Symmetry of ground states of quasilinear elliptic equations
The famous Gidas-Ni-Nirenberg result asserting the radial symmetry of nonnegative solutions of certain semilinear elliptic problems is generalized to quasilinear elliptic equations of the form \[ \nabla\cdot[A(\left|\nabla u\right|)]+f(u)=0\quad \text{on }\mathbb R^n \] subject to the limit condition \(u(x)\to 0\) as \(\left|x\right|\to 0\). Here \(n\geq 2\), \(f:[0,\infty)\to\mathbb R\) is continuous and locally Lipschitz continuous on \((0,\infty)\), \(f(0)=0\) and \(f\) is nonincresing near 0. \(A\) belongs either (regular case) to \(C^{1,1}([0,\infty))\) and \({d\over dt}tA(t)\) is positive on \([0,\infty)\) or (degenerate case) \(A\) belongs to \(C^{1,1}((0,\infty))\), \({d\over dt}tA(t)>0\) on \((0,\infty)\) and \(tA(t)\to 0\) as \(t\to 0\). In the regular case Theorem 1 of the paper states symmetry of these ``ground states'' \(u\) provided that \(\text{supp}_+(u):=\{x: u(x)>0\}\) is connected. Moreover, \(u\) as a ``function of the radius \(r\)'' satisfies \(u'(r)<0\) for those \(r\) where \(u\) is positive. As for the degenerate case, Theorem 2 yields the same assertion for these ground states provided that there is exactly one point in \(\text{supp}_+(u)\) where the gradient vanishes. Also, if \({d\over dt}tA(t)\to\infty\) as \(t\to 0\), the authors show these assertions under the hypothesis that \(\{x:\nabla u(x)>0\}\) is connected.
scientific article; zbMATH DE number 1356187