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Multiplicity of solutions for elliptic equations involving fractional operator and sign-changing nonlinearity

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Abstract

In this work, we study the existence and the multiplicity of non-negative solutions for the following problem

$$\begin{aligned} ({\mathrm{P}}_\uplambda ) \left\{ \begin{array}{ll} \mathcal {L} u = a(x) u^{q}+ \lambda b(x) u^p\quad \text {in }\Omega , \\ \\ u= 0 ,\;\; \text{ in } \,\mathbb {R}^n\setminus \Omega , \end{array} \right. \end{aligned}$$

where \(\Omega \subset \mathbb {R}^n \;(n\ge 2)\) , is a bounded smooth domain, \(\lambda , p, q\) are positive real numbers, \(s\in (0,1) \), \(a,\, b\) are continuous functions, and \( \mathcal {L}\) is a nonlocal operator defined later by (1.1). We establish the existence and we give a multiplicity of solutions by constrained minimization of the Euler-Lagrange functional corresponding to the problem \((P_\lambda )\), on suitable subsets of Nehari manifold and using the fibering maps. Precisely, we show the existence of \(\lambda _0>0,\) such that for all \(\lambda \in (0,\lambda _0)\), problem \((P_\lambda )\) has at least two non-negative solutions.

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Acknowledgements

The authors would like to thank the anonymous referees for the valuable suggestions and comments which improved the quality of the presentation.

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Correspondence to A. Ghanmi.

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Saoudi, K., Ghanmi, A. & Horrigue, S. Multiplicity of solutions for elliptic equations involving fractional operator and sign-changing nonlinearity. J. Pseudo-Differ. Oper. Appl. 11, 1743–1756 (2020). https://doi.org/10.1007/s11868-020-00357-9

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