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Multiple solutions for semilinear \(\Delta_{\gamma}\)-differential equations in \(\mathbb{R}^N\) with sign-changing potential. (English) Zbl 1433.35120

Summary: In this paper, we study the existence of infinitely many nontrivial solutions of the semilinear \(\Delta_{\gamma}\) differential equations in \(\mathbb{R}^N\) \[ - \Delta_{\gamma} u+ b(x)u=f(x,u)\quad \text{ in } \mathbb{R}^N, \quad u \in S^2_{\gamma}(\mathbb{R}^N), \] where \(\Delta_{\gamma}\) is the subelliptic operator of the type \[ \Delta_\gamma: =\sum_{j=1}^N\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \quad \partial_{x_j}: =\frac{\partial }{\partial x_j},\quad \gamma = (\gamma_1, \gamma_2,\dots, \gamma_N), \] and the potential \(b\) is allowed to be sign-changing, and the primitive of the nonlinearity \(f\) is of superquadratic growth near infinity in \(u\) and allowed to be sign-changing.

MSC:

35J62 Quasilinear elliptic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A15 Variational methods applied to PDEs
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Full Text: Euclid