September  2021, 14(9): 3067-3083. doi: 10.3934/dcdss.2021078

Existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with weight functions

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

2. 

School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, Texas 78539, USA

3. 

School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, Jiangsu 221116, China

Received  January 2020 Revised  September 2020 Published  September 2021 Early access  June 2021

Fund Project: This work is supported National Science Foundation of China No. 11871250

We consider the existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with the concave-convex nonlinearities $ f(x) |u|^{q-1} u $ and $ h(x) |u|^{p-1} u $ under certain conditions on $ f(x), \, h(x) $, $ p $ and $ q $. Applying the Nehari manifold method along with the fibering maps and the minimization method, we study the effect of $ f(x) $ and $ h(x) $ on the existence and multiplicity of nontrivial solutions for the semilinear biharmonic equation. When $ h(x)^+ \neq 0 $, we prove that the equation has at least one nontrivial solution if $ f(x)^+ = 0 $ and that the equation has at least two nontrivial solutions if $ \int_\Omega |f^+|^r\, \text{d}x \in (0, \varLambda^r) $, where $ r $ and $ \varLambda $ are explicit numbers. These results are novel, which improve and extend the existing results in the literature.

Citation: Xiyou Cheng, Zhaosheng Feng, Lei Wei. Existence and multiplicity of nontrivial solutions for a semilinear biharmonic equation with weight functions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3067-3083. doi: 10.3934/dcdss.2021078
References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[2]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475.  doi: 10.1007/BF01206962.  Google Scholar

[3]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[4]

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555-3561.  doi: 10.1090/S0002-9939-1995-1301008-2.  Google Scholar

[5]

F. BernisJ. Garcia Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 2 (1996), 219-240.   Google Scholar

[6]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[7]

X. ChengZ. Feng and L. Wei, Nontrivial solutions for a quasilinear elliptic system with weight functions, Differential Integral Equations, 33 (2020), 625-656.   Google Scholar

[8]

M. Cuesta and L. Leadi, On abstract indefinite concave-convex problems and applications to quasilinear elliptic equations, Nonlinear Differ. Equ. Appl., 24 (2017), 1-31.  doi: 10.1007/s00030-017-0444-z.  Google Scholar

[9]

F. Ebobisse and M. O. Ahmedou, On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552.  doi: 10.1016/S0362-546X(02)00273-0.  Google Scholar

[10]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[11]

F. GazzolaH.-Ch. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.  Google Scholar

[12] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., Boston, MA, 1988.   Google Scholar
[13]

S. LiS. Wu and H.-S. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224.  doi: 10.1006/jdeq.2001.4167.  Google Scholar

[14]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Reg. Conf. Ser. Math., American Mathematical Society, 1986. doi: 10.1090/cbms/065.  Google Scholar

[15]

V. D R$\check{\rm a}$dulescu and D. D. Repov$\check{\rm s}$, Combined effects for non-autonomous singular biharmonic problems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2057-2068.  doi: 10.3934/dcdss.2020158.  Google Scholar

[16]

Y. Sun and S. Li, A nonlinear elliptic equation with critical exponent: Estimates for extremal values, Nonlinear Anal., 69 (2008), 1856-1869.  doi: 10.1016/j.na.2007.07.030.  Google Scholar

[17]

G. Sweers, Positivity for the Navier bilaplace, an anti-eigenvalue and an expected lifetime, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 839-855.  doi: 10.3934/dcdss.2014.7.839.  Google Scholar

[18]

M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 705-717.  doi: 10.1017/S0308210500002614.  Google Scholar

[19]

G. Tarantello, On nonhomogeneous elliptic involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992) 281–304. doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar

[20]

Q. Wang and L. Yang, Positive solutions for a nonlinear system of fourth-order ordinary differential equations, Electron. J. Differential Equations, 45 (2020), 15 pp.  Google Scholar

[21]

L. WeiX. Cheng and Z. Feng, Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials, Discrete Contin. Dyn. Syst., 36 (2016), 7169-7189.  doi: 10.3934/dcds.2016112.  Google Scholar

[22]

L. Wei and Z. Feng, Isolated singularity for semilinear elliptic equations, Discrete Contin. Dyn. Syst., 35 (2015), 3239-3252.  doi: 10.3934/dcds.2015.35.3239.  Google Scholar

[23]

M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[24]

T.-F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270.  doi: 10.1016/j.jmaa.2005.05.057.  Google Scholar

[25]

L. Yang and X. Wang, On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, Bound. Value Probl., 2014 (2014), 117, 15 pp. doi: 10.1186/1687-2770-2014-117.  Google Scholar

[26]

