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On a critical exponential \(p \& N\) equation type: existence and concentration of changing solutions. (English) Zbl 1485.35212

Summary: In this paper we study a class of quasilinear equation with exponential critical growth. More precisely, we show existence of a family of nodal solutions, i.e, sign-changing solutions for the problem \[ \begin{cases} -\operatorname{div} \big(a (\epsilon^p |\nabla u|^p) \, \epsilon^p |\nabla u|^{p-2} \nabla u \big) \, +\, V(z) b(|u|^p) |u|^{p-2} u = f(u) \; \quad \text{in } \mathbb{R}^N, \\ u \in W^{1,p}(\mathbb{R}^N) \cap W^{1,N}(\mathbb{R}^N). \end{cases} \tag{\(P_\epsilon\)} \] Such nodal solutions concentrate on the minimum points set of the potential \(V\), changes sign exactly once in \({\mathbb{R}}^N\) and have exponential decay at infinity. Here we use variational methods and M. A. del Pino and P. L. Felmer’s technique [Calc. Var. Partial Differ. Equ. 4, No. 2, 121–137 (1996; Zbl 0844.35032)] in order to overcome the lack of compactness.

MSC:

35J62 Quasilinear elliptic equations
35B33 Critical exponents in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35J20 Variational methods for second-order elliptic equations

Citations:

Zbl 0844.35032
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Full Text: DOI

References:

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