# zbMATH — the first resource for mathematics

Qualitative properties of solutions for quasi-linear elliptic equations. (English) Zbl 1109.35349
Summary: For several classes of functions including the special case $$f(u)=u^{p-1}-u^m$$, $$m>p-1>0$$, we obtain Liouville type, boundedness and symmetry results for solutions of the non-linear $$p$$-Laplacian problem $$-\Delta_p u=f(u)$$ defined on the whole space $$\mathbb R^n$$. Suppose $$u \in C^2(\mathbb R^n)$$ is a solution. We have that either (1) if $$u$$ doesn’t change sign, then $$u$$ is a constant (hence, $$u\equiv 1$$ or $$u\equiv 0$$ or $$u\equiv -1$$); or (2) if $$u$$ changes sign, then $$u\in L^{\infty}(\mathbb R^n)$$, moreover $$u<1$$ on $$\mathbb R^n$$; or (3) if $$Du>0$$ on $$\mathbb R^n$$ and the level set $$u^{-1}(0)$$ lies on one side of a hyperplane and touches that hyperplane, i.e., there exists $$\nu \in S^{n-1}$$ and $$x_{0}\in u^{-1}(0)$$ such that $$\nu \cdot (x-x_0)\geq 0$$ for all $$x\in u^{-1}(0)$$, then $$u$$ depends on one variable only (in the direction of $$\nu$$).
##### MSC:
 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J25 Boundary value problems for second-order elliptic equations
Full Text: