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Existence and nonexistence of positive radial solutions for a class of $p$-Laplacian superlinear problems with nonlinear boundary conditions
Degenerate coercive quasilinear elliptic equations with subcritical or critical exponents in $ \mathbb{R}^N $
School of Mathematics, South China University of Technology, , Guangzhou 510640, China |
$ \!-\text{div}(g^2(u)\nabla u)\!+\!\lambda g(u)g'(u)|\nabla u|^2\!+\!V(x)u\! = \!\beta u^{(1-\gamma)(2^*-1)}\!+\!f(u), \ x\!\in\! \mathbb{R}^N, $ |
$ g(t)\in C(\mathbb{R}, \mathbb{R}) $ |
$ V(x)\in C(\mathbb{R}^N, \mathbb{R}) $ |
$ \lambda, \gamma \in \mathbb{R} $ |
$ \beta\geq 0 $ |
$ 2^* = \frac{2N}{N-2} $ |
$ N\geq3 $ |
$ g(t) $ |
$ |t| $ |
$ \lim_{|t|\rightarrow +\infty} g(t) = 0. $ |
$ g(t) $ |
$ |t| $ |
References:
[1] |
C. O. Alves, Y. J. Wang and Y. T. Shen,
Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.
doi: 10.1080/17476933.2015.1119818. |
[2] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14(4) (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
D. Arcoya, L. Boccardo and L. Orsina,
Existence of critical points for some noncoercive functionals, Ann. Inst. Henri Poincare Anal. Non Lineaire, 18 (2001), 437-457.
doi: 10.1016/S0294-1449(01)00069-5. |
[4] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations I, Arch. Ration. Mech. Anal., 82 (1983), 313-346.
doi: 10.1007/BF00250555. |
[5] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[6] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal. Theory Meth. Appl., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[7] |
Y. B. Deng, S. J. Peng and S. S. Yan,
Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differ. Equ., 258 (2015), 115-147.
doi: 10.1016/j.jde.2014.09.006. |
[8] |
Y. B. Deng, S. J. Peng and S. S. Yan,
Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differ. Equ., 260 (2016), 1228-1262.
doi: 10.1016/j.jde.2015.09.021. |
[9] |
G. di Blasio and F. Feo,
A class of nonlinear degenerate elliptic equations related to the Gauss measure, J. Math. Anal. Appl., 386 (2012), 763-779.
doi: 10.1016/j.jmaa.2011.08.037. |
[10] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $\mathbb{R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[11] |
P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. Henri Poincare Anal. Non Lineaire, (1984), 109–145,223–283. |
[12] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2002), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[13] |
J. Q. Liu, Y. Q, Wa ng and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations II, J. Differ. Equ., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[14] |
J. Marcos do Ó and U. Severo,
Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644.
doi: 10.3934/cpaa.2009.8.621. |
[15] |
A. Mercaldo and I. Peral,
Existence results for semilinear elliptic equations with some lack of coercivity, Proc. R. Soc. Edinb., 138 (2008), 569-595.
doi: 10.1017/S0308210506000126. |
[16] |
A. Mercaldo, I. Peral and A. Primo,
Results for degenerate nonlinear elliptic equations involving a Hardy potential, J. Differ. Equ., 251 (2011), 3114-3142.
doi: 10.1016/j.jde.2011.07.024. |
[17] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001. |
[18] |
Y. T. Shen and X. K. Guo,
The positive solution of degenerate variational problem and degenerate elliptic equation, Chin. J. Contemp. Math., 14 (1993), 157-165.
|
[19] |
Y. T. Shen and Y. J. Wang,
Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. Theory Meth. Appl., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[20] |
Y. T. Shen and Y. J. Wang,
A class of generalized quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 15 (2016), 853-870.
doi: 10.3934/cpaa.2016.15.853. |
[21] |
Y. T. Shen and Y. J. Wang,
Standing waves for a class of quasilinear Schrödinger equations, Complex Var. Elliptic Equ., 61 (2016), 817-842.
doi: 10.1080/17476933.2015.1119818. |
[22] |
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var., 39 (2010), 1-33.
doi: 10.1007/s00526-009-0299-1. |
[23] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.
|
[24] |
M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, 2008. |
show all references
References:
[1] |
C. O. Alves, Y. J. Wang and Y. T. Shen,
Soliton solutions for a class of quasilinear Schrödinger equations with a parameter, J. Differ. Equ., 259 (2015), 318-343.
doi: 10.1080/17476933.2015.1119818. |
[2] |
A. Ambrosetti and P. H. Rabinowitz,
Dual variational methods in critical point theory and applications, J. Funct. Anal., 14(4) (1973), 349-381.
