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Zbl 0687.35042
Boccardo, L.; Murat, F.; Puel, J.P.
Existence of bounded solutions for non linear elliptic unilateral problems. (English)
[J] Ann. Mat. Pura Appl., IV. Ser. 152, 183-196 (1988). ISSN 0373-3114; ISSN 1618-1891/e

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

MSC 2000:: *35J85 Unilateral problems; variational inequalities (elliptic type)
47H05 Monotone operators (with respect to duality)
35J65 (Nonlinear) BVP for (non)linear elliptic equations
35D05 Existence of generalized solutions of PDE
49J40 Variational methods including variational inequalities

Keywords: obstacle problem; strong nonlinearity; pseudomonotone operator

Cited in: Zbl 1097.35050

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