Nonexistence of Nontrivial Weak Solutions of Some Nonlinear Inequalities with Gradient Nonlinearity

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Abstract

In this article, we modify the results obtained by Mitidieri and Pohozaev on sufficient conditions for the absence of nontrivial weak solutions to nonlinear inequalities and systems with integer powers of|the Laplace operator and with a nonlinear term of the form a(x)|∇(Δmu)|q+ b(x)|∇u|s. We obtainoptimal a priori estimates by applying the nonlinear capacity method with an appropriate choice of testfunctions. As a result, we prove the absence of nontrivial weak solutions to nonlinear inequalities and systems by contradiction.

About the authors

V. E. Admasu

Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: galakhov@rambler.ru
Moscow, Russia

E. I. Galakhov

Peoples’ Friendship University of Russia (RUDN University)

Email: galakhov@rambler.ru
Moscow, Russia

O. A. Salieva

Moscow State Technological University “Stankin”

Email: olga.a.salieva@gmail.com
Moscow, Russia

References

  1. Галахов Е. И. О некоторых неравенствах в частных производных с градиентными слагаемыми// Тр. МИАН. - 2013. - 283.- С. 40-48.
  2. Галахов Е. И., Салиева О. А. Разрушение решений некоторых нелинейных неравенств с особенностями на неограниченных множествах// Мат. заметки. - 2015. - 98, № 2. - С. 187-195.
  3. Митидиери Э., Похожаев С. И. Априорные оценки и разрушение решений нелинейных уравнений и неравенств в частных производных// Тр. МИАН. - 2001. - 234. - С. 3-383.
  4. Похожаев С. И. Существенно нелинейные емкости, порожденные дифференциальными операторами// Докл. РАН. - 1997. - 357, № 5. - С. 592-594.
  5. Салиева О. А. Отсутствие решений некоторых нелинейных неравенств с дробными степенями оператора Лапласа// Мат. заметки. - 2017. - 101, № 4. - С. 699-703.
  6. Farina A., Serrin J. Entire solutions of completely coercive quasilinear elliptic equations// J. Differ. Equ. - 2011. - 250, № 12. - С. 4367-4408.
  7. Farina A., Serrin J. Entire solutions of completely coercive quasilinear elliptic equations II// J. Differ. Equ. - 2011. - 250, № 12. - С. 4409-4436.
  8. Filippucci R., Pucci P., Rigoli M. Nonlinear weighted p-Laplacian elliptic inequalities with gradient terms// Commun. Contemp. Math. - 2010. - 12, № 3. - С. 501-535.
  9. Galakhov E., Salieva O. On blow-up of solutions to differential inequalities with singularities on unbounded sets// J. Math. Anal. Appl. - 2013. - 408, № 1. - С. 102-113.
  10. Galakhov E., Salieva O. Nonexistence of solutions of some inequalities with gradient non-linearities and fractional Laplacian// В сб.: «Proc. Int. Conf. Equadiff 2017». - Bratislava: Spektrum STU Publishing, 2017. - С. 157-162.
  11. Galakhov E., Salieva O. Uniqueness of the trivial solution of some inequalities with fractional Laplacian// Electron. J. Qual. Theory Differ. Equ. - 2019. - 2019, № 1. - С. 1-8.
  12. Li X., Li F. Nonexistence of solutions for singular quasilinear differential inequalities with a gradient nonlinearity// Nonlinear Anal. - 2012. - 75, № 2. - С. 2812-2822.

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