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Unilateral global interval bifurcation for Kirchhoff type problems and its applications
A nonlinear eigenvalue problem with $ p(x) $-growth and generalized Robin boundary value condition
1. | Institute of Mathematics "Simion Stoilow" of the Romanian Academy, P.O. Box 1-764,014700 Bucharest, Romania, Department of Mathematics, University of Craiova, Street A.I. Cuza 13,200585 Craiova, Romania |
2. | School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran |
$\begin{cases} -{\rm div}\,(a(x,\nabla u))=λ b(x,u)&\mbox{in} \ Ω\\\dfrac{\partial A}{\partial n}+β(x) c(x,u)=0&\mbox{on}\\partialΩ.\end{cases}$ |
$\begin{cases} -{\rm div}\,(a_0(x) |\nabla u|^{p(x)-2}\nabla u)=λ b_0(x)|u|^{q(x)-2}u&\mbox{in} \ Ω\\|\nabla u|^{p(x)-2}\dfrac{\partial u}{\partial n}+β(x)|u|^{r(x)-2} u=0&\mbox{on}\\partialΩ.\end{cases}$ |
References:
[1] |
R. Agarwal, M. B. Ghaemi and S. Saiedinezhad,
The existence of weak solution for degenerate $ \sum {{\Delta _{{p_i}(x)}}} $-equation, J. Comput. Anal. Appl., 13 (2011), 629-641.
|
[2] |
C. Alves and Marco A. S. Souto, Existence of solutions for a class of problems in $ {\mathbb R}^N $ involving the p(x)-Laplacian, in Contributions to nonlinear analysis, Birkhäuser Basel, (2005), 17-32.
doi: 10.1007/3-7643-7401-2_2. |
[3] |
R. Aronson,
Boundary conditions for diffusion of light, J. Opt. Soc. Am. A, 12 (1995), 2532-2539.
|
[4] |
F. Browder,
On the eigenfunctions and eigenvalues of the general linear elliptic differential operator, Proc. Nat. Acad. Sci. USA, 39 (1953), 433-439.
|
[5] |
F. Browder,
Lusternik-Schnirelmann category and nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 644-648.
doi: 10.1090/S0002-9904-1965-11378-7. |
[6] |
F. Browder,
Variational methods for nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 176-183.
doi: 10.1090/S0002-9904-1965-11275-7. |
[7] |
F. Browder,
Existence theorems for nonlinear partial differential equations, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), pp. 1-60, Amer. Math. Soc., Providence, R. I. |
[8] |
S.-G. Deng,
Eigenvalues of the $ p (x) $-Laplacian Steklov problem, J. Math. Anal. Appl., 339 (2008), 925-937.
doi: 10.1016/j.jmaa.2007.07.028. |
[9] |
X. Fan,
Remarks on eigenvalue problems involving the $ p (x) $-Laplacian, J. Math. Anal. Appl., 352 (2009), 85-98.
doi: 10.1016/j.jmaa.2008.05.086. |
[10] |
X. Fan, Q. Zhang and D. Zhao,
Eigenvalues of $ p (x) $-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306-317.
doi: 10.1016/j.jmaa.2003.11.020. |
[11] |
R. Filippucci, P. Pucci and V.D. Rădulescu,
Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Communications in Partial Differential Equations, 33 (2008), 706-717.
doi: 10.1080/03605300701518208. |
[12] |
Y. Fu and Y. Shan,
On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlinear Anal., 5 (2016), 121-132.
doi: 10.1515/anona-2015-0055. |
[13] |
O. Kovacik and J. Rakosnik,
On spaces $ L^{p (x)} $ and $ W^{k, p (x)} $, Czechoslovak Mathematical Journal, 41 (1991), 592-618.
|
[14] |
A. Le,
Eigenvalue problems for the $ p $-Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 1057-1099.
doi: 10.1016/j.na.2005.05.056. |
[15] |
L. A. Lusternik and L. G. Schnirelmann, Topological Methods in Variational Problems, Trudy
Inst. Mat. Mech. Moscow State Univ. (1930), 1-68. |
[16] |
M. Mihailescu and V. Răadulescu,
A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2625-2641.
