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A fractional Kirchhoff system with singular nonlinearities

Research paper by K. Saoudi

Indexed on: 10 Oct '18Published on: 03 Oct '18Published in: Analysis and Mathematical Physics



Abstract

Abstract In this work we consider the following fractional Kirchhoff equations with singular nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} M\Big ( \int _{\mathbb {R}^{2N}}\frac{ u(x)-u(y) ^{2}}{ x-y ^{N+2s}}dx dy\Big )(-\Delta )^s u = \lambda a(x) u ^{q-2}u +\frac{1-\alpha }{2-\alpha -\beta } c(x) u ^{-\alpha } v ^{1-\beta }, \quad \text {in }\Omega ,\\ M\Big ( \int _{\mathbb {R}^{2N}}\frac{ v(x)-v(y) ^{2}}{ x-y ^{N+2s}}dx dy\Big ) (-\Delta )^s v= \mu b(x) v ^{q-2}v +\frac{1-\beta }{2-\alpha -\beta } c(x) u ^{1-\alpha } v ^{-\beta }, \quad \text {in }\Omega ,\\ u=v = 0,\;\; \text{ in } \,\mathbb {R}^N\setminus \Omega , \end{array} \right. \end{aligned}$$ where \(\Omega \) is a bounded domain in \(\mathbb {R}^n\) with smooth boundary \(\partial \Omega \) , \(N> 2s\) , \(s \in (0,1)\) , \(0<\alpha<1,\;0<\beta <1,\) \(0<\alpha +\beta<2\theta<q<2^*_s,\) \(2^*_s=\frac{2N}{N-2s}\) is the fractional Sobolev exponent, \(\lambda , \mu \) are two parameters, \(a,\, b, \,c \in C({\overline{\Omega }})\) are non-negative weight functions, M is a continuous function, given by \(M(t)=k+lt^{\theta -1}\) \(k>0,\,l,\,\theta \ge 1,\) and \((-\Delta )^s\) is the fractional Laplacien operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter \(\lambda \) and \(\mu \) .