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We consider a class of stationary viscous Hamilton-Jacobi equations aswhere , is a bounded and uniformly elliptic matrix and is convex in and grows at most like , with and . Under such growth conditions solutions are in general unbounded, and there is not uniqueness of usual weak solutions. We prove that uniqueness holds in the restricted class of solutions satisfying a suitable energy-type estimate,
, for a certain (optimal) exponent . This completes the recent results...
For and either or , we prove the existence of solutions of in a cone , with vertex 0 and opening , vanishing on , of the form . The problem reduces to a quasilinear elliptic equation on and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.
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