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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 \(\nu\)).
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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