Y. Zhang, Positive solutions of semilinear biharmonic equations with critical Sobolev exponents, Nonlinear Anal., 75 (2012), 55-67.  doi: 10.1016/j.na.2011.07.065.  Google Scholar

show all references

References:
[1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.   Google Scholar
[2]

S. Alama and G. Tarantello, On semilinear elliptic equations with indefinite nonlinearities, Calc. Var. Partial Differential Equations, 1 (1993), 439-475.  doi: 10.1007/BF01206962.  Google Scholar

[3]

A. AmbrosettiH. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal., 122 (1994), 519-543.  doi: 10.1006/jfan.1994.1078.  Google Scholar

[4]

T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555-3561.  doi: 10.1090/S0002-9939-1995-1301008-2.  Google Scholar

[5]

F. BernisJ. Garcia Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order, Adv. Differential Equations, 2 (1996), 219-240.   Google Scholar

[6]

K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Differential Equations, 193 (2003), 481-499.  doi: 10.1016/S0022-0396(03)00121-9.  Google Scholar

[7]

X. ChengZ. Feng and L. Wei, Nontrivial solutions for a quasilinear elliptic system with weight functions, Differential Integral Equations, 33 (2020), 625-656.   Google Scholar

[8]

M. Cuesta and L. Leadi, On abstract indefinite concave-convex problems and applications to quasilinear elliptic equations, Nonlinear Differ. Equ. Appl., 24 (2017), 1-31.  doi: 10.1007/s00030-017-0444-z.  Google Scholar

[9]

F. Ebobisse and M. O. Ahmedou, On a nonlinear fourth-order elliptic equation involving the critical Sobolev exponent, Nonlinear Anal., 52 (2003), 1535-1552.  doi: 10.1016/S0362-546X(02)00273-0.  Google Scholar

[10]

I. Ekeland, On the variational principle, J. Math. Anal. Appl., 47 (1974), 324-353.  doi: 10.1016/0022-247X(74)90025-0.  Google Scholar

[11]

F. GazzolaH.-Ch. Grunau and M. Squassina, Existence and nonexistence results for critical growth biharmonic elliptic equations, Calc. Var. Partial Differential Equations, 18 (2003), 117-143.  doi: 10.1007/s00526-002-0182-9.  Google Scholar

[12] D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Inc., Boston, MA, 1988.   Google Scholar
[13]

S. LiS. Wu and H.-S. Zhou, Solutions to semilinear elliptic problems with combined nonlinearities, J. Differential Equations, 185 (2002), 200-224.  doi: 10.1006/jdeq.2001.4167.  Google Scholar

[14]

P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Reg. Conf. Ser. Math., American Mathematical Society, 1986. doi: 10.1090/cbms/065.  Google Scholar

[15]

V. D R$\check{\rm a}$dulescu and D. D. Repov$\check{\rm s}$, Combined effects for non-autonomous singular biharmonic problems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2057-2068.  doi: 10.3934/dcdss.2020158.  Google Scholar

[16]

Y. Sun and S. Li, A nonlinear elliptic equation with critical exponent: Estimates for extremal values, Nonlinear Anal., 69 (2008), 1856-1869.  doi: 10.1016/j.na.2007.07.030.  Google Scholar

[17]

G. Sweers, Positivity for the Navier bilaplace, an anti-eigenvalue and an expected lifetime, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 839-855.  doi: 10.3934/dcdss.2014.7.839.  Google Scholar

[18]

M. Tang, Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 705-717.  doi: 10.1017/S0308210500002614.  Google Scholar

[19]

G. Tarantello, On nonhomogeneous elliptic involving critical Sobolev exponent, Ann. Inst. H. Poincaré Anal. Non Linéaire, 9 (1992) 281–304. doi: 10.1016/S0294-1449(16)30238-4.  Google Scholar

[20]

Q. Wang and L. Yang, Positive solutions for a nonlinear system of fourth-order ordinary differential equations, Electron. J. Differential Equations, 45 (2020), 15 pp.  Google Scholar

[21]

L. WeiX. Cheng and Z. Feng, Exact behavior of positive solutions to elliptic equations with multi-singular inverse square potentials, Discrete Contin. Dyn. Syst., 36 (2016), 7169-7189.  doi: 10.3934/dcds.2016112.  Google Scholar

[22]

L. Wei and Z. Feng, Isolated singularity for semilinear elliptic equations, Discrete Contin. Dyn. Syst., 35 (2015), 3239-3252.  doi: 10.3934/dcds.2015.35.3239.  Google Scholar

[23]

M. Willem, Minimax Theorems, Birkhäuser, Basel, 1996. doi: 10.1007/978-1-4612-4146-1.  Google Scholar

[24]

T.-F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl., 318 (2006), 253-270.  doi: 10.1016/j.jmaa.2005.05.057.  Google Scholar

