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Existence and non-existence results for ground states of quasilinear elliptic equations. (English) Zbl 0795.35027
Summary: We consider the quasilinear equation $\Delta u+ u^ p- | \nabla u|^ q=0 \qquad \text{in }\mathbb{R}^ n, \tag{I}$ where $$p$$ and $$q$$ are given positive exponents and $$n\geq 3$$. Of particular interest is the question of existence and non-existence of radial ground states, that is, nonnegative, nontrivial radial solutions defined for all $$r\geq 0$$. Using shooting methods, general Poincaré inequalities, and Pokhozaev-Pucci- Serrin type identities, we establish existence if either $\text{(i)} \quad p>\ell, \qquad \text{(ii)} \quad p=\ell,\;q<p, \qquad \text{(iii)} \quad p<\ell,\;q< {{2p} \over {p+1}},$ where $$\ell$$ is the critical value $$(n+2)/ (n-2)$$. On the other hand, putting $$\ell_ 1= n/ (n-2)$$, we show that if either $\text{(i)} \quad q\geq {{2p} \over {p+1}},\;0<p<1, \qquad \text{(ii)} \quad q> {{2p} \over {p+1}},\;1\leq p\leq \ell_ 1,\qquad \text{(iii)} \quad q\geq \overline{q},\;\ell_ 1<p\leq \ell,$ then radial ground states for (I) cannot exist, where $$\overline{q}$$ is a function of $$p$$ and $$n$$ such that $$2p/ (p+1)< \overline{q} <p$$ for $$\ell_ 1<p<\ell$$ and $$\overline{q}=p$$ for $$p=\ell$$.
Finally, if $$p<q<2p/ (p+1)$$ and $$0<p<1$$, then radial ground states have compact support, while if $$q\geq 1$$ or $$p\geq\ell$$, they are necessarily everywhere positive.
The cases $$n=1,2$$ can also be treated, with corresponding results except that $$\ell_ 1= \ell=\infty$$.

##### MSC:
 35J60 Nonlinear elliptic equations 35A05 General existence and uniqueness theorems (PDE) (MSC2000) 35B40 Asymptotic behavior of solutions to PDEs
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