An Exact Multiplicity Result for Singular Subcritical Elliptic Problems

Abstract

For a bounded and smooth enough domain $\Omega$ in $\mathbb{R}^{n}$, with $n\geq2,$ we consider the problem $-\Delta u=au^{-\beta}+\lambda h\left( .,u\right) $ in $\Omega,$ $u=0$ on $\partial\Omega,$ $u>0$ in $\Omega,$ where $\lambda>0,$ $0<\beta<3,$ $a\in L^{\infty}\left( \Omega\right) ,$ $ess\,inf\,\,(a)>0,$ and with $h=h\left( x,s\right) \in C\left( \overline{\Omega}\times\left[ 0,\infty\right) \right) $ positive on $\Omega\times\left( 0,\infty\right) $ and such that, for any $x\in\Omega,$ $h\left( x,.\right) $ is strictly convex on $\left( 0,\infty\right) $, nondecreasing, belongs to $C^{2}\left( 0,\infty\right) ,$ and satisfies, for some $p\in\left( 1,\frac{n+2}{n-2}\right) ,$ that $\lim_{s\rightarrow\infty }\frac{h_{s}\left( x,s\right) }{s^{p}}=0$ and $\lim_{s\rightarrow\infty }\frac{h\left( x,s\right) }{s^{p}}=k\left( x\right) ,$ in both limits uniformly respect to $x\in\overline{\Omega}$, and with $k\in C\left( \overline{\Omega}\right)$ such that $\min_{\overline{\Omega}}k>0.$ Under these assumptions it is known the existence of $\Sigma > 0 $ such that for $ \lambda =0 $ and $ \lambda = \Sigma $ the above problem has exactly a weak solution, whereas for $ \lambda \in \left( 0, \Sigma \right) $ it has at least two weak solutions, and no weak solutions exist if $ \lambda > \Sigma $. For such a $ \Sigma $ we prove that for $ \lambda \in \left( 0, \Sigma \right) $ the considered problem has it has exactly two weak solutions.

Keywords

• Bifurcation problems
• Implicit function theorem
• Positive solutions
• Singular elliptic problem
• Sub and supersolutions method

References

1. [1] A. Callegari and A. Nachman, A nonlinear singular boundary-value problem in the theory of pseudoplastic fluids, SIAM J. Appl. Math., 38(2) (1980), 275–281. $\href{https://doi.org/10.1137/0138024}{\mbox{[CrossRef]}} $
2. [2] J.I. Diaz, J.M. Morel and L. Oswald, An elliptic equation with singular nonlinearity, Commun. Partial Differ. Equ., 12(2) (1987), 1333–1344. $\href{https://doi.org/10.1080/03605308708820531}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0000980547&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22An+elliptic+equation+with+singular+nonlinearity%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=2}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1987K621400002}{\mbox{[Web of Science]}} $
3. [3] W. Fulks and J.S. Maybee, A singular nonlinear equation, Osaka J. Math., 12(1) (1960), 1–19. $ \href{https://projecteuclid.org/journals/osaka-mathematical-journal/volume-12/issue-1/A-singular-non-linear-equation/ojm/1200689813.full}{\mbox{[Web]}} $
4. [4] D.S. Cohen and H.B. Keller, Some positive problems suggested by nonlinear heat generators, J. Math. Mech., 16(2) (1967), 1361–1376. $ \href{https://authors.library.caltech.edu/records/k15e3-18z28}{\mbox{[Web]}} $
5. [5] M.G. Crandall, P.H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Partial Differ. Equ., 2(2) (1977), 193–222. $\href{https://doi.org/10.1080/03605307708820029}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84863910600&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+Dirichlet+problem+with+a+singular+nonlinearity%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=1}{\mbox{[Scopus]}} $
6. [6] C.A. Stuart, Existence and approximation of solutions of nonlinear elliptic problems, Math. Z., 147 (1976), 53-63. $ \href{https://doi.org/10.1007/BF01214274}{\mbox{[CrossRef]}} $
7. [7] A.C. Lazer and P.J. McKenna, On a singular nonlinear elliptic boundary value problem, Proc. Amer. Math. Soc., 111(3) (1991), 721–730. $\href{https://doi.org/10.2307/2048410}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84968514134&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+singular+nonlinear+elliptic+boundary+value+problem%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1991FE26200018}{\mbox{[Web of Science]}} $
8. [8] M.M. Coclite and G. Palmieri, On a singular nonlinear Dirichlet problem, Commun. Partial Differ. Equ., 14(10) (1989), 1315–1327. $ \href{https://doi.org/10.1080/03605308908820656}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84946263908&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+singular+nonlinear+Dirichlet+problem%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=3}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1989AX75900001}{\mbox{[Web of Science]}} $

