Advanced Search
Article Contents
Article Contents

Positive solutions to a Dirichlet problem with $p$-Laplacian and concave-convex nonlinearity depending on a parameter

Abstract Related Papers Cited by
  • A nonlinear elliptic equation with $p$-Laplacian, concave-convex reaction term depending on a parameter $\lambda>0$, and homogeneous boundary condition, is investigated. A bifurcation result, which describes the set of positive solutions as $\lambda$ varies, is obtained through variational methods combined with truncation and comparison techniques.
    Mathematics Subject Classification: Primary: 35J25, 35J92; Secondary: 49J40.


    \begin{equation} \\ \end{equation}
  • [1]

    S. Aizicovici, N. S. Papageorgiou and V. Staicu, Degree theory for operators of monotone type and nonlinear elliptic equations with inequality constraints, Mem. Amer. Math. Soc., 196 (2008).


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


    D. Arcoya and D. Ruiz, The Ambrosetti-Prodi problem for the $p$-Laplace operator, Comm. Partial Differential Equations, 31 (2006), 849-865.doi: 10.1080/03605300500394447.


    D. Averna, S. A. Marano and D. Motreanu, Multiple solutions for a Dirichlet problem with $p$-Laplacian and set-valued nonlinearity, Bull. Austral. Math. Soc., 77 (2008), 285-303.doi: 10.1017/S0004972708000282.


    L. Boccardo, M. Escobedo and I. Peral, A Dirichlet problem involving critical exponents, Nonlinear Anal., 24 (1995), 1639-1648.doi: 10.1016/0362-546X(94)E0054-K.


    G. Bonanno and G. Molica Bisci, Infinitely many solutions for a Dirichlet problem involving the $p$-Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 140 (2010), 737-752.doi: 10.1017/S0308210509000845.


    L. Gasiński and N. S. Papageorgiou, "Nonlinear Analysis," Ser. Math. Anal. Appl., 9, Chapman and Hall/CRC Press, Boca Raton, 2006.


    L. Gasiński and N. S. Papageorgiou, "Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems," Ser. Math. Anal. Appl., 8, Chapman and Hall/CRC Press, Boca Raton, 2005.


    J. P. Garcia Azorero, J. J. Manfredi and I. Peral Alonso, Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations, Comm. Contemp. Math., 2 (2000), 385-404.doi: 10.1142/S0219199700000190.


    M. Guedda and L. Veron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear Anal., 13 (1989), 879-902.doi: 10.1016/0362-546X(89)90020-5.


    S. Hu and N. S. Papageorgiou, Multiplicity of solutions for parametric $p$-Laplacian equations with nonlinearity concave near the origin, Tohoku Math. J., 62 (2010), 137-162.doi: 10.2748/tmj/1270041030.


    An Lê, Eigenvalue problems for the $p$-Laplacian, Nonlinear Anal., 64 (2006), 1057-1099.doi: 10.1016/j.na.2005.05.056.


    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.


    G. Li and C. Yang, The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti-Rabinowitz condition, Nonlinear Anal., 72 (2010), 4602-4613.doi: 10.1016/j.na.2010.02.037.


    P. Lindqvist, On the equation div$(|\nabla u|^{p-2}\nabla u) +\lambda |u|^{p-2}u=0$, Proc. Amer. Math. Soc., 109 (1990), 157-164.doi: 10.1090/S0002-9939-1990-1007505-7.


    O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz condition, J. Differential Equations, 245 (2008), 3628-3638.doi: 10.1016/j.jde.2008.02.035.


    I. Peral, Some results on quasilinear elliptic equations: growth versus shape, in "Nonlinear Functional Analysis and Applications to Differential Equations (Trieste 1997)" (A. Ambrosetti, K.-C. Chang and I. Ekeland eds.), World Sci. Publ., River Edge, NJ, (1998), 153-202.


    J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.doi: 10.1007/BF01449041.

  • 加载中

Article Metrics

HTML views() PDF downloads(273) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint