April  2021, 20(4): 1347-1361. doi: 10.3934/cpaa.2021023

On problems with weighted elliptic operator and general growth nonlinearities

University of Texas, Rio Grande Valley, Edinburg, TX 78539, USA

Received  August 2020 Revised  December 2020 Published  April 2021

Fund Project: The author is partially supported by the Simons Foundation Collaboration Grants for Mathematicians 524335

This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form
$ \begin{equation*} -div (|x|^{a} D u ) = f(x,u), \; u > 0,\, \mbox{ in } \Omega, \end{equation*} $
where
$ N \geq 3 $
,
$ \Omega $
is an open domain in
$ \mathbb{R}^N $
containing the origin,
$ N-2+a > 0 $
and
$ f $
satisfies structural conditions, including certain growth properties. The first main result is a non-existence theorem for boundary-value problems in bounded domains star-shaped with respect to the origin, provided
$ f $
exhibits supercritical growth. A consequence of this is the existence of positive entire solutions to the equation for
$ f $
exhibiting the same growth. A Liouville-type theorem is then established, which asserts no positive solution of the equation in
$ \Omega = \mathbb{R}^N $
exists provided the growth of
$ f $
is subcritical. The results are then extended to systems of the form
$ \begin{equation*} -div (|x|^{a} D u_1) \! = \! f_{1}(x,u_1,u_2), -div (|x|^{a} D u_2) \! = \! f_{2}(x,u_1,u_2), u_1, u_2 \!>\! 0,\, \mbox{ in } \Omega, \end{equation*} $
but after overcoming additional obstacles not present in the single equation. Specific cases of our results recover classical ones for a renowned problem connected with finding best constants in Hardy-Sobolev and Caffarelli-Kohn-Nirenberg inequalities as well as existence results for well-known elliptic systems.
Citation: John Villavert. On problems with weighted elliptic operator and general growth nonlinearities. Communications on Pure & Applied Analysis, 2021, 20 (4) : 1347-1361. doi: 10.3934/cpaa.2021023
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D. G. De Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21 (1994), 387-397.   Google Scholar

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Z. Guo and F. Wan, Further study of a weighted elliptic equation, Sci. China Math., 60 (2017), 2391-2406.  doi: 10.1007/s11425-017-9134-7.  Google Scholar

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C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems, Proc. Amer. Math. Soc., 144 (2016), 3731-3740.  doi: 10.1090/proc/13166.  Google Scholar

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C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth, Commun. Partial Differ. Equ., 41 (2016), 1029-1039.  doi: 10.1080/03605302.2016.1190376.  Google Scholar

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J. LiuY. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differ. Equ., 19 (2006), 256-270.   Google Scholar

[16]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${R}^{N}$, Differ. Integral Equ., 9 (1996), 465-480.   Google Scholar

[17]

Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy–Hénon systems, Adv. Differ. Equ., 17 (2012), 605-634.   Google Scholar

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P. PoláčikP. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

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W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equ., 161 (2000), 219-243.  doi: 10.1006/jdeq.1999.3700.  Google Scholar

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J. Serrin and H. Zou, Non-existence of positive solutions of Lane–Emden systems, Differ. Integral Equ., 9 (1996), 635-653.   Google Scholar

[21]

J. Serrin and H. Zou, Existence of positive solutions of the Lane–Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1996), 369-380.   Google Scholar

[22]

Ph. Souplet, The proof of the Lane–Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[23]

J. Villavert, Shooting with degree theory: {A}nalysis of some weighted poly-harmonic systems, J. Differ. Equ., 257 (2014), 1148-1167.  doi: 10.1016/j.jde.2014.05.003.  Google Scholar

[24]

J. Villavert, Classification of radial solutions to equations related to Caffarelli-Kohn-Nirenberg inequalities, Ann. Mat. Pura Appl., 199 (2020), 299-315.  doi: 10.1007/s10231-019-00879-0.  Google Scholar

show all references

References:
[1]

M. F. Bidaut-Veron and H. Giacomini, A new dynamical approach of Emden-Fowler equations and systems, Adv. Differ. Equ., 15 (2010), 1033-1082.   Google Scholar

[2]

J. Busca and R. Manásevich, A Liouville-type theorem for Lane–Emden systems, Indiana Univ. Math. J., 51 (2002), 37-51.  doi: 10.1512/iumj.2002.51.2160.  Google Scholar