9. [9] M.A. del Pino, A global estimate for the gradient in a singular elliptic boundary value problem, Proc. Roy. Soc. Edinburgh Sect. A, 122(3-4) (1992), 341–352. $\href{https://doi.org/10.1017/S0308210500021144}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84971141957&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+global+estimate+for+the+gradient+in+a+singular+elliptic+boundary+value+problem%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:A1992KD35000009}{\mbox{[Web of Science]}} $
10. [10] B. Bougherara, J. Giacomoni and J. Hernandez, Existence and regularity of weak solutions for singular elliptic problems, 2014 Madrid Conference on Applied Mathematics in honor of Alfonso Casal, Electron. J. Differ. Equ. Conf., 22 (2015), 19–30. $\href{https://www.webofscience.com/wos/woscc/full-record/WOS:000455675900003}{\mbox{[Web of Science]}} $
11. [11] H. Maagli, Asymptotic behavior of positive solutions of a semilinear Dirichlet problem, Nonlinear Anal., 74(9) (2011), 2941–2947. $ \href{https://doi.org/10.1016/j.na.2011.01.011}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-79952573551&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Asymptotic+behavior+of+positive+solutions+of+a+semilinear+Dirichlet+problem%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=4}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000288239400005}{\mbox{[Web of Science]}} $
12. [12] F. Oliva and F. Petitta, Finite and infinite energy solutions of singular elliptic problems: Existence and uniqueness, J. Diff. Equ., 264(1) (2018), 311–340. $\href{https://doi.org/10.1016/j.jde.2017.09.008}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85029703451&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Finite+and+infinite+energy+solutions+of+singular+elliptic+problems%3A+Existence+and+uniqueness%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000413886600012}{\mbox{[Web of Science]}} $
13. [13] J. Shi and M. Yao, On a singular nonlinear semilinear elliptic problem, Proc. Roy. Soc. Edinburgh Sect. A, 128(6) (1998), 1389–1401. $ \href{https://doi.org/10.1017/S0308210500027384}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-22444455186&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+singular+nonlinear+semilinear+elliptic+problem%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000078031900014}{\mbox{[Web of Science]}} $
14. [14] M. Ghergu and V.D. Radulescu, Multi-parameter bifurcation and asymptotics for the singular Lane-Emden-Fowler equation with a convection term, Proc. Roy. Soc. Edinburgh Sect. A, 135(1) (2005), 61–84. $\href{https://doi.org/10.1017/S0308210500003760}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-14644399904&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Multi-parameter+bifurcation+and+asymptotics+for+the+singular+Lane-Emden-Fowler+equation+with+a+convection+term%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000227283700004}{\mbox{[Web of Science]}} $
15. [15] L. Dupaigne, M. Ghergu and V. Radulescu, Lane-Emden-Fowler equations with convection and singular potential, J. Math. Pures Appl., 87(6) (2007), 563–581. $ \href{https://doi.org/10.1016/j.matpur.2007.03.002}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-34250218210&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Lane-Emden-Fowler+equations+with+convection+and+singular+potential%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=1}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000247996000001}{\mbox{[Web of Science]}}$
16. [16] F. Cˆırstea, M. Ghergu and V. Radulescu, Combined effects of asymptotically linear and singular nonlinearities in bifurcation problems of Lane-Emden-Fowler type, J. Math. Pures Appl., 84(4) (2005), 493–508. $\href{https://doi.org/10.1016/j.matpur.2004.09.005}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-16244423685&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Combined+effects+of+asymptotically+linear+and+singular+nonlinearities+in+bifurcation+problems+of+Lane-Emden-Fowler+type%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000228630500003}{\mbox{[Web of Science]}} $
17. [17] P. Agarwal, J.Merker and G. Schuldt, Singular integral Neumann boundary conditions for semilinear elliptic PDEs, Axioms, 10(2) 74 (2021), 1-9. $\href{https://doi.org/10.3390/axioms10020074}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85105253195&origin=resultslist&sort=plf-f&src=s&sid=aafb701277b63cc409c69edb52846e26&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Singular+Integral+Neumann+Boundary+Conditions+for+Semilinear+Elliptic+PDEs%22%29&sl=91&sessionSearchId=aafb701277b63cc409c69edb52846e26&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000665174000001}{\mbox{[Web of Science]}} $
