# American Institute of Mathematical Sciences

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, 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.   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. Caffarelli, R. 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. 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.  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. Liu, Y. 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. 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.  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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##### 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. Caffarelli, R. 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. 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.  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. Liu, Y. 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. 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.  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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