Remarks on least energy solutions for quasilinear elliptic problems in \(\mathbb R^N\). (English) Zbl 1109.35318

The authors consider the quasilinear elliptic problem \[ -\Delta_p w= g(w),\quad\text{ in }\mathbb R^N,\tag{1} \] where \(\Delta_p u= \text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian operator and \(1< p\leq N\). Using variational methods more precisely by a constrained minimization argument, they show the existence of ground states solutions (or least energy solutions) for the (1) in both cases, \(1< p< N\) and \(p= N\). They prove also that the mountain-pass value gives the least energy level and obtain the exponential decay of the derivatives of the solutions of (1).


35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
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