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

The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form $\mathrm{div}a\left(\nabla u\right)+F\left[u\right]\left(x\right)=0,$ over the functions $u\in W^{1,1}\left(\Omega \right)$ that assume given boundary values on ∂Ω. The vector field $a:{ℝ}^n\to {ℝ}^n$ satisfies an ellipticity condition and for a fixed denotes a non-linear functional of In considering the same problem, Hartman and Stampacchia [ (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions when satisfies...

Given a compact manifold $N^n\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p\<\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s,p}(Q^m;N^n)$ when $N^n$ is $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ when $N^n$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s,p}\cap W^{1,sp}$.

Given a compact manifold $N^n$, an integer $k\in {\mathbb{N}}_{*}$ and an exponent $1\le p<\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is dense with respect to the strong topology in the Sobolev space $W^{k,p}(Q^m;N^n)$ when the homotopy group ${\pi }_{\lfloor kp\rfloor }\left(N^n\right)$ of order $\lfloor kp\rfloor$ is trivial. We also prove density of maps that are smooth except for a set of dimension $m-\lfloor kp\rfloor -1$, without any restriction on the homotopy group of $N^n$.