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Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials
Existence of solutions for a class of p-Laplacian type equation with critical growth and potential vanishing at infinity
1. | Department of Mathematics, Huazhong Normal University, Wuhan 430079 |
2. | Department of Mathematics and Statistics, Wright State University, Dayton, OH 45435, USA, United States |
3. | Department of Mathematics, Huazhong Normal University, Wuhan, 430079 |
References:
[1] |
C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potentials vanishing at infinitly,, J. Diff. Eqns., 254 (2013), 1977.
doi: 10.1016/j.jde.2012.11.013. |
[2] |
A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinitly,, J. Eur. Math. Soc., 7 (2005), 117.
doi: 10.4171/JEMS/24. |
[3] |
A. Ambrosetti and P. H. Rabinowitz, Dual varitional methods in critical point theory and applications,, J. Funct. Analysis, 14 (1973), 349.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, Diff. Integ. Eqns., 18 (2005), 1321.
|
[5] |
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[6] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Ration. Mech. Anal., 82 (1983), 313.
doi: 10.1007/BF00250555. |
[7] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Commun. Pure. Appl. Math., 36 (1983), 437.
doi: 10.1002/cpa.3160360405. |
[8] |
D. Bonheure and J. Van Schaftingen, Ground states for the nonlinear Schrödinger equation with potentials vanishing at infinitly,, Annali Mat. Pura Appl., 189 (2010), 273.
doi: 10.1007/s10231-009-0109-6. |
[9] |
F. Catrina, M. Furtado and M. Montenegro, Positive solutions for nonlinear elliptic equations with fast increasing weights,, Proc. Roy. Soc. Edinb. A, 137 (2007), 1157.
doi: 10.1017/S0308210506000795. |
[10] |
Y. B. Deng, The existence and nodal character of the solutions in $\mathbbR^N$ for semilinear elliptic equation involving critical Sobolev exponent,, Acta Math. Sci., 9 (1989), 385.
|
[11] |
Y. B. Deng, L. Y. Jin and S. J. Peng, Solutions of Schrodinger Equations with Inverse Square Potential and Critical Nonlinearity,, J. Diff. Eqns., 253 (2012), 1376.
doi: 10.1016/j.jde.2012.05.009. |
[12] |
Y. B. Deng, Z. H. Guo and G. S. Wang, Nodal solutions for $p-$Laplace equations with critical growth,, Nonlin. Analysis, 54 (2003), 1121.
doi: 10.1016/S0362-546X(03)00129-9. |
[13] |
Y. B. Deng and Y. Li, On the existence of multiple positive solutions for a semilinear problem in exterior domains,, J. Diff. Eqns., 181 (2002), 197.
doi: 10.1006/jdeq.2001.4077. |
[14] |
M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of heat equations,, Nonlin Analysis, 11 (1987), 1103.
doi: 10.1016/0362-546X(87)90001-0. |
[15] |
M. F. Furtado, J. P. P. da Silva and M. S. Xavier, Multiplicity of self-similar solutions for a critical equation,, J. Diff. Eqns., 254 (2013), 2732.
doi: 10.1016/j.jde.2013.01.007. |
[16] |
G. B. Li, The existence of a weak solution of quasilinear elliptic equation with critical Sobolev exponent on unbounded domain,, Acta Math. Sci., 14 (1994), 64.
|
[17] |
P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part I,, Rev. Mat. Iber., 1 (1985), 145.
doi: 10.4171/RMI/6. |
[18] |
P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part II,, Rev. Mat. Iber., 1 (1985), 45.
doi: 10.4171/RMI/12. |
[19] |
B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Research Notes in Mathematics Series, (1990).
|
[20] |
J. Serrin, Local behavior of solutions of quasilinear equations,, Acta. Math., 111 (1964), 247.
doi: 10.1007/BF02391014. |
[21] |
D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities,, Trans. Amer. Math. Soc., 357 (2005), 2909.
doi: 10.1090/S0002-9947-04-03769-9. |
[22] |
W. A. Strauss, Existence of solitary in higher dimensions,, Commun. Math. Phys., 55 (1977), 149.
doi: 10.1007/BF01626517. |
[23] |
C. A. Swanson and L. S. Yu., Critical p-laplacian problems in $\mathbbR^N$,, Annali Mat. Pura Appl., 169 (1995), 233.
doi: 10.1007/BF01759355. |
[24] |
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Diff. Eqns., 51 (1984), 126.
doi: 10.1016/0022-0396(84)90105-0. |
[25] |
M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (1996).
doi: 10.1007/978-1-4612-4146-1. |
[26] |
X. P. Zhu, Nontrivial solution of quasilinear elliptic equation involving critical Sobolev exponent,, Sci. Sinica., 31 (1990), 1161. Google Scholar |
show all references
References:
[1] |
C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potentials vanishing at infinitly,, J. Diff. Eqns., 254 (2013), 1977.
doi: 10.1016/j.jde.2012.11.013. |
[2] |
A. Ambrosetti, V. Felli and A. Malchiodi, Ground states of nonlinear Schrödinger equations with potentials vanishing at infinitly,, J. Eur. Math. Soc., 7 (2005), 117.
