(p_1, \ldots, p_n)$-Laplacian, Bull. Korean Math. Soc., 47 (2010), 1235-1250.
doi: 10.4134/BKMS.2010.47.6.1235. |
[5] |
C. Amrouche, Singular boundary conditions and regularity for the biharmonic problem in the half-space, Commun. Pure Appl. Anal. 6 (2007), 957-982.
doi: 10.3934/cpaa.2007.6.957. |
[6] |
D. Averna and G. Bonanno, A three critical points theorem and its applications to the ordinary Dirichlet problem, Topol. Methods Nonlinear Anal., 22 (2003), 93-103. |
[7] |
D. Averna and G. Bonanno, A mountain pass theorem for a suitable class of functions, Rocky Mountain J. Math., 39 (2009), 707-727.
doi: 10.1216/RMJ-2009-39-3-707. |
[8] |
M. B. Ayed and M. Hammami, On a fourth order elliptic equation with critical nonlinearity in dimension six, Nonlinear Anal., 64 (2006), 924-957.
doi: 10.1016/j.na.2005.05.050. |
[9] |
M. Ayed and A. Selmi, Asymptotic behavior and existence results for a biharmonic equation involving the critical Sobolev exponent in a five-dimensional domain, Commun. Pure Appl. Anal., 9 (2012), 1705-1722. |
[10] |
Z. Bai and H. Wang, On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl., 270 (2002), 357-368.
doi: 10.1016/S0022-247X(02)00071-9. |
[11] |
L. Boccardo and D. Figueiredo, Some remarks on a system of quasilinear elliptic equations, Nonlinear Differential Equations Appl., 9 (2002), 309-323.
doi: 10.1007/s00030-002-8130-0. |
[12] |
G. Bonanno and B. Di Bella, A boundary value problem for fourth-order elastic beam equations, J. Math. Anal. Appl., 343 (2008), 1166-1176.
doi: 10.1016/j.jmaa.2008.01.049. |
[13] |
Y. Bozhkov and E. Mitidieri, Existence of multiple solutions for quasilinear systems via fibering method, J. Differential Equations, 190 (2003), 239-267.
doi: 10.1016/S0022-0396(02)00112-2. |
[14] |
A. Cabada, J. A. Cid and L. Sanchez, Positivity and lower and upper solutions for fourth-order boundary value problems, Nonlinear Anal., 67 (2007), 1599-1612.
doi: 10.1016/j.na.2006.08.002. |
[15] |
J. Chabrowski and J. Marcos do Ó, On some fourth-order semilinear elliptic problems in $R^N$, Nonlinear Anal., 49 (2002), 861-884.
doi: 10.1016/S0362-546X(01)00144-4. |
[16] |
C. Cowan, P. Esposito and N. Ghoussoub, Regularity of extremal solutions in fourth order nonlinear eigenvalue problems on general domains, Discrete Contin. Dyn. Syst., 28 (2010), 1033-1050.
doi: 10.3934/dcds.2010.28.1033. |
[17] |
A. Djellit and S. Tas, On some nonlinear elliptic systems, Nonlinear Anal., 59 (2004), 695-706. |
[18] |
A. Djellit and S. Tas, Quasilinear elliptic systems with critical Sobolev exponents in $R^N$, Nonlinear Anal., 66 (2007), 1485-1497.
doi: 10.1016/j.na.2006.02.005. |
[19] |
P. Drábek, N. M. Stavrakakis and N. B. Zographopoulos, Multiple nonsemitrivial solutions for quasilinear elliptic systems, Differential Integral Equations, 16 (2003), 1519-1531. |
[20] |
S. Federica, A biharmonic equation in $R^4$ involving nonlinearities with critical exponential growth, Commun. Pure Appl. Anal., 12 (2013), 405-428. |
[21] |
M. R. Grossinho, L. Sanchez and S. A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation, Appl. Math. Lett., 18 (2005), 439-444.
doi: 10.1016/j.aml.2004.03.011. |
[22] |
G. Han and Z. Xu, Multiple solutions of some nonlinear fourth-order beam equation, Nonlinear Anal., 68 (2008), 3646-3656.
doi: 10.1016/j.na.2007.04.007. |
[23] |
S. Heidarkhani and Y. Tian, Multiplicity results for a class of gradient systems depending on two parameters, Nonlinear Anal., 73 (2010), 547-554.
doi: 10.1016/j.na.2010.03.051. |
[24] |
A. Kristály, Existence of two non-trivial solutions for a class of quasilinear elliptic variational systems on strip-like domains, Proc. Edinb. Math. Soc., 48 (2005), 465-477.