18. [18] H. Zhou, L. Yang, and P. Agarwal, Solvability for fractional p-Laplacian differential equations with multipoint boundary conditions at resonance on infinite interval, J. Appl. Math. Comput. 53(1-2) (2017), 51–76. $\href{https://doi.org/10.1007/s12190-015-0957-8}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84946866450&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Solvability+for+fractional+p-Laplacian+differential+equations+with+multipoint+boundary+conditions+at+resonance+on+infinite+interval%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000392287300004}{\mbox{[Web of Science]}} $
19. [19] C. Aranda and T. Godoy, Existence and multiplicity of positive solutions for a singular problem associated to the p-Laplacian operator, Electron. J. Differential. Equations, 132 (2004), 1–15. $\href{https://www.scopus.com/record/display.uri?eid=2-s2.0-8844267684&origin=resultslist&sort=plf-f&src=s&sid=aafb701277b63cc409c69edb52846e26&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Existence+and+multiplicity+of+positive+solutions+for+a+singular+problem+associated+to+the+p-Laplacian+operator%22%29&sl=91&sessionSearchId=aafb701277b63cc409c69edb52846e26&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000208969900132}{\mbox{[Web of Science]}} $
20. [20] R. Guefaifia, J. Zuo, S. Boulaaras and P. Agarwal, Existence and multiplicity of positive weak solutions for a new class of (p; q)-Laplacian systems, Miskolc Math. Notes, 21(2) (2020), 861–872. $\href{https://doi.org/10.18514/MMN.2020.3378}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85099993325&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Existence+and+multiplicity+of+positive+weak+solutions+for+a+new+class+of+%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000602739200023}{\mbox{[Web of Science]}} $
21. [21] K. Saoudi, P. Agarwal and M. Mursaleen, A multiplicity result for a singular problem with subcritical nonlinearities, J. Nonlinear Funct. Anal., 2017 (2017), Article ID 33, 1–18. $\href{http://dx.doi.org/10.23952/jnfa.2017.33}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85028619160&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+multiplicity+result+for+a+singular+problem+with+subcritical+nonlinearities%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000439569900011}{\mbox{[Web of Science]}} $
22. [22] J. Giacomoni, I. Schindler and P. Takac, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., Series 5, 6(1) (2007), 117–158. $\href{http://www.numdam.org/item/ASNSP_2007_5_6_1_117_0.pdf}{\mbox{[Web]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-35448932803&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22+local+minimizers+and+existence+of+multiple+solutions+for+a+singular+quasilinear+equation%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000246901600007}{\mbox{[Web of Science]}} $
23. [23] J. Giacomoni, and K. Saoudi, Multiplicity of positive solutions for a singular and critical problem, Nonlinear Analysis, Theory, methods, and applications, 71(9) (2009), 4060–4077. $\href{https://doi.org/10.1016/j.na.2009.02.087}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-67349150688&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Multiplicity+of+positive+solutions+for+a+singular+and+critical+problem%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000267621000042}{\mbox{[Web of Science]}} $
24. [24] R. Dhanya, J. Giacomoni, S. Prashanth and K. Saoudi, Global bifurcation and local multiplicity results for elliptic equations with singular nonlinearity of super exponential growth in R2, Adv. Differ. Equ. 17(3-4) (2012), 369–400. $\href{http://dx.doi.org/10.57262/ade/1355703090}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84893783288&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Global+bifurcation+and+local+multiplicity+results+for+elliptic+equations+with+singular+nonlinearity+of+super+exponential+growth+in%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000299854300006}{\mbox{[Web of Science]}} $
25. [25] K. Saoudi, and M. Kratou, Existence of multiple solutions for a negular and quasilinear equation, Complex Var. Elliptic Equ., 60(7) (2015), 893–925. $ \href{https://doi.org/10.1080/17476933.2014.981169}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84929948414&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Existence+of+multiple+solutions+for+a+singular+and+quasilinear+equation%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000355108000001}{\mbox{[Web of Science]}} $
26. [26] W. Liu, G. Dai, N. S. Papageorgiou, and P. Winkert, Existence of solutions for singular double phase problems via the Nehari manifold method, Anal. Math. Phys., 12(3) (2022), Artcle 75, 1-25. $\href{https://doi.org/10.1007/s13324-022-00686-6}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85128437960&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Existence+of+solutions+for+singular+double+phase+problems+via+the+Nehari+manifold+method%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000783477100001}{\mbox{[Web of Science]}} $
