Qualitative analysis of solutions to a nonlocal Choquard–Kirchhoff diffusion equations in ℝN$$ {\mathbb{R}}^N $$

Abstract

This paper deals with a nonlocal Choquard-Kirchhoff diffusion equations involving the fractional p-Laplacian in R N , and the conditions on global existence, polynomial and exponential decays, and finite time blow-up of solutions are studied. The results improve the recent studies on this model.

Ground state solutions of Pohožaev type for Kirchhoff‐type problems with general convolution nonlinearity and variable potential

“…Remark Let $$$ \alpha \to 0 $$$; the results obtained in this paper also cover many results in the literature, for example, the ones in previous studies, 19,21,22,30,35,39,40 in which the nonlinearity is a special form of $$$ \left({I}_{\alpha}\ast F(u)\right)f(u) $$$. Besides, the nonlinearity $$$ F $$$ in this paper only needs to satisfy the Berestycki‐Lion type conditions (S1)–(S3), which seem more simpler.…”

“…We must point out that when $$$ f(u)={\left|u\right|}^{p-2}u $$$, the results obtained in Chen et al 42 also fill the gap with $$$ p\in \left(1+\alpha /3,2\right] $$$ and cover many results in the literature, such as the results in Li and Ye, 30 Tang and Chen, 35 and Guo 39 when $$$ \alpha \to 0 $$$. Besides, the results obtained in Chen et al 42 are more general than those in Chen and Liu 40 since the nonlinearity $$$ F $$$ is more general than that of Chen and Liu 40 .…”

“…Kirchhoff Equation () has received more and more attention from mathematical community after Lions 17 brought forward an abstract functional analysis framework for studying it. Via the variational method, there have been many important results on the existence and multiplicity of solutions for problem (), when the nonlinearity $$$ g $$$ is autonomous or nonautonomous and satisfies different kinds of conditions; see, for example, previous studies 18–36 and the references therein. A classical way to deal with () is to use the mountain‐pass theorem.…”

“…Li and Ye 30 first established the existence of positive ground state solutions when $$$ 2<p\le 4 $$$ and $$$ V(x)\equiv 1 $$$ by applying a minimizing argument on a new manifold $$$$ \tilde{M}=\left\{u\in {H}^1\left({\mathbb{R}}^3\right):\left\langle {\Phi}_0^{\prime }(u),u\right\rangle +{P}_0(u)=0\right\}, $$$$ where $$$ {\Phi}_0(u) $$$ and $$$ {P}_0(u) $$$ are the energy functional and the Pohožaev equality for problem (), respectively. The idea used in Li and Ye 30 comes from Ruiz, 38 in which a kind of nonlinear Schrödinger‐Poisson system was investigated. Subsequently, Guo 39 generalized the results of Li and Ye 30 to problem ().…”

“…The idea used in Li and Ye 30 comes from Ruiz, 38 in which a kind of nonlinear Schrödinger‐Poisson system was investigated. Subsequently, Guo 39 generalized the results of Li and Ye 30 to problem (). Later, Chen and Tang 19 and Tang and Chen 35 improved the above results to problem () under (V1), some standard growth assumptions on $$$ g $$$, and the following additional conditions: $$$ V\in {C}^1\left({\mathbb{R}}^3,\mathbb{R}\right) $$$, and there is $$$ {\theta}^{\prime}\in \left(0,1\right) $$$ such that $$$$ \nabla V(x)\cdotp x\le \frac{\theta^{\prime }a}{2{\left|x\right|}^2},\kern0.4em \mathrm{a}.\mathrm{e}.\kern0.4em x\in {\mathbb{R}}^3\backslash \left\{0\right\}; $$$$ $$$ g\in C\left(\mathbb{R},\mathbb{R}\right) $$$, and $$$ \frac{g(t)t+6G(t)}{t\mid t\mid } $$$ is nondecreasing on …”