Advanced Search
Article Contents
Article Contents

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

This work is supported National Science Foundation of China No. 11871250
Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Mathematics Subject Classification: 35J35, 35J40, 35J65.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] R. A. AdamsSobolev 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. LakshmikanthamNonlinear 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.
  • 加载中

Article Metrics

HTML views(326) PDF downloads(254) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint