We consider a nonlinear elliptic equation of the form div [a(∇u)] + F[u] = 0 on a domain Ω, subject to a Dirichlet boundary condition tru = φ. We do not assume that the higher order term a satisfies growth conditions from above. We prove the existence of continuous solutions either when Ω is convex and φ satisfies a one-sided bounded slope condition, or when a is radial: $a\left(\xi \right)=\frac{l\left(\right|\xi \left|\right)}{\left|\xi \right|}\xi $ for some increasing l:ℝ^{+} → ℝ^{+}.

Keywords: nonlinear elliptic equations, continuity of solutions, lower bounded slope condition, Lavrentiev phenomenon

@article{COCV_2013__19_1_1_0, author = {Bousquet, Pierre}, title = {Continuity of solutions of a nonlinear elliptic equation}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1--19}, publisher = {EDP-Sciences}, volume = {19}, number = {1}, year = {2013}, doi = {10.1051/cocv/2011194}, mrnumber = {3023057}, zbl = {1271.35028}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv/2011194/} }

TY - JOUR AU - Bousquet, Pierre TI - Continuity of solutions of a nonlinear elliptic equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 1 EP - 19 VL - 19 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2011194/ DO - 10.1051/cocv/2011194 LA - en ID - COCV_2013__19_1_1_0 ER -

%0 Journal Article %A Bousquet, Pierre %T Continuity of solutions of a nonlinear elliptic equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 1-19 %V 19 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2011194/ %R 10.1051/cocv/2011194 %G en %F COCV_2013__19_1_1_0

Bousquet, Pierre. Continuity of solutions of a nonlinear elliptic equation. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 1, pp. 1-19. doi : 10.1051/cocv/2011194. http://archive.numdam.org/articles/10.1051/cocv/2011194/

[1] The lower bounded slope condition. J. Convex Anal. 1 (2007) 119 − 136. | MR | Zbl

,[2] Local Lipschitz continuity of solutions of non-linear elliptic differential-functional equations. ESAIM Control Optim. Calc. Var. 13 (2007) 707 − 716. | Numdam | MR | Zbl

,[3] Continuity of solutions of a problem in the calculus of variations. Calc. Var. Partial Differential Equations 41 (2011) 413 − 433. | MR | Zbl

,[4] Continuity of solutions to a basic problem in the calculus of variations. Ann. Scvola Norm. Super. Pisa Cl. Sci. (5) 4 (2005) 511 − 530. | Numdam | MR | Zbl

,[5] On the Euler-Lagrange equation for functionals of the calculus of variations without upper growth conditions. SIAM J. Control Optim. 48 (2009) 2857 − 2870. | MR | Zbl

and ,[6] Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin (2001) Reprint of the 1998 edition. | MR | Zbl

and ,[7] On the bounded slope condition. Pac. J. Math. 18 (1966) 495 − 511. | MR | Zbl

,[8] On some non-linear elliptic differential-functional equations. Acta Math. 115 (1966) 271 − 310. | MR | Zbl

and ,[9] Linear and quasilinear elliptic equations. Academic Press, New York (1968). | MR | Zbl

and ,[10] Multiple integrals in the calculus of variations. Springer-Verlag, New York (1966). | MR | Zbl

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