## Existence of bounded solutions for non linear elliptic unilateral problems.(English)Zbl 0687.35042

Let $$\Omega$$ be a bounded domain in $${\mathbb{R}}^ n$$, $$\psi$$ a measurable function on $$\Omega$$, $$p>1$$, and $$A(u)=div a(x,u,Du)+a_ 0(x,u,Du)$$ an elliptic quasilinear differential operator whose coefficients a, $$a_ 0$$ satisfy natural regularity and growth conditions which in particular guarantee that A is a continuous, pseudomonotone operator from the Sobolev space $$W_ 0^{1,p}(\Omega)$$ into its dual. The authors prove the existence of a solution $$u\in W_ 0^{1,p}(\Omega)\cap L^{\infty}(\Omega)$$ of the variational inequality $$u\geq \psi$$, $$<A(u),v-u>+\int H(x,u,Du)(v-u)dx\geq 0$$ for all $$v\in W_ 0^{1,p}(\Omega)\cap L^{\infty}(\Omega)$$ such that $$v\geq \psi$$. Here it is important to notice that the inhomogeneous term H is allowed to grow like $$| Du|^ p$$. The proof is carried out by an approximation of H by bounded functions $$H_{\epsilon}$$ for which a solution $$u_{\epsilon}$$ of the corresponding problem is known to exist. Then it is shown that the family $$u_{\epsilon}$$ is compact in $$W_ 0^{1,p}(\Omega)$$.
Reviewer: F.Tomi

### MSC:

 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 47H05 Monotone operators and generalizations 35J65 Nonlinear boundary value problems for linear elliptic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 49J40 Variational inequalities
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### References:

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