Electron. J. Differential Equations, Vol. 2017 (2017), No. 115, pp. 1-13.

Existence of solutions for degenerate Kirchhoff type problems with fractional p-Laplacian

Nemat Nyamoradi, Lahib Ibrahim Zaidan

Abstract:
In this article, by using the Fountain theorem and Mountain pass theorem in critical point theory without Palais-Smale (PS) condition, we show the existence and multiplicity of solutions to the degenerate Kirchhoff type problem with the fractional p-Laplacian
$$\displaylines{ \Big(a + b\int\int_{\mathbb{R}^{2N}} \frac{|u (x) - u (y)|^p}{|x - y|^{N + ps}} \, dx \, dy\Big) (- \Delta)_p^s u = f (x, u ) \quad \text{in } \Omega,\cr u = 0 \quad \text{in } \mathbb{R}^N \setminus \Omega, }$$
where $(- \Delta)_p^s$ is the fractional p-Laplace operator with $0 < s < 1 < p < \infty$, $\Omega$ is a smooth bounded domain of $\mathbb{R}^N$, N > 2 s, a, b > 0 are constants and $f: \Omega \times \mathbb{R} \to \mathbb{R}$ is a continuous function.

Submitted February 23, 2017. Published April 27, 2017.
Math Subject Classifications: 34B27, 35J60, 35B05.
Key Words: Kirchhoff nonlocal operators; fractional differential equations; fountain theorem; mountain pass theorem; critical point theory.

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Nemat Nyamoradi
Department of Mathematics
Faculty of Sciences, Razi University
67149 Kermanshah, Iran
email: nyamoradi@razi.ac.ir, neamat80@yahoo.com
Lahib Ibrahim Zaidan
Department of Mathematics, Faculty of Sciences, Razi University
67149 Kermanshah, Iran
email: lahibzaidan75@yahoo.com; lahibzaidan@uobabylon.edu.iq

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