Illinois Journal of Mathematics

Solution to biharmonic equation with vanishing potential

Waldemar D. Bastos, Olimpio H. Miyagaki, and Rônei S. Vieira

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We establish a result on the existence of nontrivial solution for the following class of biharmonic elliptic equation

\[(\mathrm{P})\quad \Bigl\{\begin{array}{l@{\quad}l}\Delta^{2}u+V(x)u=K(x)f(u)&\mbox{in }R^{N},\\u\neq0,&\mbox{in }R^{N},u\in\mathcal{D}^{2,2}(R^{N}),\end{array}\]

where $\Delta^{2}u=\Delta(\Delta u)$, $V$ and $K$ are nonnegative potentials. $K$ vanishes at infinity and $f$ has a subcritical growth at infinity. The technique used here is the variational approach.

Article information

Illinois J. Math., Volume 57, Number 3 (2013), 839-854.

First available in Project Euclid: 3 November 2014

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Zentralblatt MATH identifier

Primary: 35J30: Higher-order elliptic equations [See also 31A30, 31B30] 35J60: Nonlinear elliptic equations 35J70: Degenerate elliptic equations 35J75: Singular elliptic equations


Bastos, Waldemar D.; Miyagaki, Olimpio H.; Vieira, Rônei S. Solution to biharmonic equation with vanishing potential. Illinois J. Math. 57 (2013), no. 3, 839--854. doi:10.1215/ijm/1415023513.

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  • C. O. Alves and J. M. do Ó, Positive solutions of a fourth order semilinear problem involving critical growth, Adv. Nonlinear Stud. 2 (2002), no. 4, 437–458.
  • C. O. Alves, J. M. do Ó and O. H. Miyagaki, On a class of singular biharmonic problems involving critical exponents, J. Math. Anal. Appl. 277 (2003), 12–26.
  • C. O. Alves, J. M. do Ó and O. H. Miyagaki, Nontrivial solutions for a class of semilinear biharmonic problems involving critical exponents, Nonlinear Anal. 46 (2001), 121–133.
  • C. O. Alves, O. H. Miyagaki and S. H. M. Soares, On the existence and concentration of positive solutions to a class of quasilinear elliptic problems on R, Math. Nachr. 284 (2011), 1784–1795.
  • C. O. Alves and M. A. S. Souto, Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity, J. Differential Equations 254 (2013), 1977–1991.
  • H. Berestycki and P.-L. Lions, Nonlinear scalar field quations, I: Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983), 313–346.
  • H. Brezis and L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437–477.
  • P. C. Carrião, R. Demarque and O. H. Miyagaki, Nonlinear biharmonic problems with singular potentials, Commun. Pure Appl. Anal. 13 (2014), 2141–2154.
  • J. Chabrowski and J. M. do Ó, On some fourth order semilinear elliptic problems in $R^N$, Nonlinear Anal. 49 (2002), 861–884.
  • R. Demarque and O. H. Miyagaki, Radial solutions of inhomogeneous fourth order elliptic equations and weighted Sobolev embeddings, to appear.
  • D. E. Edmunds, D. Fortunato and E. Jannelli, Critical exponents, critical dimensions and the biharmonic operator, Arch. Ration. Mech. Anal. 112 (1990), 269–289.
  • F. Gazzola and H.-C. Grunau, Radial entire solutions for supercritical biharmonic equations, Math. Ann. 334 (2006), 905–936.
  • V. I. Karpman, Influence of high-order dispersion on self-focusing. I. Qualitative investigation, Phys. Lett. A 160 (1991), 531–537.
  • V. I. Karpman, Stabilization of soliton instabilities by higher-order dispersion: Fourth order nonlinear Schrödinger-type equations, Phys. Rev. E (2) 53 (1996), no. 2, 1336–1339.
  • V. I. Karpman and A. G. Shagalov, Stability of soliton described by nonlinear Schrödinger-type equations with higher-order dispersion, Phys. D 144 (2000), 194–210.
  • P. L. Lions, The concentration-compactness principle in the calculus of variations. the limit case, part 1, Rev. Mat. Iberoam. 1 (1985), 145–201.
  • W. G. Maz'ja, Sobolev spaces, Springer, Berlin, 1985.
  • O. H. Miyagaki and M. A. S. Souto, Superlinear problems without Ambrosetti and Rabinowitz growth condition, J. Differential Equations 245 (2008), 3628–3638.
  • E. S. Noussair, C. A. Swanson and J. Yang, Transcritical biharmonic equations in $R^N$, Funkcial. Ekvac. 35 (1992), 533–543.
  • B. Pausader, The cubic fourth-order Schrödinger equation, J. Funct. Anal. 256 (2009), 2473–2517.
  • M. T. O. Pimenta and S. H. M. Soares, Existence and concentration of solutions for a class of biharmonic equations, J. Math. Anal. Appl. 390 (2012), 274–289.
  • Y. Shen and Y. Wang, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations 246 (2009), 3109–3125.
  • Y. Ye and C.-L. Tang, Infinitely many solutions for fourth order elliptic equations, J. Math. Anal. Appl. 394 (2012), 841–854.