doi: 10.4171/JEMS/24. |
[3] |
A. Ambrosetti and P. H. Rabinowitz, Dual varitional methods in critical point theory and applications,, J. Funct. Analysis, 14 (1973), 349.
doi: 10.1016/0022-1236(73)90051-7. |
[4] |
A. Ambrosetti and Z. Q. Wang, Nonlinear Schrödinger equations with vanishing and decaying potentials,, Diff. Integ. Eqns., 18 (2005), 1321.
|
[5] |
H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals,, Proc. Amer. Math. Soc., 88 (1983), 486.
doi: 10.1090/S0002-9939-1983-0699419-3. |
[6] |
H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, Arch. Ration. Mech. Anal., 82 (1983), 313.
doi: 10.1007/BF00250555. |
[7] |
H. Brézis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents,, Commun. Pure. Appl. Math., 36 (1983), 437.
doi: 10.1002/cpa.3160360405. |
[8] |
D. Bonheure and J. Van Schaftingen, Ground states for the nonlinear Schrödinger equation with potentials vanishing at infinitly,, Annali Mat. Pura Appl., 189 (2010), 273.
doi: 10.1007/s10231-009-0109-6. |
[9] |
F. Catrina, M. Furtado and M. Montenegro, Positive solutions for nonlinear elliptic equations with fast increasing weights,, Proc. Roy. Soc. Edinb. A, 137 (2007), 1157.
doi: 10.1017/S0308210506000795. |
[10] |
Y. B. Deng, The existence and nodal character of the solutions in $\mathbbR^N$ for semilinear elliptic equation involving critical Sobolev exponent,, Acta Math. Sci., 9 (1989), 385.
|
[11] |
Y. B. Deng, L. Y. Jin and S. J. Peng, Solutions of Schrodinger Equations with Inverse Square Potential and Critical Nonlinearity,, J. Diff. Eqns., 253 (2012), 1376.
doi: 10.1016/j.jde.2012.05.009. |
[12] |
Y. B. Deng, Z. H. Guo and G. S. Wang, Nodal solutions for $p-$Laplace equations with critical growth,, Nonlin. Analysis, 54 (2003), 1121.
doi: 10.1016/S0362-546X(03)00129-9. |
[13] |
Y. B. Deng and Y. Li, On the existence of multiple positive solutions for a semilinear problem in exterior domains,, J. Diff. Eqns., 181 (2002), 197.
doi: 10.1006/jdeq.2001.4077. |
[14] |
M. Escobedo and O. Kavian, Variational problems related to self-similar solutions of heat equations,, Nonlin Analysis, 11 (1987), 1103.
doi: 10.1016/0362-546X(87)90001-0. |
[15] |
M. F. Furtado, J. P. P. da Silva and M. S. Xavier, Multiplicity of self-similar solutions for a critical equation,, J. Diff. Eqns., 254 (2013), 2732.
doi: 10.1016/j.jde.2013.01.007. |
[16] |
G. B. Li, The existence of a weak solution of quasilinear elliptic equation with critical Sobolev exponent on unbounded domain,, Acta Math. Sci., 14 (1994), 64.
|
[17] |
P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part I,, Rev. Mat. Iber., 1 (1985), 145.
doi: 10.4171/RMI/6. |
[18] |
P. L. Lions, The concentration-compactness principle in calculus of variations, The limit case Part II,, Rev. Mat. Iber., 1 (1985), 45.
doi: 10.4171/RMI/12. |
[19] |
B. Opic and A. Kufner, Hardy-Type Inequalities,, Pitman Research Notes in Mathematics Series, (1990).
|
[20] |
J. Serrin, Local behavior of solutions of quasilinear equations,, Acta. Math., 111 (1964), 247.
doi: 10.1007/BF02391014. |
[21] |
D. Smets, Nonlinear Schrödinger equations with Hardy potential and critical nonlinearities,, Trans. Amer. Math. Soc., 357 (2005), 2909.
doi: 10.1090/S0002-9947-04-03769-9. |
[22] |
W. A. Strauss, Existence of solitary in higher dimensions,, Commun. Math. Phys., 55 (1977), 149.
doi: 10.1007/BF01626517. |
[23] |
C. A. Swanson and L. S. Yu., Critical p-laplacian problems in $\mathbbR^N$,, Annali Mat. Pura Appl., 169 (1995), 233.
doi: 10.1007/BF01759355. |
[24] |
P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations,, J. Diff. Eqns., 51 (1984), 126.
doi: 10.1016/0022-0396(84)90105-0. |
[25] |
M. Willem, Minimax Theorems,, Progress in Nonlinear Differential Equations and Their Applications, (1996).
doi: 10.1007/978-1-4612-4146-1. |
[26] |
X. P. Zhu, Nontrivial solution of quasilinear elliptic equation involving critical Sobolev exponent,, Sci. Sinica., 31 (1990), 1161. Google Scholar |
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