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Multiplicity of symmetrically distinct sequences of solutions for a quasilinear problem in \({\mathbb{R}}^N\). (English) Zbl 1143.35045

Summary: The present paper is concerned with an elliptic problem in \({\mathbb{R}}^N\) which involves the \(p\)-Laplacian, \(p > N\), (\(N = 4\) or \(N \geq 6\)), while the nonlinear term has an oscillatory behaviour and is odd near an arbitrarily small neighborhood of the origin. A direct variational argument and a careful group-theoretical construction show the existence of at least \(\left[\frac{N-3}{2}\right] + (-1)^N\) sequences of arbitrary small, non-radial, sign-changing solutions such that elements in different sequences are distinguished by their symmetry properties.

MSC:

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
35J70 Degenerate elliptic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
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