[3]

L. CaffarelliR. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.   Google Scholar

[4]

F. Catrina and Z. Wang, On the Caffarelli–Kohn–Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Commun. Pure Appl. Math., 54 (2001), 229-258.  doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.  Google Scholar

[5]

W. Chen and C. Li, An integral system and the Lane–Emden conjecture, Discrete Contin. Dyn. S., 4 (2009), 1167-1184.  doi: 10.3934/dcds.2009.24.1167.  Google Scholar

[6]

K. S. Chou and C. W. Chu, On the best constant for a weighted Sobolev–Hardy inequality, J. Lond. Math. Soc., 2 (1993), 137-151.  doi: 10.1112/jlms/s2-48.1.137.  Google Scholar

[7]

E. N. DancerY. Du and Z. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differ. Equ., 250 (2011), 3281-3310.  doi: 10.1016/j.jde.2011.02.005.  Google Scholar

[8]

Y. Du and Z. Guo, Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic equations, Adv. Differ. Equ., 18 (2013), 737-768.   Google Scholar

[9]

Y. Du and Z. Guo, Finite Morse index solutions of weighted elliptic equations and the critical exponents, Calc. Var. Partial Differ. Equ., 54 (2015), 3116-3181.  doi: 10.1007/s00526-015-0897-z.  Google Scholar

[10]

M. Fazly and N. Ghoussoub, On the Hénon-Lane-Emden conjecture, Discrete Contin. Dyn. S., 34 (2014), 2513-2533.  doi: 10.3934/dcds.2014.34.2513.  Google Scholar

[11]

D. G. De Figueiredo and P. L. Felmer, A Liouville-type theorem for elliptic systems, Ann. Sc. Norm. Sup. Pisa, 21 (1994), 387-397.   Google Scholar

[12]

Z. Guo and F. Wan, Further study of a weighted elliptic equation, Sci. China Math., 60 (2017), 2391-2406.  doi: 10.1007/s11425-017-9134-7.  Google Scholar

[13]

C. Li and J. Villavert, A degree theory framework for semilinear elliptic systems, Proc. Amer. Math. Soc., 144 (2016), 3731-3740.  doi: 10.1090/proc/13166.  Google Scholar

[14]

C. Li and J. Villavert, Existence of positive solutions to semilinear elliptic systems with supercritical growth, Commun. Partial Differ. Equ., 41 (2016), 1029-1039.  doi: 10.1080/03605302.2016.1190376.  Google Scholar

[15]

J. LiuY. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differ. Equ., 19 (2006), 256-270.   Google Scholar

[16]

E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${R}^{N}$, Differ. Integral Equ., 9 (1996), 465-480.   Google Scholar

[17]

Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy–Hénon systems, Adv. Differ. Equ., 17 (2012), 605-634.   Google Scholar

[18]

P. PoláčikP. Quittner and Ph. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I: Elliptic equations and systems, Duke Math. J., 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.  Google Scholar

[19]

W. Reichel and H. Zou, Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differ. Equ., 161 (2000), 219-243.  doi: 10.1006/jdeq.1999.3700.  Google Scholar

[20]

J. Serrin and H. Zou, Non-existence of positive solutions of Lane–Emden systems, Differ. Integral Equ., 9 (1996), 635-653.   Google Scholar

[21]

J. Serrin and H. Zou, Existence of positive solutions of the Lane–Emden system, Atti Sem. Mat. Fis. Univ. Modena, 46 (1996), 369-380.   Google Scholar

[22]

Ph. Souplet, The proof of the Lane–Emden conjecture in four space dimensions, Adv. Math., 221 (2009), 1409-1427.  doi: 10.1016/j.aim.2009.02.014.  Google Scholar

[23]

J. Villavert, Shooting with degree theory: {A}nalysis of some weighted poly-harmonic systems, J. Differ. Equ., 257 (2014), 1148-1167.  doi: 10.1016/j.jde.2014.05.003.  Google Scholar

[24]

J. Villavert, Classification of radial solutions to equations related to Caffarelli-Kohn-Nirenberg inequalities, Ann. Mat. Pura Appl., 199 (2020), 299-315.  doi: 10.1007/s10231-019-00879-0.  Google Scholar

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