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\(N\)-Laplacian equations in \(\mathbb{R}^N\) with critical growth. (English) Zbl 0932.35076

Summary: We study the existence of nontrivial solutions to the following problem: \[ \begin{cases} u\in W^{1,N}(\mathbb{R}^N),\;u\geq 0\text{ and}\\ -\text{div}(|\nabla u|^{N- 2}\nabla u)+ a(x)|u|^{N- 2}u= f(x,u)\text{ in }\mathbb{R}^N\;(N\geq 2),\end{cases} \] where \(a\) is a continuous function which is coercive, i.e., \(a(x)\to\infty\) as \(|x|\to\infty\) and the nonlinearity \(f\) behaves like \(\exp(\alpha|x|^{N/(N- 1)})\) when \(|u|\to\infty\).

MSC:

35J60 Nonlinear elliptic equations
35A15 Variational methods applied to PDEs
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