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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 $

Yaotian Shen and  Youjun Wang ,

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.
Keywords: Degenerate coercive, quasilinear elliptic equations, critical.
Mathematics Subject Classification: Primary: 35J20, 35J62; Secondary: 35Q55.
Citation: Yaotian Shen, Youjun Wang. Degenerate coercive quasilinear elliptic equations with subcritical or critical exponents in $ \mathbb{R}^N $. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4667-4697. doi: 10.3934/cpaa.2020197
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar

[24]

M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, 2008.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

H. Berestycki and P. L. Lions, Nonlinear scalar field equations I, Arch. Ration. Mech. Anal., 82 (1983), 313-346.  doi: 10.1007/BF00250555.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer, 2001.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

W. A. Strauss, Existence of solitary waves in higher dimensions, Commun. Math. Phys., 55 (1977), 149-162.   Google Scholar

[24]

M. Struwe, Variational Methods, Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer, 2008.  Google Scholar

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