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Maximal saddle solution of a nonlinear elliptic equation involving the \(p\)-Laplacian. (English) Zbl 1291.35071

Summary: A saddle solution is called maximal saddle solution if its absolute value is not smaller than those absolute values of any solutions that vanish on the Simons cone \(\mathcal{C}=\{s=t\}\) and have the same sign as \(s - t\). We prove the existence of a maximal saddle solution of the nonlinear elliptic equation involving the \(p\)-Laplacian, by using the method of monotone iteration, \[ \Delta_{p}u=f(u) \quad \text{in} \quad\mathbb R^{2m}, \] where \(2m \geq p > 2\).

MSC:

35J60 Nonlinear elliptic equations
35J70 Degenerate elliptic equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
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