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Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents
Nonlinear Neumann equations driven by a nonhomogeneous differential operator
1. | College of Mathematics, Shandong Normal University, Jinan, Shandong |
2. | Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780 |
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, "Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints," Memoirs of AMS, vol. 196, no. 915, 2008. |
[2] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Annali di Mat Pura Appl., 188 (2009), 679-719.
doi: doi:10.1007/s10231-009-0096-7. |
[3] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, The spectrum and an index formula for the Neumann $p$-Laplacian and multiple solutions for problems with a crossing nonlinearity, Discrete Contin. Dynamical Systems, 25 (2009), 431-456. |
[4] |
P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlin. Anal., 7 (1983), 981-1012.
doi: doi:10.1016/0362-546X(83)90115-3. |
[5] |
T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal., 28 (1997), 419-441.
doi: doi:10.1016/0362-546X(95)00167-T. |
[6] |
H. Brezis and Louis Nirenberg, $H^1$-versus $C^1$ local minimizers, CRAS Paris, Ser. J. Math., 317 (1993), 465-472. |
[7] |
S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2002), 613-625.
doi: doi:10.1155/S1085337502207010. |
[8] |
E. Casas and L. Fernandez, A Green's formula for quasilinear elliptic operators, J. Math. Anal. Appl., 142 (1989), 62-73.
doi: doi:10.1016/0022-247X(89)90164-9. |
[9] |
K. C. Chang, "Infinite Dimensional Morse theory and Multiple Solution Problems," Birkhauser, Boston, 1993. |
[10] |
J. A. Cid and P. J. Torres, Solvability for some boundary value problems with $p$-Laplacian operators, Discrete Contin. Dynamical Systems, 23 (2009), 727-732. |
[11] |
D. Costa and C. Magalhaes, Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal., 24 (1995), 409-418.
doi: doi:10.1016/0362-546X(94)E0046-J. |
[12] |
L. Damascellli, Comparision theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincare Analyse Non linenire, 15 (1998), 493-516. |
[13] |
N. Dancer and K. Perera, Some remarks on the Fučik spectrum of the $p$-Laplacian and critical groups, J. Math. Anal. Appl., 254 (2001), 164-177.
doi: doi:10.1006/jmaa.2000.7228. |
[14] |
P. De Mapoli and C. Mariani, Mountain pass solutions to equations of $p$-Laplacian type, Nonlinear Anal., 54 (2003), 1205-1219.
doi: doi:10.1016/S0362-546X(03)00105-6. |
[15] |
F. de Paiva and H. R. Quoirin, Resonance and nonresonance for $p$-Laplacian problems with weighted eigenvalues conditions, Discrete Contin. Dynamical Systems, 25 (2009), 1219-1227. |
[16] |
G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electronic J. Diff. Equas., 8 (2002), 1-12. |
[17] |
M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nontrivial solutions with precise sign data for a $p$-Laplacian equation, Discrete Contin. Dynamical Systems, 25 (2009), 405-440. |
[18] |
J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quarsilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. |
[19] |
L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Elliptic Boundary Value Problems," Chapman & Hall / CRC Press, Baca Raton, 2005. |
[20] |
L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006. |
[21] |
L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian type equations, Adv. Nonlin. Studies, 8 (2008), 843-870. |
[22] |
A. Granas and J. Dugundji, "Fixed Point Theory," Springer, New York, 2003. |
[23] |
Z. Guo and Z. Zhang, $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Anal., 286 (2003), 32-50.
doi: doi:10.1016/S0022-247X(03)00282-8. |
[24] |
A. Kristaly, H. Lisei and C. Varga, Multiple solutions for $p$-Laplacian type equations, Nonlinear Anal., 68 (2008), 1375-1381.
doi: doi:10.1016/j.na.2006.12.031. |
[25] |
O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Acad. Press., New York, 1968. |
[26] |
G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.
doi: doi:10.1016/0362-546X(88)90053-3. |
[27] |
S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236.
doi: doi:10.1016/j.jmaa.2005.04.034. |
[28] |
J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600.
doi: doi:10.1112/S0024609304004023. |
[29] |
M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations, Nonlinear Anal., 37 (1991), 431-448.
doi: doi:10.1016/S0362-546X(98)00057-1. |
[30] |
D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Differential Eqns., 232 (2007), 1-35.
doi: doi:10.1016/j.jde.2006.09.008. |
[31] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J., 58 (2009), 1257-1279.
doi: doi:10.1512/iumj.2009.58.3565. |
[32] |
N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009. |
[33] |
E. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian, J. Funct. Anal., 244 (2007), 63-77.
