June  2022, 21(6): 2253-2269. doi: 10.3934/cpaa.2022056

Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities

Department of Mathematics and Statistics, Masaryk University, Building 08, Kotlářská 2, Brno, 611 37, Czech Republic

Received  September 2021 Revised  February 2022 Published  June 2022 Early access  March 2022

Fund Project: The author acknowledge the support from Czech Science Foundation, project GJ19-14413Y

This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of
$ p $
-
$ q $
type and singular nonlinearities
$ \left\{ \begin{alignedat}{2} {} - \mathcal{L}_{p,q} u & {} = \lambda \frac{f(u)}{u^\gamma}, \ u>0 && \quad\mbox{ in } \, \Omega, \\ u & {} = 0 && \quad\mbox{ on } \partial\Omega, \end{alignedat} \right. $
where
$ \Omega $
is a bounded domain in
$ \mathbb{R}^N $
with
$ C^2 $
boundary,
$ N \geq 1 $
,
$ \lambda >0 $
is a real parameter,
$ \mathcal{L}_{p,q} u : = {\rm{div}}(|\nabla u|^{p-2} \nabla u + |\nabla u|^{q-2} \nabla u), $
$ 1<p<q< \infty $
,
$ \gamma \in (0,1) $
, and
$ f $
is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann [1], we prove existence of three positive solutions in the positive cone of
$ C_\delta(\overline{\Omega}) $
and in a certain range of
$ \lambda $
.
Citation: Rakesh Arora. Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2253-2269. doi: 10.3934/cpaa.2022056
References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM review, 18 (1976), 620-709.  doi: 10.1137/1018114.

[2]

R. AroraJ. Giacomoni and G. Warnault, Regularity results for a class of nonlinear fractional Laplacian and singular problems, Nonlinear Differ. Equ. Appl., 28 (2021), 35 pp.  doi: 10.1007/s00030-021-00693-9.

[3]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.

[4]

M. G. CrandallP. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Partial Differ. Equ., 2 (1977), 193-222.  doi: 10.1080/03605307708820029.

[5]

R. DhanyaE. Ko and R. Shivaji, A three solution theorem for singular nonlinear elliptic boundary value problems, J. Math. Anal. Appl., 424 (2015), 598-612.  doi: 10.1016/j.jmaa.2014.11.012.

[6] M. Ghergu and V. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford University Press, 2008. 
[7]

J. GiacomoniD. Kumar and K. Sreenadh, Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems, Calc. Var. Partial Differ. Equ., 60 (2021), 33 pp.  doi: 10.1007/s00526-021-01994-8.

[8]

J. GiacomoniT. Mukherjee and K. Sreenadh, Existence of three positive solutions for a nonlocal singular Dirichlet boundary problem, Adv. Nonlinear Stud., 19 (2019), 333-352.  doi: 10.1515/ans-2018-0011.

[9]

J. GiacomoniI. Schindler and P. Taká$\breve{\rm{c}}$, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 117-158.  doi: 10.2422/2036-2145.2007.1.07.

[10]

C. He and G. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing p & q-laplacians, Ann. Acad. Sci. Fenn. Math, 33 (2008), 337-371. 

[11]

E. KoK. L. Eun and R. Shivaji, Multiplicity results for classes of infinite positone problems, Z. Anal. Anwend., 30 (2011), 305-318.  doi: 10.4171/ZAA/1436.

[12]

D. KumarV. D. Rădulescu and K. Sreenadh, Singular elliptic problems with unbalanced growth and critical exponent, Nonlinearity, 33 (2020), 3336-3369.  doi: 10.1088/1361-6544/ab81ed.

[13]

A. Lê, On the local Hölder continuity of the inverse of the $p$-laplace operator, Proc. Amer. Math. Soc., 135 (2007), 3553-3560.  doi: 10.1090/S0002-9939-07-08913-7.

[14]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Urall'tseva for elliptic equations, Commun. Partial Differ. Equ., 16 (1991), 311-361.  doi: 10.1080/03605309108820761.

[15]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p$, $q$–growth conditions, J. Differ. Equ., 90 (1991), 1-30.  doi: 10.1016/0022-0396(91)90158-6.

[16]

N. S. PapageorgiouV. D. Rădulescu and D. D. Repov$\breve{\rm{s}}$, Nonlinear nonhomogeneous singular problems, Calc. Var., 59 (2020), 1-31.  doi: 10.1007/s00526-019-1667-0.

[17]

P. Pucci and J. B. Serrin, The Maximum Principle, Springer Science and Business Media, 2007. doi: 10.1007/978-3-7643-8145-5.