doi: 10.1098/rspa.2005.1633. |
[17] |
C. V. Pao,
Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[18] |
V. Răadulescu,
Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 121 (2015), 336-369.
doi: 10.1016/j.na.2014.11.007. |
[19] |
V. Răadulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015.
doi: 10.1201/b18601. |
[20] |
D. Repovš,
Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.), 13 (2015), 645-661.
doi: 10.1142/S0219530514500420. |
[21] |
M. Ruzicka,
Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science & Business Media, New York, 2000. |
[22] |
S. Samko,
On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms and Special Functions, 16 (2005), 461-482.
doi: 10.1080/10652460412331320322. |
[23] |
O. Scherzer (Ed. ),
Handbook of Mathematical Methods in Imaging, Springer, Berlin, 2011. |
[24] |
J. Simon, Régularité de la solution d'une équation non linéaire dans $ {\mathbb R}^N $, Journées d'Analyse
Non Linéaire (Proc. Conf., Besan¸con, 1977), pp. 205-227, Lecture Notes in Math., 665, Springer, Berlin, 1978. |
[25] |
Z. Yücedag,
Solutions of nonlinear problems involving $ p(x) $-Laplacian operator, Adv. Nonlinear Anal., 4 (2015), 285-293.
doi: 10.1515/anona-2015-0044. |
[26] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅲ. Variational Methods and Optimization, Springer Science & Business Media, New York, 2013.
doi: 10.1007/978-1-4612-5020-3. |
[27] |
E. Zeidler,
The Lusternik-Schnirelmann theory for indefinite and not necessarily odd nonlinear operators and its applications, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 451-489.
doi: 10.1016/0362-546X(80)90085-1. |
[28] |
Q. Zhang,
Existence of solutions for $ p (x) $-Laplacian equations with singular coefficients in $ {\mathbb R}^N $, J. Math. Anal. Appl., 348 (2008), 38-50.
doi: 10.1016/j.jmaa.2008.06.026. |
show all references
References:
[1] |
R. Agarwal, M. B. Ghaemi and S. Saiedinezhad,
The existence of weak solution for degenerate $ \sum {{\Delta _{{p_i}(x)}}} $-equation, J. Comput. Anal. Appl., 13 (2011), 629-641.
|
[2] |
C. Alves and Marco A. S. Souto, Existence of solutions for a class of problems in $ {\mathbb R}^N $ involving the p(x)-Laplacian, in Contributions to nonlinear analysis, Birkhäuser Basel, (2005), 17-32.
doi: 10.1007/3-7643-7401-2_2. |
[3] |
R. Aronson,
Boundary conditions for diffusion of light, J. Opt. Soc. Am. A, 12 (1995), 2532-2539.
|
[4] |
F. Browder,
On the eigenfunctions and eigenvalues of the general linear elliptic differential operator, Proc. Nat. Acad. Sci. USA, 39 (1953), 433-439.
|
[5] |
F. Browder,
Lusternik-Schnirelmann category and nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 644-648.
doi: 10.1090/S0002-9904-1965-11378-7. |
[6] |
F. Browder,
Variational methods for nonlinear elliptic eigenvalue problems, Bull. Amer. Math. Soc., 71 (1965), 176-183.
doi: 10.1090/S0002-9904-1965-11275-7. |
[7] |
F. Browder,
Existence theorems for nonlinear partial differential equations, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., 1968), pp. 1-60, Amer. Math. Soc., Providence, R. I. |
[8] |
S.-G. Deng,
Eigenvalues of the $ p (x) $-Laplacian Steklov problem, J. Math. Anal. Appl., 339 (2008), 925-937.
doi: 10.1016/j.jmaa.2007.07.028. |
[9] |
X. Fan,
Remarks on eigenvalue problems involving the $ p (x) $-Laplacian, J. Math. Anal. Appl., 352 (2009), 85-98.