doi: doi:10.1016/j.jfa.2006.11.015. |
[34] |
E. Papageorgiou and N. S. Papageorgiou, Multiplicity of solutions for a class of resonant $p$-Laplacian Dirichlet problems, Pacific J. Math., 241 (2009), 309-328.
doi: doi:10.2140/pjm.2009.241.309. |
[35] |
N. S. Papageorgiou, E. M. Rocha and V. Staicu, A multiplicity theorem for hemivariational inequalities with a $p$-Laplacian-like differential operator, Nonlin. Anal., 69 (2008), 1150-1163.
doi: doi:10.1016/j.na.2007.06.023. |
[36] |
N. S. Trudinger and XuJia Wang, Quasilinear elliptic equations with signed measure, Discrete Contin. Dynamical Systems, 23 (2009), 477-494. |
[37] |
Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincare Analyse Non Lineaire, 8 (1991), 43-58. |
[38] | |
[39] |
Q. Zhang, A strong maximum principle for differential equations with nonstandard $p(x)$-growth condition, J. Math. Anal. Appl., 312 (2005), 24-32.
doi: doi:10.1016/j.jmaa.2005.03.013. |
[40] |
Z. Zhang, J. Q. Chen and S. Li, Construction of pseudo-gradient vector field and sign-changing multiple solutions involving $p$-Laplacian, J. Differential Eqns., 201 (2004), 287-303.
doi: doi:10.1016/j.jde.2004.03.019. |
show all references
References:
[1] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, "Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints," Memoirs of AMS, vol. 196, no. 915, 2008. |
[2] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, Existence of multiple solutions with precise sign information for superlinear Neumann problems, Annali di Mat Pura Appl., 188 (2009), 679-719.
doi: doi:10.1007/s10231-009-0096-7. |
[3] |
S. Aizicovici, N. S. Papageorgiou and V. Staicu, The spectrum and an index formula for the Neumann $p$-Laplacian and multiple solutions for problems with a crossing nonlinearity, Discrete Contin. Dynamical Systems, 25 (2009), 431-456. |
[4] |
P. Bartolo, V. Benci and D. Fortunato, Abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity, Nonlin. Anal., 7 (1983), 981-1012.
doi: doi:10.1016/0362-546X(83)90115-3. |
[5] |
T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlin. Anal., 28 (1997), 419-441.
doi: doi:10.1016/0362-546X(95)00167-T. |
[6] |
H. Brezis and Louis Nirenberg, $H^1$-versus $C^1$ local minimizers, CRAS Paris, Ser. J. Math., 317 (1993), 465-472. |
[7] |
S. Carl and K. Perera, Sign-changing and multiple solutions for the $p$-Laplacian, Abstr. Appl. Anal., 7 (2002), 613-625.
doi: doi:10.1155/S1085337502207010. |
[8] |
E. Casas and L. Fernandez, A Green's formula for quasilinear elliptic operators, J. Math. Anal. Appl., 142 (1989), 62-73.
doi: doi:10.1016/0022-247X(89)90164-9. |
[9] |
K. C. Chang, "Infinite Dimensional Morse theory and Multiple Solution Problems," Birkhauser, Boston, 1993. |
[10] |
J. A. Cid and P. J. Torres, Solvability for some boundary value problems with $p$-Laplacian operators, Discrete Contin. Dynamical Systems, 23 (2009), 727-732. |
[11] |
D. Costa and C. Magalhaes, Existence results for perturbations of the $p$-Laplacian, Nonlinear Anal., 24 (1995), 409-418.
doi: doi:10.1016/0362-546X(94)E0046-J. |
[12] |
L. Damascellli, Comparision theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Inst. H. Poincare Analyse Non linenire, 15 (1998), 493-516. |
[13] |
N. Dancer and K. Perera, Some remarks on the Fučik spectrum of the $p$-Laplacian and critical groups, J. Math. Anal. Appl., 254 (2001), 164-177.
doi: doi:10.1006/jmaa.2000.7228. |
[14] |
P. De Mapoli and C. Mariani, Mountain pass solutions to equations of $p$-Laplacian type, Nonlinear Anal., 54 (2003), 1205-1219.
