## Monotonicity and symmetry results for $$p$$-Laplace equations and applications.(English)Zbl 1002.35045

Summary: We prove monotonicity and symmetry properties of positive solutions of the equation $$-\text{div} (|Du|^{p-2}Du)= f(u)$$, $$1<p<2$$, in a smooth bounded domain $$\Omega$$ satisfying the boundary condition $$u=0$$ on $$\partial\Omega$$. We assume $$f$$ locally Lipschitz continuous only in $$(0,\infty)$$ and either $$f\geq 0$$ in $$[0,\infty]$$ or $$f$$ satisfying a growth condition near zero. In particular we can treat the case of $$f(s)= s^\alpha-cs^q$$, $$\alpha>0$$, $$c\geq 0$$, $$q\geq p-1$$. As a consequence we get an extension to the $$p$$-Laplacian case of a symmetry theorem of Serrin for an overdetermined problem in bounded domains. Finally we apply the results obtained to the problem of finding the best constants for the classical isoperimetric inequality and for some Sobolev embeddings.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35B50 Maximum principles in context of PDEs 35J70 Degenerate elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs