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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
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