doi: 10.1017/S0013091504000112. |
[25] |
A. C. Lazer and P. J. McKenna, Large amplitude periodic oscillations in suspension bridges: Some new connections with nonlinear analysis, SIAM Rev., 32 (1990), 537-578.
doi: 10.1137/1032120. |
[26] |
C. Li and C.-L. Tang, Three solutions for a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian, Nonlinear Anal., 69 (2008), 3322-3329.
doi: 10.1016/j.na.2007.09.021. |
[27] |
C. Li and C.-L. Tang, Three solutions for a Navier boundary value problem involving the p-biharmonic, Nonlinear Anal., 72 (2010), 1339-1347.
doi: 10.1016/j.na.2009.08.011. |
[28] |
L. Li and C.-L. Tang, Existence of three solutions for (p,q)-biharmonic systems, Nonlinear Anal., 73 (2010), 796-805.
doi: 10.1016/j.na.2010.04.018. |
[29] |
X.-L. Liu and W.-T. Li, Existence and multiplicity of solutions for fourth-order boundary value problems with parameters, J. Math. Anal. Appl., 327 (2007), 362-375.
doi: 10.1016/j.jmaa.2006.04.021. |
[30] |
S. Liu and M. Squassina, On the existence of solutions to a fourth-order quasilinear resonant problem, Abstr. Appl. Anal., 7 (2002), 125-133.
doi: 10.1155/S1085337502000805. |
[31] |
A. M. Micheletti and A. Pistoia, Multiplicity results for a fourth-order semilinear elliptic problem, Nonlinear Anal., 31 (1998), 895-908.
doi: 10.1016/S0362-546X(97)00446-X. |
[32] |
S. I. Pokhozhaev, On a constructive method of the calculus of variations, (Russian) Dokl. Akad. Nauk SSSR, 298 (1988), 1330-1333; translation in Soviet Math. Dokl., 37 (1988), 274-277. |
[33] |
B. Ricceri, On a three critical points theorem, Arch. Math. (Basel), 75 (2000), 220-226. |
[34] |
B. Ricceri, A three critical points theorem revisited, Nonlinear Anal., 70 (2009), 3084-3089.
doi: 10.1016/j.na.2008.04.010. |
[35] |
J. Simon, Regularitè de la solution d'une equation non lineaire dans $R^N$, in "Journes d'Analyse Non Linaire" (P. Bénilan, J. Robert, eds.), Proc. Conf., Besanon, 1977, Lecture Notes in Math., 665, pp. 205-227, Springer, Berlin-Heidelberg-New York, 1978. |
[36] |
J. Su and Z. Liu, A bounded resonance problem for semilinear elliptic equations, Discrete Contin. Dyn. Syst., 19 (2007), 431-445.
doi: 10.3934/dcds.2007.19.431. |
[37] |
T. Teramoto, On Positive radial entire solutions of second-order quasilinear elliptic systems, J. Math. Anal. Appl., 282 (2003), 531-552.
doi: 10.1016/S0022-247X(03)00153-7. |
[38] |
W. Wang and P. Zhao, Nonuniformly nonlinear elliptic equations of p-biharmonic type, J. Math. Anal. Appl., 348 (2008), 730-738.
doi: 10.1016/j.jmaa.2008.07.068. |
[39] |
Z. Wang, Nonradial positive solutions for a biharmonic critical growth problem, Commun. Pure Appl. Anal., 11 (2012), 517-545.
doi: 10.3934/cpaa.2012.11.517. |
[40] |
G. Warnault, Regularity of the extremal solution for a biharmonic problem with general nonlinearity, Commun. Pure Appl. Anal., 8 (2009), 1709-1723.
