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September  2020, 19(9): 4667-4697. doi: 10.3934/cpaa.2020197

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

*Corresponding author

Received  September 2019 Revised  March 2020 Published  June 2020

Fund Project: Supported by the Fundamental Research Funds for the Central Universities (No.2018MS59) and Natural Science Foundation of Guangdong (No.2018A0303130196)

We study the existence of positive solutions of the following degenerate coercive quasilinear elliptic equations:
$ \!-\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, $
where
$ 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 $
and
$ 2^* = \frac{2N}{N-2} $
,
$ N\geq3 $
. The novelty of this paper is that
$ g(t) $
is non-increasing with respect to
$ |t| $
and
$ \lim_{|t|\rightarrow +\infty} g(t) = 0. $
The main results of this paper can be regarded as a supplement to the case that
$ g(t) $
is non-decreasing with respect to
$ |t| $
which has been extensively studied recently.
Citation: Yaotian Shen, Youjun Wang. Degenerate coercive quasilinear elliptic equations with subcritical or critical exponents in $ \mathbb{R}^N $. Communications on Pure and Applied Analysis, 2020, 19 (9) : 4667-4697. doi: 10.3934/cpaa.2020197
References:
[1]

C. O. AlvesY. 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. ArcoyaL. 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. DengS. 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. DengS. 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. LiuY. QWa 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. MercaldoI. 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. AlvesY. 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. ArcoyaL. 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. DengS. 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. DengS. 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. LiuY. QWa 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. MercaldoI. 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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