AbstractThe 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}_{p}u+V(x)|u|^{p-2}u]=\lambda f(x,u)%
+\Bigg{(}\int_{\mathbb{R}^{N}}\frac{|u|^{p_{\mu,s}^{*}}}{|x-y|^{\mu}}\,dy%
\Biggr{)}|u|^{p_{\mu,s}^{*}-2}u\quad\text{in }\mathbb{R}^{N},where\|u\|_{s}=\Bigg{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}%
\,dx\,dy+\int_{\mathbb{R}^{N}}V(x)|u|^{p}\,dx\Biggr{)}^{\frac{1}{p}},{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}\rightarrow\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}\rightarrow\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 λ
is large enough. Then, via the Ekeland variational principle, existence of
nonnegative solutions is investigated when f is sublinear at infinity and λ 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.
Splitting Type Variational Problems with Linear Growth Conditions
