### Nonlinear equations with natural growth terms and measure data.

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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, ${(1+|u\left|\right)}^{\overline{q}-1}\phantom{\rule{0.166667em}{0ex}}u\in {H}_{0}^{1}\left(\Omega \right)$, for a certain (optimal) exponent $\overline{q}$. This completes the recent results...

For $0<p-1<q$ and either $\u03f5=1$ or $\u03f5=-1$, we prove the existence of solutions of $-{\Delta}_{p}u=\u03f5{u}^{q}$ 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.

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