# American Institute of Mathematical Sciences

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 Early access  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 and Applied Analysis, 2021, 20 (4) : 1347-1361. doi: 10.3934/cpaa.2021023
##### 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. [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. [3] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275. [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. [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. [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. [7] E. N. Dancer, Y. 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. [8] Y. Du and Z. Guo, Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic equations, Adv. Differ. Equ., 18 (2013), 737-768. [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. [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. [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. [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. [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. [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. [15] J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differ. Equ., 19 (2006), 256-270. [16] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${R}^{N}$, Differ. Integral Equ., 9 (1996), 465-480. [17] Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy–Hénon systems, Adv. Differ. Equ., 17 (2012), 605-634. [18] P. Poláčik, P. 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. [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. [20] J. Serrin and H. Zou, Non-existence of positive solutions of Lane–Emden systems, Differ. Integral Equ., 9 (1996), 635-653. [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. [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. [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. [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.

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##### 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. [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. [3] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275. [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. [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. [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. [7] E. N. Dancer, Y. 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. [8] Y. Du and Z. Guo, Finite Morse-index solutions and asymptotics of weighted nonlinear elliptic equations, Adv. Differ. Equ., 18 (2013), 737-768. [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. [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. [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. [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. [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. [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. [15] J. Liu, Y. Guo and Y. Zhang, Existence of positive entire solutions for polyharmonic equations and systems, J. Partial Differ. Equ., 19 (2006), 256-270. [16] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in ${R}^{N}$, Differ. Integral Equ., 9 (1996), 465-480. [17] Q. H. Phan, Liouville-type theorems and bounds of solutions for Hardy–Hénon systems, Adv. Differ. Equ., 17 (2012), 605-634. [18] P. Poláčik, P. 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. [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. [20] J. Serrin and H. Zou, Non-existence of positive solutions of Lane–Emden systems, Differ. Integral Equ., 9 (1996), 635-653. [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. [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. [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. [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.
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