doi: 10.1016/0022-1236(73)90051-7. |
[3] |
D. Arcoya, L. Boccardo and L. Orsina,
Existence of critical points for some noncoercive functionals, Ann. Inst. Henri Poincare Anal. Non Lineaire, 18 (2001), 437-457.
doi: 10.1016/S0294-1449(01)00069-5. |
[4] |
H. Berestycki and P. L. Lions,
Nonlinear scalar field equations I, Arch. Ration. Mech. Anal., 82 (1983), 313-346.
doi: 10.1007/BF00250555. |
[5] |
H. Brézis and L. Nirenberg,
Positive solutions of nonlinear elliptic equations involving critical Sobolev exponent, Commun. Pure Appl. Math., 36 (1983), 437-477.
doi: 10.1002/cpa.3160360405. |
[6] |
M. Colin and L. Jeanjean,
Solutions for a quasilinear Schrödinger equations: A dual approach, Nonlinear Anal. Theory Meth. Appl., 56 (2004), 213-226.
doi: 10.1016/j.na.2003.09.008. |
[7] |
Y. B. Deng, S. J. Peng and S. S. Yan,
Positive soliton solutions for generalized quasilinear Schrödinger equations with critical growth, J. Differ. Equ., 258 (2015), 115-147.
doi: 10.1016/j.jde.2014.09.006. |
[8] |
Y. B. Deng, S. J. Peng and S. S. Yan,
Critical exponents and solitary wave solutions for generalized quasilinear Schrödinger equations, J. Differ. Equ., 260 (2016), 1228-1262.
doi: 10.1016/j.jde.2015.09.021. |
[9] |
G. di Blasio and F. Feo,
A class of nonlinear degenerate elliptic equations related to the Gauss measure, J. Math. Anal. Appl., 386 (2012), 763-779.
doi: 10.1016/j.jmaa.2011.08.037. |
[10] |
L. Jeanjean and K. Tanaka,
A remark on least energy solutions in $\mathbb{R}^N$, Proc. Amer. Math. Soc., 131 (2003), 2399-2408.
doi: 10.1090/S0002-9939-02-06821-1. |
[11] |
P. L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. Henri Poincare Anal. Non Lineaire, (1984), 109–145,223–283. |
[12] |
J. Q. Liu and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations I, Proc. Amer. Math. Soc., 131 (2002), 441-448.
doi: 10.1090/S0002-9939-02-06783-7. |
[13] |
J. Q. Liu, Y. Q, Wa ng and Z. Q. Wang,
Soliton solutions for quasilinear Schrödinger equations II, J. Differ. Equ., 187 (2003), 473-493.
doi: 10.1016/S0022-0396(02)00064-5. |
[14] |
J. Marcos do Ó and U. Severo,
Quasilinear Schrödinger equations involving concave and convex nonlinearities, Commun. Pure Appl. Anal., 8 (2009), 621-644.
doi: 10.3934/cpaa.2009.8.621. |
[15] |
A. Mercaldo and I. Peral,
Existence results for semilinear elliptic equations with some lack of coercivity, Proc. R. Soc. Edinb., 138 (2008), 569-595.
doi: 10.1017/S0308210506000126. |
[16] |
A. Mercaldo, I. Peral and A. Primo,
Results for degenerate nonlinear elliptic equations involving a Hardy potential, J. Differ. Equ., 251 (2011), 3114-3142.
doi: 10.1016/j.jde.2011.07.024. |
[17] |
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001. |
[18] |
Y. T. Shen and X. K. Guo,
The positive solution of degenerate variational problem and degenerate elliptic equation, Chin. J. Contemp. Math., 14 (1993), 157-165.
|
[19] |
Y. T. Shen and Y. J. Wang,
Soliton solutions for generalized quasilinear Schrödinger equations, Nonlinear Anal. Theory Meth. Appl., 80 (2013), 194-201.
doi: 10.1016/j.na.2012.10.005. |
[20] |
Y. T. Shen and Y. J. Wang,
A class of generalized quasilinear Schrödinger equations, Commun. Pure Appl. Anal., 15 (2016), 853-870.
doi: 10.3934/cpaa.2016.15.853. |
[21] |
Y. T. Shen and Y. J. Wang,
Standing waves for a class of quasilinear Schrödinger equations, Complex Var. Elliptic Equ., 61 (2016), 817-842.
doi: 10.1080/17476933.2015.1119818. |
[22] |
E. A. B. Silva and G. F. Vieira,
Quasilinear asymptotically periodic Schrödinger equations with critical growth, Calc. Var., 39 (2010), 1-33.
doi: 10.1007/s00526-009-0299-1. |
[23] |
W. A. Strauss,
Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.
|
[24] |
M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, 2008. |
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