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Existence results for Schrödinger-Choquard-Kirchhoff equations involving the fractional \(p\)-Laplacian. (English) Zbl 1431.35233

Summary: The paper is concerned with existence of nonnegative solutions of a Schrödinger-Choquard-Kirchhoff-type fractional \(p\)-equation. As a consequence, the results can be applied to the special case \[(a+b \|u\|_s^{p(\theta-1)})[(-\Delta)^s_pu+V(x)|u|^{p-2}u]=\lambda f(x,u)+\Bigl(\int_{\mathbb{R}^N}\frac{|u|^{p_{\mu,s}^*}}{|x-y|^\mu}dy\Bigr)|u|^{p_{\mu,s}^*-2}u\quad\text{in } \mathbb{R}^N,\] where \[\|u\|_{s}=\Bigl(\iint_{\mathbb{R}^{2N}} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}},dx\,dy+\int_{\mathbb{R}^N} V(x)|u|^pdx\Bigr)^{\frac 1p},\] \(a,b\in\mathbb{R}^+_0\), with \(a+b>0\), \(\lambda>0\) is a parameter, \(s\in(0,1)\), \(N>ps\), \(\theta \in[1,N/(N-ps))\), \((-\Delta)^s_p\) is the fractional \(p\)-Laplacian, \(V: \mathbb{R}^N\to\mathbb{R}^+\) is a potential function, \(0<\mu<N\), \(p_{\mu,s}^*=(pN-p\mu/2)/(N-ps)\) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, and \(f:\mathbb{R}^N\times \mathbb{R}\to\mathbb{R}\) is a Carathéodory function. First, via the Mountain Pass theorem, existence of nonnegative solutions is obtained when \(f\) satisfies superlinear growth conditions and \(\lambda\) is large enough. Then, via the Ekeland variational principle, existence of nonnegative solutions is investigated when \(f\) is sublinear at infinity and \(\lambda\) is small enough. More intriguingly, the paper covers a novel feature of Kirchhoff problems, which is the fact that the parameter \(a\) can be zero. Hence the results of the paper are new even for the standard stationary Kirchhoff problems.

MSC:

35R11 Fractional partial differential equations
35A15 Variational methods applied to PDEs
47G20 Integro-differential operators
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