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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 \(\ell_ 1= \ell=\infty\).

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