## Existence via partial regularity for degenerate systems of variational inequalities with natural growth.(English)Zbl 0774.49008

Let $$\Omega\subset \mathbb{R}^ n$$, $$n\geq 2$$, be a bounded open set, $$K\subset \mathbb{R}^ N$$, $$N\geq 1$$, be a convex compact set with $$C^ 2$$ boundary, $$p\geq 2$$, $$H^{1,p}(\Omega,\mathbb{R}^ N)$$ and $$\mathring H^{1,p}(\Omega,\mathbb{R}^ N)$$ be the usual Sobolev spaces, $$u_ 0(\cdot)\in H^{1,p}(\Omega,\mathbb{R}^ N)$$ with $$u_ 0(\cdot)\in K$$ a.e. and $$\mathbb{K}:=\{w\in H^{1,p}(\Omega,\mathbb{R}^ N): w-u_ 0\in\mathring H^{1,p}(\Omega,\mathbb{R}^ N)$$, $$w(x)\in K$$ a.e.}. Let $$f: \Omega\times \mathbb{R}^ N\times \mathbb{R}^{n\cdot N}\to\mathbb{R}^ N$$ be a continuous function satisfying $$| f(x,y,Q)|\leq a| Q|^ p$$ where $$a$$ is a positive real for which $$a\cdot\text{diam }K<1$$ holds. The following result is proved:
Theorem. Under the above assumptions the variational inequality “Find $$u\in\mathbb{K}$$ such that $$\int_ \Omega\bigl\{| Du|^{p-2} Du\cdot D(v-u)-f(\cdot,u,Du)(v-u)\bigr\} dx\geq 0$$ holds for all $$v\in \mathbb{K}$$” has at least one solution $$u\in \mathbb{K}$$. For $$p<n$$, there exists a relatively closed subset $$\Sigma$$ of $$\Omega$$ such that $${\mathcal H}^{n- p}(\Sigma)=0$$ and with the additional property that $$u$$ is of class $$C^{1,\varepsilon}$$ (for some $$\varepsilon\in (0,1))$$ on $$\Omega\backslash \Sigma$$. If $$p\geq n$$, then $$\Sigma=\emptyset$$.
As an application the existence of a solution to a nonlinear elliptic Dirichlet problem is proved.

### MSC:

 49J40 Variational inequalities
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