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Existence results for nonlinear problems in \(\mathbb R^N\) involving the \(p\)-Laplacian. (English) Zbl 1302.35186
Summary: We establish an existence theorem for a problem of the type \[ \begin{cases}\-\Delta_pu+\lambda(x)|u|^{p-2}u=f(x,u)+g(x,u)\text{ in }\mathbb R^N\\ \lim\limits_{|x|\to+\infty}u(x)=0\end{cases} \] where \(f,g:\mathbb R^N\times\mathbb R\rightarrow\mathbb R\) are Carth‘eodory functions, \(\lambda\in L^\infty(\Omega)\), with \(\mathrm{ess }\inf_\Omega\lambda>0\), and \(\Delta_p\) is the \(p\)-Laplacian operator with \(p>N\). This result extends to the case of \(\mathbb R^N\) a previous result [G. Anello and G. Cordaro, Arch. Math. 79, No. 4, 274–287 (2002; Zbl 1091.35025)] related to a Neumann problem in bounded domains.
MSC:
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J20 Variational methods for second-order elliptic equations
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