Bounded solutions of fourth-order nonlinear elliptic equations with convection terms

ABSTRACT

We study a Dirichlet problem associated to a nonlinear fourth-order equation with a convective lower order term and with coefficients of principal part satisfying a strengthened coercivity condition. We obtain the existence of a weak solution and its Hölder's continuity.

Keywords:

• High-order equations
• convection term
• Hölder's continuity

2010 Mathematics Subject Classifications:

• 35J30
• 35J60
• 35B65

Acknowledgements

The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 We note that $|R_2\left(w\right)|\le |{\psi }^{\mathrm{\prime }\mathrm{\prime }}\left(w\right)|\sum _{|\text{β}|=1}|{\mathrm{D}}^{\text{β}}w{|}^2\text{a.e. in }\mathrm{\Omega }.$

2 Here $p_{\alpha }^{\mathrm{\prime }}$ is the conjugate exponent of $p_{\alpha }$

Additional information

Funding

This paper was supported by FIR 2018 and Piano Triennale della Ricerca 2016-2018 (UNICT).