doi: 10.1016/j.jmaa.2008.05.086. |
[10] |
X. Fan, Q. Zhang and D. Zhao,
Eigenvalues of $ p (x) $-Laplacian Dirichlet problem, J. Math. Anal. Appl., 302 (2005), 306-317.
doi: 10.1016/j.jmaa.2003.11.020. |
[11] |
R. Filippucci, P. Pucci and V.D. Rădulescu,
Existence and non-existence results for quasilinear elliptic exterior problems with nonlinear boundary conditions, Communications in Partial Differential Equations, 33 (2008), 706-717.
doi: 10.1080/03605300701518208. |
[12] |
Y. Fu and Y. Shan,
On the removability of isolated singular points for elliptic equations involving variable exponent, Adv. Nonlinear Anal., 5 (2016), 121-132.
doi: 10.1515/anona-2015-0055. |
[13] |
O. Kovacik and J. Rakosnik,
On spaces $ L^{p (x)} $ and $ W^{k, p (x)} $, Czechoslovak Mathematical Journal, 41 (1991), 592-618.
|
[14] |
A. Le,
Eigenvalue problems for the $ p $-Laplacian, Nonlinear Analysis: Theory, Methods & Applications, 64 (2006), 1057-1099.
doi: 10.1016/j.na.2005.05.056. |
[15] |
L. A. Lusternik and L. G. Schnirelmann, Topological Methods in Variational Problems, Trudy
Inst. Mat. Mech. Moscow State Univ. (1930), 1-68. |
[16] |
M. Mihailescu and V. Răadulescu,
A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 462 (2006), 2625-2641.
doi: 10.1098/rspa.2005.1633. |
[17] |
C. V. Pao,
Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. |
[18] |
V. Răadulescu,
Nonlinear elliptic equations with variable exponent: old and new, Nonlinear Anal., 121 (2015), 336-369.
doi: 10.1016/j.na.2014.11.007. |
[19] |
V. Răadulescu and D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, CRC Press, Taylor & Francis Group, Boca Raton FL, 2015.
doi: 10.1201/b18601. |
[20] |
D. Repovš,
Stationary waves of Schrödinger-type equations with variable exponent, Anal. Appl. (Singap.), 13 (2015), 645-661.
doi: 10.1142/S0219530514500420. |
[21] |
M. Ruzicka,
Electrorheological Fluids: Modeling and Mathematical Theory, Springer Science & Business Media, New York, 2000. |
[22] |
S. Samko,
On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators, Integral Transforms and Special Functions, 16 (2005), 461-482.
doi: 10.1080/10652460412331320322. |
[23] |
O. Scherzer (Ed. ),
Handbook of Mathematical Methods in Imaging, Springer, Berlin, 2011. |
[24] |
J. Simon, Régularité de la solution d'une équation non linéaire dans $ {\mathbb R}^N $, Journées d'Analyse
Non Linéaire (Proc. Conf., Besan¸con, 1977), pp. 205-227, Lecture Notes in Math., 665, Springer, Berlin, 1978. |
[25] |
Z. Yücedag,
Solutions of nonlinear problems involving $ p(x) $-Laplacian operator, Adv. Nonlinear Anal., 4 (2015), 285-293.
doi: 10.1515/anona-2015-0044. |
[26] |
E. Zeidler, Nonlinear Functional Analysis and Its Applications, Ⅲ. Variational Methods and Optimization, Springer Science & Business Media, New York, 2013.
doi: 10.1007/978-1-4612-5020-3. |
[27] |
E. Zeidler,
The Lusternik-Schnirelmann theory for indefinite and not necessarily odd nonlinear operators and its applications, Nonlinear Analysis: Theory, Methods & Applications, 4 (1980), 451-489.
doi: 10.1016/0362-546X(80)90085-1. |
[28] |
Q. Zhang,
Existence of solutions for $ p (x) $-Laplacian equations with singular coefficients in $ {\mathbb R}^N $, J. Math. Anal. Appl., 348 (2008), 38-50.
doi: 10.1016/j.jmaa.2008.06.026. |
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