27. [27] N.S. Papageorgiou, P. Winkert, Positive solutions for weighted singular p-Laplace equations via Nehari manifolds, Appl. Anal., 100(11) (2021), 2436–2448. $ \href{https://doi.org/10.1080/00036811.2019.1688791}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85075068557&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Positive+solutions+for+weighted+singular+p-Laplace+equations+via+Nehari+manifolds%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000496326600001}{\mbox{[Web of Science]}} $
28. [28] N.S. Papageorgiou, D.D. Repovs and C. Vetro, Positive solutions for singular double phase problems, J. Math. Anal. Appl., 501(1) (2021), 123896. $ \href{https://doi.org/10.1016/j.jmaa.2020.123896}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85078417352&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Positive+solutions+for+singular+double+phase+problems%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000651636400031}{\mbox{[Web of Science]}} $
29. [29] N.S. Papageorgiou and V.D. Radulescu, Combined effects of singular and sublinear nonlinearities in some elliptic problems, Nonlinear Anal., 109 (2014), 236–244. $ \href{https://doi.org/10.1016/j.na.2014.06.016}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84905279931&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Combined+effects+of+singular+and+sublinear+nonlinearities+in+some+elliptic+problems%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000340940000017}{\mbox{[Web of Science]}}$
30. [30] M. Ghergu and V.D. Radulescu, Bifurcation and asymptotics for the Lane-Emden-Fowler equation, C. R. Math. Acad. Sci. Paris, 337(4) (2003), 259–264. $\href{https://doi.org/10.1016/S1631-073X(03)00335-2}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0141738636&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Bifurcation+and+asymptotics+for+the+Lane-Emden-Fowler+equation%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000185912100008}{\mbox{[Web of Science]}} $
31. [31] M. Ghergu and V.D. Radulescu, Sublinear singular elliptic problems with two parameters, J. Differ. Equ., 195(2) (2003), 520–536. $ \href{https://doi.org/10.1016/S0022-0396(03)00105-0}{\mbox{[CrossRef]}} \href{https://www.scopus.com/results/results.uri?sort=plf-f&src=s&st1=%22Functional+Analysis%2CSobolev+Spaces+and+Partial+Differential+Equations%22&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&sl=87&s=TITLE-ABS-KEY%28%22Sublinear+singular+elliptic+problems+with+two+parameters%22%29&origin=searchbasic&editSaveSearch=&yearFrom=Before+1960&yearTo=Present&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&limit=10}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000186784000010}{\mbox{[Web of Science]}} $
32. [32] K.S. Yijing and Z. Duanzhi, The role of the power 3 for elliptic equations with negative exponents, Calc. Var. Partial Differ. Equ., 49(3-4), (2014), 909–922. $ \href{https://doi.org/10.1007/s00526-013-0604-x}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84894539079&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22The+role+of+the+power+3+for+elliptic+equations+with+negative+exponents%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000334679400001}{\mbox{[Web of Science]}} $
33. [33] M. Ghergu and V.D. Radulescu, Ground state solutions for the singular Lane-Emden-Fowler equation with sublinear convection term, J. Math. Anal. Appl., 333(1) (2007), 265–273. $\href{https://doi.org/10.1016/j.jmaa.2006.09.074}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-34248159929&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22ground+state+solutions+for+the+singular+Lane-Emden-Fowler+equation+with+sublinear+convection+term%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000247325900019}{\mbox{[Web of Science]}} $
34. [34] J.I. Diaz and J. Hernandez, Positive and free boundary solutions to singular nonlinear elliptic problems with absorption; An overview and open problems, Variational and Topological Methods: Theory, Applications, Numerical Simulations, and Open Problems (2012). Electron. J. Differ. Equ., Conf., 21 (2014), 31–44. $ \href{https://ejde.math.txstate.edu/conf-proc/21/d1/diaz.pdf}{\mbox{[Web]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000455674400005}{\mbox{[Web of Science]}} $
35. [35] T. Godoy and A. Guerin, Multiplicity of weak solutions to subcritical singular elliptic Dirichlet problems, Electron. J. Qual. Theory Differ. Equ., 2017(100) (2017), 1–30. $\href{https://doi.org/10.14232/ejqtde.2017.1.100}{\mbox{[CrossRef]}} $
