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Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate

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Abstract

In this paper we consider a p-Laplacian equation with strong Allee effect growth rate and Dirichlet boundary condition

$$\left\{\begin{array}{ll} {\rm div} (|\nabla u|^{p-2} \nabla u) + \lambda f(x,u)=0, &\quad x \in \Omega, \\ u=0, &\quad x \in \partial \Omega, \qquad \qquad ^ {(P_\lambda)} \end{array}\right.$$

where Ω is a bounded smooth domain in \({\mathbb{R}^N}\) for \({N \ge 1, p > 1}\), and λ is a positive parameter. By using variational methods and a suitable truncation technique, we prove that problem (P λ) has at least two positive solutions for large parameter and it has no positive solutions for small parameter. In addition, a nonexistence result is investigated.

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Authors and Affiliations

  1. Department of Mathematics, College of William and Mary, Williamsburg, VI, 23187-8795, USA

    Chan-Gyun Kim & Junping Shi

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  1. Chan-Gyun Kim
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  2. Junping Shi
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Correspondence to Chan-Gyun Kim.

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Partially supported by National Research Foundation of Korea Grant funded by the Korean Government (Ministry of Education, Science and Technology, NRF-2011-357-C00006).

Partially supported by NSF grant DMS-1022648.

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Kim, CG., Shi, J. Existence and Multiplicity of Positive Solutions to a Quasilinear Elliptic Equation with Strong Allee Effect Growth Rate. Results. Math. 64, 165–173 (2013). https://doi.org/10.1007/s00025-013-0306-x

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