doi: doi:10.1016/S0362-546X(03)00105-6. |
[15] |
F. de Paiva and H. R. Quoirin, Resonance and nonresonance for $p$-Laplacian problems with weighted eigenvalues conditions, Discrete Contin. Dynamical Systems, 25 (2009), 1219-1227. |
[16] |
G. Fei, On periodic solutions of superquadratic Hamiltonian systems, Electronic J. Diff. Equas., 8 (2002), 1-12. |
[17] |
M. Filippakis, A. Kristaly and N. S. Papageorgiou, Existence of five nontrivial solutions with precise sign data for a $p$-Laplacian equation, Discrete Contin. Dynamical Systems, 25 (2009), 405-440. |
[18] |
J. Garcia Azorero, J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quarsilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404. |
[19] |
L. Gasinski and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Elliptic Boundary Value Problems," Chapman & Hall / CRC Press, Baca Raton, 2005. |
[20] |
L. Gasinski and N. S. Papageorgiou, "Nonlinear Analysis," Chapman & Hall/CRC, Boca Raton, 2006. |
[21] |
L. Gasinski and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann $p$-Laplacian type equations, Adv. Nonlin. Studies, 8 (2008), 843-870. |
[22] |
A. Granas and J. Dugundji, "Fixed Point Theory," Springer, New York, 2003. |
[23] |
Z. Guo and Z. Zhang, $W^{1,p}$ versus $C^1$ local minimizers and multiplicity results for quasilinear elliptic equations, J. Math. Anal. Anal., 286 (2003), 32-50.
doi: doi:10.1016/S0022-247X(03)00282-8. |
[24] |
A. Kristaly, H. Lisei and C. Varga, Multiple solutions for $p$-Laplacian type equations, Nonlinear Anal., 68 (2008), 1375-1381.
doi: doi:10.1016/j.na.2006.12.031. |
[25] |
O. Ladyzhenskaya and N. Uraltseva, "Linear and Quasilinear Elliptic Equations," Acad. Press., New York, 1968. |
[26] |
G. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Anal., 12 (1988), 1203-1219.
doi: doi:10.1016/0362-546X(88)90053-3. |
[27] |
S. Liu, Multiple solutions for coercive $p$-Laplacian equations, J. Math. Anal. Appl., 316 (2006), 229-236.
doi: doi:10.1016/j.jmaa.2005.04.034. |
[28] |
J. Liu and S. Liu, The existence of multiple solutions to quasilinear elliptic equations, Bull. London Math. Soc., 37 (2005), 592-600.
doi: doi:10.1112/S0024609304004023. |
[29] |
M. Montenegro, Strong maximum principles for supersolutions of quasilinear elliptic equations, Nonlinear Anal., 37 (1991), 431-448.
doi: doi:10.1016/S0362-546X(98)00057-1. |
[30] |
D. Motreanu and N. S. Papageorgiou, Existence and multiplicity of solutions for Neumann problems, J. Differential Eqns., 232 (2007), 1-35.
doi: doi:10.1016/j.jde.2006.09.008. |
[31] |
D. Motreanu, V. V. Motreanu and N. S. Papageorgiou, Nonlinear Neumann problems near resonance, Indiana Univ. Math. J., 58 (2009), 1257-1279.
doi: doi:10.1512/iumj.2009.58.3565. |
[32] |
N. S. Papageorgiou and S. Th. Kyritsi, "Handbook of Applied Analysis," Springer, New York, 2009. |
[33] |
E. Papageorgiou and N. S. Papageorgiou, A multiplicity theorem for problems with the $p$-Laplacian, J. Funct. Anal., 244 (2007), 63-77.
doi: doi:10.1016/j.jfa.2006.11.015. |
[34] |
E. Papageorgiou and N. S. Papageorgiou, Multiplicity of solutions for a class of resonant $p$-Laplacian Dirichlet problems, Pacific J. Math., 241 (2009), 309-328.
doi: doi:10.2140/pjm.2009.241.309. |
[35] |
N. S. Papageorgiou, E. M. Rocha and V. Staicu, A multiplicity theorem for hemivariational inequalities with a $p$-Laplacian-like differential operator, Nonlin. Anal., 69 (2008), 1150-1163.
doi: doi:10.1016/j.na.2007.06.023. |
[36] |
N. S. Trudinger and XuJia Wang, Quasilinear elliptic equations with signed measure, Discrete Contin. Dynamical Systems, 23 (2009), 477-494. |
[37] |
Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincare Analyse Non Lineaire, 8 (1991), 43-58. |
[38] | |
[39] |
Q. Zhang, A strong maximum principle for differential equations with nonstandard $p(x)$-growth condition, J. Math. Anal. Appl., 312 (2005), 24-32.
doi: doi:10.1016/j.jmaa.2005.03.013. |
[40] |
Z. Zhang, J. Q. Chen and S. Li, Construction of pseudo-gradient vector field and sign-changing multiple solutions involving $p$-Laplacian, J. Differential Eqns., 201 (2004), 287-303.
doi: doi:10.1016/j.jde.2004.03.019. |
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