[25]

L. Yang and X. Wang, On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, Bound. Value Probl., 2014 (2014), 117, 15 pp. doi: 10.1186/1687-2770-2014-117.  Google Scholar

[26]

Y. Zhang, Positive solutions of semilinear biharmonic equations with critical Sobolev exponents, Nonlinear Anal., 75 (2012), 55-67.  doi: 10.1016/j.na.2011.07.065.  Google Scholar

[1]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the heat equation with concave-convex nonlinearity and initial data in weak-$L^p$ spaces. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1715-1732. doi: 10.3934/cpaa.2011.10.1715

[2]

Junping Shi, Ratnasingham Shivaji. Exact multiplicity of solutions for classes of semipositone problems with concave-convex nonlinearity. Discrete & Continuous Dynamical Systems, 2001, 7 (3) : 559-571. doi: 10.3934/dcds.2001.7.559

[3]

Salvatore A. Marano, Nikolaos S. Papageorgiou. Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter. Communications on Pure & Applied Analysis, 2013, 12 (2) : 815-829. doi: 10.3934/cpaa.2013.12.815

[4]

Boumediene Abdellaoui, Abdelrazek Dieb, Enrico Valdinoci. A nonlocal concave-convex problem with nonlocal mixed boundary data. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1103-1120. doi: 10.3934/cpaa.2018053

[5]

M. L. M. Carvalho, Edcarlos D. Silva, C. Goulart. Choquard equations via nonlinear rayleigh quotient for concave-convex nonlinearities. Communications on Pure & Applied Analysis, 2021, 20 (10) : 3445-3479. doi: 10.3934/cpaa.2021113

[6]

Hongyu Ye. Positive high energy solution for Kirchhoff equation in $\mathbb{R}^{3}$ with superlinear nonlinearities via Nehari-Pohožaev manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3857-3877. doi: 10.3934/dcds.2015.35.3857

[7]

Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709

[8]

Qingfang Wang. The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2261-2281. doi: 10.3934/cpaa.2018108

[9]

Jia-Feng Liao, Yang Pu, Xiao-Feng Ke, Chun-Lei Tang. Multiple positive solutions for Kirchhoff type problems involving concave-convex nonlinearities. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2157-2175. doi: 10.3934/cpaa.2017107

[10]

Caisheng Chen, Qing Yuan. Existence of solution to $p-$Kirchhoff type problem in $\mathbb{R}^N$ via Nehari manifold. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2289-2303. doi: 10.3934/cpaa.2014.13.2289

[11]

Yaoping Chen, Jianqing Chen. Existence of multiple positive weak solutions and estimates for extremal values for a class of concave-convex elliptic problems with an inverse-square potential. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1531-1552. doi: 10.3934/cpaa.2017073

[12]

Miao-Miao Li, Chun-Lei Tang. Multiple positive solutions for Schrödinger-Poisson system in $\mathbb{R}^{3}$ involving concave-convex nonlinearities with critical exponent. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1587-1602. doi: 10.3934/cpaa.2017076

[13]

Jinguo Zhang, Dengyun Yang. Fractional $ p $-sub-Laplacian operator problem with concave-convex nonlinearities on homogeneous groups. Electronic Research Archive, 2021, 29 (5) : 3243-3260. doi: 10.3934/era.2021036

[14]

Tran Ngoc Thach, Nguyen Huy Tuan, Donal O'Regan. Regularized solution for a biharmonic equation with discrete data. Evolution Equations & Control Theory, 2020, 9 (2) : 341-358. doi: 10.3934/eect.2020008

[15]

Xiaoming He, Marco Squassina, Wenming Zou. The Nehari manifold for fractional systems involving critical nonlinearities. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1285-1308. doi: 10.3934/cpaa.2016.15.1285

[16]

Ziqing Yuana, Jianshe Yu. Existence and multiplicity of nontrivial solutions of biharmonic equations via differential inclusion. Communications on Pure & Applied Analysis, 2020, 19 (1) : 391-405. doi: 10.3934/cpaa.2020020

[17]

Fouad Hadj Selem, Hiroaki Kikuchi, Juncheng Wei. Existence and uniqueness of singular solution to stationary Schrödinger equation with supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4613-4626. doi: 10.3934/dcds.2013.33.4613

[18]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335

[19]

Dimitra Antonopoulou, Georgia Karali. Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 31-55. doi: 10.3934/dcdsb.2011.16.31

[20]

A. Pankov. Gap solitons in periodic discrete nonlinear Schrödinger equations II: A generalized Nehari manifold approach. Discrete & Continuous Dynamical Systems, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (160)
  • HTML views (140)
  • Cited by (0)

Other articles
by authors

[Back to Top]