A nonhomogeneous fractional \(p\)-Kirchhoff type problem involving critical exponent in \(\mathbb{R}^N\). (English) Zbl 1372.35348
Summary: This paper concerns itself with the nonexistence and multiplicity of solutions for the following fractional Kirchhoff-type problem involving the critical Sobolev exponent:
\[
\left[a+b\left(\iint_{\mathbb{R}^{2N}}\frac{| u(x)-u(y)|^p}{| x-y|^{N+ps}}dxdy\right)^{\theta-1}\right](-\Delta)^s_p u=| u |^{p^\ast_s-2}u+\lambda f(x)\quad \text{in }\mathbb{R}^N,
\]
where \(a\geq0\), \(b>0\), \(\theta>1\), \((-\Delta)_p^s\) is the fractional \(p\)-Laplacian with \(0<s<1\) and \(1<p<N/s\), \(p_s^\ast=Np/(N-ps)\) is the critical Sobolev exponent, \(\lambda\geq0\) is a parameter, and \(f\in L^{p_s^\ast/(p_s^\ast-1)}(\mathbb{R}^N)\setminus\{0\}\) is a nonnegative function. When \(\lambda=0\), we show that the multiplicity and nonexistence of solutions for the above problem are related with \(N\), \(\theta\), \(s\), \(p\), \(a\), and \(b\). When \(\lambda>0\), by using Ekeland’s variational principle and the mountain pass theorem, we show that there exists \(\lambda^{**}>0\) such that the above problem admits at least two nonnegative solutions for all \(\lambda\in (0,\lambda^{**})\). In the latter case, in order to overcome the loss of compactness, we derive a fractional version of the principle of concentration compactness in the setting of the fractional \(p\)-Laplacian.
MSC:
| 35R11 | Fractional partial differential equations |
| 35A15 | Variational methods applied to PDEs |
| 35B33 | Critical exponents in context of PDEs |
| 47G20 | Integro-differential operators |
Keywords:
fractional \(p\)-Kirchhoff problem; critical Sobolev exponent; principle of concentration compactnessReferences:
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