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 and 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. 
[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.

[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.

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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.

[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. 

[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.

[7]

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

[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.

[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.

[10]

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

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

show all references

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

[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.

[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.

[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. 

[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.

[7]

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

[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.

[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.

[10]

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

[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.

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[23]

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

[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.

[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.

[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.

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