36. [36] T. Godoy and A. Guerin, Multiple finite-energy positive weak solutions to singular elliptic problems with a parameter, AIMS Math., 3(1) (2018), 233–252. $ \href{https://doi.org/10.3934/Math.2018.1.233}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85068501705&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Multiple+finite-energy+positive+weak+solutions+to+singular+elliptic+problems+with+a+parameter%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000432884700014}{\mbox{[Web of Science]}}$
37. [37] T. Godoy and A. Guerin, Regularity of the lower positive branch for singular elliptic bifurcation problems, Electron. J. Differ. Equ., 2019(49) (2019), 1–32. $ \href{https://ejde.math.txstate.edu/Volumes/2019/49/godoy.pdf}{\mbox{[Web]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85068511534&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Regularity+of+the+lower+positive+branch+for+singular+elliptic+bifurcation+problems%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000465612200001}{\mbox{[Web of Science]}} $
38. [38] M. Chhetri, P. Drabek, and R. Shivaji, Analysis of positive solutions for classes of quasilinear singular problems on exterior domains, Adv. Nonlinear Anal., 6(4) (2017), 447–459. $ \href{https://doi.org/10.1515/anona-2015-0143}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85032980186&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Analysis+of+positive+solutions+for+classes+of+quasilinear+singular+problems+on+exterior+domains%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000414657300005}{\mbox{[Web of Science]}} $
39. [39] M. Ghergu and V. D. Radulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, No 37 (2008). $ \href{http://dx.doi.org/10.1093/oso/9780195334722.001.0001}{\mbox{[CrossRef]}} $
40. [40] V.D. Radulescu, Singular phenomena in nonlinear elliptic problems. From blow-up boundary solutions to equations with singular nonlinearities, In: Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 4 (M. Chipot, Editor), North-Holland Elsevier Science, Amsterdam, 2007, pp. 483–591. $\href{https://doi.org/10.1016/S1874-5733(07)80010-6}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-67649958959&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Singular+phenomena+in+nonlinear+elliptic+problems.+From+blow-up+boundary+solutions+to+equations+with+singular+nonlinearities%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000311386600008}{\mbox{[Web of Science]}} $
41. [41] B. Gidas and J. Spruck, A Priori Bounds for Positive Solutions of Nonlinear Elliptic Equations, Commun. Partial Differ. Equ., 6(8) (1981), 883–901. $ \href{https://doi.org/10.1080/03605308108820196}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84939873114&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+Priori+Bounds+for+Positive+Solutions+of+Nonlinear+Elliptic+Equations%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=1}{\mbox{[Scopus]}}$
42. [42] A.C. Ponce, Selected Problems on Elliptic Equations Involving Measures, ArXiv: 1204.0668v2 [Math.AP], (2014). $ \href{https://doi.org/10.48550/arXiv.1204.0668}{\mbox{[CrossRef]}} $
43. [43] T. Godoy, Principal eigenvalues of elliptic problems with singular potential and bounded weight function, Construc. Math. Anal., 6(2) (2023), 107–127. $\href{https://doi.org/10.33205/cma.1272110}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85167894843&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Principal+eigenvalues+of+elliptic+problems+with+singular+potential+and+bounded+weight+function%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001106119900002}{\mbox{[Web of Science]}} $
44. [44] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second Order Springer-Verlag, Berlin Heidelberg New York, (2001). $\href{https://doi.org/10.1007/978-3-642-61798-0}{\mbox{[CrossRef]}} $
45. [45] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, 2011. $ \href{https://doi.org/10.1007/978-0-387-70914-7}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000284734900012}{\mbox{[Web of Science]}} $
46. [46] N.H. Loc and K. Schmitt, Boundary value problems for singular elliptic equations, Rocky Mountain J. Math., 41(2) (2011), 555–572. $ \href{http://dx.doi.org/10.1216/RMJ-2011-41-2-555}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-79958709502&origin=resultslist&sort=plf-f&src=s&sid=ad3bb1fca4dec66d86ecf2536e1e693a&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22boundary+value+problems+for+singular+elliptic+equations%22%29&sl=87&sessionSearchId=ad3bb1fca4dec66d86ecf2536e1e693a&relpos=1}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000291250200012}{\mbox{[Web of Science]}} $