In this paper, we study a diffusion model of Kirchhoff-type driven by a nonlocal integro-differential operator. As a particular case, we consider the following diffusion problem where [ u ] _s is the Gagliardo seminorm of u , is a bounded domain with Lipschitz boundary, is the fractional Laplacian with , is the initial function, and is continuous. Under some appropriate conditions, the local existence of nonnegative solutions is obtained by employing the Galerkin method. Then, by virtue of a differential inequality technique, we prove that the local nonnegative solutions blow-up in finite time with arbitrary negative initial energy and suitable initial values. Moreover, we give an estimate for the lower and upper bounds of the blow-up time. The main novelty is that our results cover the degenerate case, that is, the coefficient of could be zero at the origin.

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 ⁢ ( θ - 1 ) ) ⁢ [ ( - Δ ) p s ⁢ u + V ⁢ ( x ) ⁢ | u | p - 2 ⁢ u ] = λ ⁢ f ⁢ ( x , u ) + ( ∫ ℝ N | u | p μ , s * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | p μ , s * - 2 ⁢ u in ⁢ ℝ N , (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 = ( ∬ ℝ 2 ⁢ N | u ⁢ ( x ) - u ⁢ ( y ) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N V ⁢ ( x ) ⁢ | u | p ⁢ 𝑑 x ) 1 p , \|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 ∈ ℝ 0 + {a,b\in\mathbb{R}^{+}_{0}} , with a + b > 0 {a+b>0} , λ > 0 {\lambda>0} is a parameter, s ∈ ( 0 , 1 ) {s\in(0,1)} , N > p ⁢ s {N>ps} , θ ∈ [ 1 , N / ( N - p ⁢ s ) ) {\theta\in[1,N/(N-ps))} , ( - Δ ) p s {(-\Delta)^{s}_{p}} is the fractional p-Laplacian, V : ℝ N → ℝ + {V:\mathbb{R}^{N}\rightarrow\mathbb{R}^{+}} is a potential function, 0 < μ < N {0<\mu<N} , p μ , s * = ( p ⁢ N - p ⁢ μ / 2 ) / ( N - p ⁢ s ) {p_{\mu,s}^{*}=(pN-p\mu/2)/(N-ps)} is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality, and f : ℝ N × ℝ → ℝ {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.

The aim of this paper is to establish the multiplicity of weak solutions for a Kirchhoff-type problem driven by a fractional p -Laplacian operator with homogeneous Dirichlet boundary conditions: where is an open bounded subset of with Lipshcitz boundary , is the fractional p -Laplacian operator with 0 < s < 1 < p < N such that sp < N , M is a continuous function and f is a Carathéodory function satisfying the Ambrosetti–Rabinowitz-type condition. When f satisfies the suplinear growth condition, we obtain the existence of a sequence of nontrivial solutions by using the symmetric mountain pass theorem; when f satisfies the sublinear growth condition, we obtain infinitely many pairs of nontrivial solutions by applying the Krasnoselskii genus theory. Our results cover the degenerate case in the fractional setting: the Kirchhoff function M can be zero at zero.

The paper deals with the existence and multiplicity of solutions of the fractional Schrödinger-Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case $\begin{equation*} (a+b[u]_{s,A}^{2θ-2})(-Δ)_A^su+V(x)u=f(x,|u|)u\,\, \text{in $\mathbb{R}^N$},\end{equation*}$ where $s∈ (0,1)$, $N>2s$, $a∈ \mathbb{R}^+_0$, $b∈ \mathbb{R}^+_0$, $θ∈[1,N/(N-2s))$, $A:\mathbb{R}^N\to\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\to \mathbb{R}^+$ is an electric potential, $(-Δ )_A^s$ is the fractional magnetic operator. In the super-and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.