# American Institute of Mathematical Sciences

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. Ambrosetti, H. 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. Bernis, J. 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. Cheng, Z. 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. Gazzola, H.-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. Li, S. 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. Wei, X. 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. Ambrosetti, H. 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. Bernis, J. 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. Cheng, Z. 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. Gazzola, H.-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. Li, S. 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. Wei, X. 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.
 [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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2007, 19 (2) : 419-430. doi: 10.3934/dcds.2007.19.419

2020 Impact Factor: 2.425