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


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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