[18]

M. Ramaswamy and R. Shivaji, Multiple positive solutions for classes of p-laplacian equations, Differ. Int. Equ., 17 (2004), 1255-1261. 

[19]

R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear analysis and applications (Arlington, Tex., 1986), 561-566, Lecture Notes in Pure and Appl. Math., 109, Dekker, New York, 1987.

[20]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv., 29 (1987), 33 pp.  doi: 10.1070/im1987v029n01abeh000958.

show all references

References:
[1]

H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM review, 18 (1976), 620-709.  doi: 10.1137/1018114.

[2]

R. AroraJ. Giacomoni and G. Warnault, Regularity results for a class of nonlinear fractional Laplacian and singular problems, Nonlinear Differ. Equ. Appl., 28 (2021), 35 pp.  doi: 10.1007/s00030-021-00693-9.

[3]

M. Colombo and G. Mingione, Regularity for double phase variational problems, Arch. Ration. Mech. Anal., 215 (2015), 443-496.  doi: 10.1007/s00205-014-0785-2.

[4]

M. G. CrandallP. H. Rabinowitz and L. Tartar, On a Dirichlet problem with a singular nonlinearity, Commun. Partial Differ. Equ., 2 (1977), 193-222.  doi: 10.1080/03605307708820029.

[5]

R. DhanyaE. Ko and R. Shivaji, A three solution theorem for singular nonlinear elliptic boundary value problems, J. Math. Anal. Appl., 424 (2015), 598-612.  doi: 10.1016/j.jmaa.2014.11.012.

[6] M. Ghergu and V. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford University Press, 2008. 
[7]

J. GiacomoniD. Kumar and K. Sreenadh, Sobolev and Hölder regularity results for some singular nonhomogeneous quasilinear problems, Calc. Var. Partial Differ. Equ., 60 (2021), 33 pp.  doi: 10.1007/s00526-021-01994-8.

[8]

J. GiacomoniT. Mukherjee and K. Sreenadh, Existence of three positive solutions for a nonlocal singular Dirichlet boundary problem, Adv. Nonlinear Stud., 19 (2019), 333-352.  doi: 10.1515/ans-2018-0011.

[9]

J. GiacomoniI. Schindler and P. Taká$\breve{\rm{c}}$, Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 6 (2007), 117-158.  doi: 10.2422/2036-2145.2007.1.07.

[10]

C. He and G. Li, The regularity of weak solutions to nonlinear scalar field elliptic equations containing p & q-laplacians, Ann. Acad. Sci. Fenn. Math, 33 (2008), 337-371. 

[11]

E. KoK. L. Eun and R. Shivaji, Multiplicity results for classes of infinite positone problems, Z. Anal. Anwend., 30 (2011), 305-318.  doi: 10.4171/ZAA/1436.

[12]

D. KumarV. D. Rădulescu and K. Sreenadh, Singular elliptic problems with unbalanced growth and critical exponent, Nonlinearity, 33 (2020), 3336-3369.  doi: 10.1088/1361-6544/ab81ed.

[13]

A. Lê, On the local Hölder continuity of the inverse of the $p$-laplace operator, Proc. Amer. Math. Soc., 135 (2007), 3553-3560.  doi: 10.1090/S0002-9939-07-08913-7.

[14]

G. M. Lieberman, The natural generalization of the natural conditions of Ladyzhenskaya and Urall'tseva for elliptic equations, Commun. Partial Differ. Equ., 16 (1991), 311-361.  doi: 10.1080/03605309108820761.

[15]

P. Marcellini, Regularity and existence of solutions of elliptic equations with $p$, $q$–growth conditions, J. Differ. Equ., 90 (1991), 1-30.  doi: 10.1016/0022-0396(91)90158-6.

[16]

N. S. PapageorgiouV. D. Rădulescu and D. D. Repov$\breve{\rm{s}}$, Nonlinear nonhomogeneous singular problems, Calc. Var., 59 (2020), 1-31.  doi: 10.1007/s00526-019-1667-0.

[17]

P. Pucci and J. B. Serrin, The Maximum Principle, Springer Science and Business Media, 2007. doi: 10.1007/978-3-7643-8145-5.

[18]

M. Ramaswamy and R. Shivaji, Multiple positive solutions for classes of p-laplacian equations, Differ. Int. Equ., 17 (2004), 1255-1261. 

[19]

R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear analysis and applications (Arlington, Tex., 1986), 561-566, Lecture Notes in Pure and Appl. Math., 109, Dekker, New York, 1987.

[20]

V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izv., 29 (1987), 33 pp.  doi: 10.1070/im1987v029n01abeh000958.

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