doi: 10.3934/cpaa.2009.8.1709. |
[41] |
E. Zeidler, "Nonlinear Functional Analysis and its Applications," Vol. II, Berlin-Heidelberg-New York, 1985. |
[42] |
G. Q. Zhang, X. P. Liu and S. Y. Liu, Remarks on a class of quasilinear elliptic systems involving the $(p,q)$-Laplacian, Electron. J. Differential Equations, 20 (2005), 1-10. |
[43] |
J. Zhang and S. Li, Multiple nontrivial solutions for some fourth-order semilinear elliptic problems, Nonlinear Anal., 60 (2005), 221-230. |
[1] |
Zheng-Hai Huang, Nan Lu. Global and global linear convergence of smoothing algorithm for the Cartesian $P_*(\kappa)$-SCLCP. Journal of Industrial and Management Optimization, 2012, 8 (1) : 67-86. doi: 10.3934/jimo.2012.8.67 |
[2] |
Nadia Hazzam, Zakia Kebbiche. A primal-dual interior point method for $ P_{\ast }\left( \kappa \right) $-HLCP based on a class of parametric kernel functions. Numerical Algebra, Control and Optimization, 2021, 11 (4) : 513-531. doi: 10.3934/naco.2020053 |
[3] |
Mohan Mallick, R. Shivaji, Byungjae Son, S. Sundar. Bifurcation and multiplicity results for a class of $n\times n$ $p$-Laplacian system. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1295-1304. doi: 10.3934/cpaa.2018062 |
[4] |
Yuxiang Zhang, Shiwang Ma. Some existence results on periodic and subharmonic solutions of ordinary $P$-Laplacian systems. Discrete and Continuous Dynamical Systems - B, 2009, 12 (1) : 251-260. doi: 10.3934/dcdsb.2009.12.251 |
[5] |
Vladimir Ejov, Anatoli Torokhti. How to transform matrices $U_1, \ldots, U_p$ to matrices $V_1, \ldots, V_p$ so that $V_i V_j^T= {\mathbb O} $ if $ i \neq j $?. Numerical Algebra, Control and Optimization, 2012, 2 (2) : 293-299. doi: 10.3934/naco.2012.2.293 |
[6] |
Hui Liu, Ling Zhang. Multiplicity of closed Reeb orbits on dynamically convex $ \mathbb{R}P^{2n-1} $ for $ n\geq2 $. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1801-1816. doi: 10.3934/dcds.2021172 |
[7] |
Marcin Dumnicki, Tomasz Szemberg, Halszka Tutaj-Gasińska. New results on fat points schemes in $\mathbb{P}^2$. Electronic Research Announcements, 2013, 20: 51-54. doi: 10.3934/era.2013.20.51 |
[8] |
Pasquale Candito, Giovanni Molica Bisci. Multiple solutions for a Navier boundary value problem involving the $p$--biharmonic operator. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 741-751. doi: 10.3934/dcdss.2012.5.741 |
[9] |
Jiří Benedikt. Continuous dependence of eigenvalues of $p$-biharmonic problems on $p$. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1469-1486. doi: 10.3934/cpaa.2013.12.1469 |
[10] |
Leszek Gasiński, Nikolaos S. Papageorgiou. Three nontrivial solutions for periodic problems with the $p$-Laplacian and a $p$-superlinear nonlinearity. Communications on Pure and Applied Analysis, 2009, 8 (4) : 1421-1437. doi: 10.3934/cpaa.2009.8.1421 |
[11] |
Vincenzo Ambrosio, Teresa Isernia. Multiplicity and concentration results for some nonlinear Schrödinger equations with the fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5835-5881. doi: 10.3934/dcds.2018254 |
[12] |
Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177 |
[13] |
Jorge Garcia Villeda. A computable formula for the class number of the imaginary quadratic field $ \mathbb Q(\sqrt{-p}), \ p = 4n-1 $. Electronic Research Archive, 2021, 29 (6) : 3853-3865. doi: 10.3934/era.2021065 |
[14] |
Giuseppina Barletta, Gabriele Bonanno. Multiplicity results to elliptic problems in $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems - S, 2012, 5 (4) : 715-727. doi: 10.3934/dcdss.2012.5.715 |
[15] |
Claudianor O. Alves, J. V. Gonçalves, Olimpio Hiroshi Miyagaki. Remarks on multiplicity of positive solutions of nonlinear elliptic equations in $IR^N$ with critical growth. Conference Publications, 1998, 1998 (Special) : 51-57. doi: 10.3934/proc.1998.1998.51 |
[16] |
Jie Xiao. On the variational $p$-capacity problem in the plane. Communications on Pure and Applied Analysis, 2015, 14 (3) : 959-968. doi: 10.3934/cpaa.2015.14.959 |
[17] |
Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure and Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729 |
[18] |
Mohamed Karim Hamdani, Lamine Mbarki, Mostafa Allaoui, Omar Darhouche, Dušan D. Repovš. Existence and multiplicity of solutions involving the $ p(x) $-Laplacian equations: On the effect of two nonlocal terms. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022129 |
[19] |
Jean Dolbeault, Marta García-Huidobro, Rául Manásevich. Interpolation inequalities in $ \mathrm W^{1,p}( {\mathbb S}^1) $ and carré du champ methods. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 375-394. doi: 10.3934/dcds.2020014 |
[20] |
Zhongliang Wang. Nonradial positive solutions for a biharmonic critical growth problem. Communications on Pure and Applied Analysis, 2012, 11 (2) : 517-545. doi: 10.3934/cpaa.2012.11.517 |
2021 Impact Factor: 1.273
Tools
Metrics
Other articles
by authors
[Back to Top]