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We consider a class of stationary viscous Hamilton-Jacobi equations aswhere $\lambda \ge 0$, $A\left(x\right)$ is a bounded and uniformly elliptic matrix and $H(x,\xi )$ is convex in $\xi$ and grows at most like ${\left|\xi \right|}^q+f\left(x\right)$, with $1\<q\<2$ and $f\in L^{N/q^{\text{'}}}\left(\Omega \right)$. 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, ${(1+|u\left|\right)}^{\overline{q}-1}u\in H_0^1\left(\Omega \right)$, for a certain (optimal) exponent $\overline{q}$. This completes the recent results...

For $0<p-1<q$ and either $ϵ=1$ or $ϵ=-1$, we prove the existence of solutions of $-{\Delta }_{p}u=ϵu^q$ in a cone $C_S$, with vertex 0 and opening $S$, vanishing on $\partial C_S$, of the form $u\left(x\right)={\left|x\right|}^{-\beta }\omega (x/\left|x\right|)$. The problem reduces to a quasilinear elliptic equation on $S$ 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.