On the nonlocal Schrödinger‐poisson type system in the Heisenberg group
Abstract: This paper is concerned with a class of nonlocal Schrödinger‐Poisson type system with sublinear term. With the aid of the Ekeland's variational principle and the mountain pass theorem, the existence of negative energy solution, positive energy solution, and positive ground state solution is obtained, respectively. Moreover, we also obtain the multiplicity of solutions by using the Clark theorem. The novelty of this paper is that the equation has not only nonlocal term but also nonlocal coefficient.
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Cited by 3 publications
References 37 publications
This article is devoted to the study of the combined effects of logarithmic and critical nonlinearities for the Kirchhoff-Poisson system − M ∫ Ω ∣ ∇ H u ∣ 2 d ξ Δ H u + μ ϕ u = λ ∣ u ∣ q − 2 u ln ∣ u ∣ 2 + ∣ u ∣ 2 u in Ω , − Δ H ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω , \left\{\begin{array}{ll}-M\left(\mathop{\displaystyle \int }\limits_{\Omega }| {\nabla }_{H}u{| }^{2}{\rm{d}}\xi \right){\Delta }_{H}u+\mu \phi u=\lambda | u{| }^{q-2}u\mathrm{ln}| u{| }^{2}+| u{| }^{2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}\phi ={u}^{2}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\phi =0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ H {\Delta }_{H} is the Kohn-Laplacian operator in the first Heisenberg group H 1 {{\mathbb{H}}}^{1} , Ω \Omega is a smooth bounded domain of H 1 {{\mathbb{H}}}^{1} , q ∈ ( 2 θ , 4 ) q\in \left(2\theta ,4) , μ ∈ R \mu \in {\mathbb{R}} , and λ > 0 \lambda \gt 0 are some real parameters. Under suitable assumptions on the Kirchhoff function M M , which cover the degenerate case, we prove the existence of nontrivial solutions for the above problem when λ > 0 \lambda \gt 0 is sufficiently large. Moreover, our results are new even in the Euclidean case.
This article is devoted to the study of the combined effects of logarithmic and critical nonlinearities for the Kirchhoff-Poisson system − M ∫ Ω ∣ ∇ H u ∣ 2 d ξ Δ H u + μ ϕ u = λ ∣ u ∣ q − 2 u ln ∣ u ∣ 2 + ∣ u ∣ 2 u in Ω , − Δ H ϕ = u 2 in Ω , u = ϕ = 0 on ∂ Ω , \left\{\begin{array}{ll}-M\left(\mathop{\displaystyle \int }\limits_{\Omega }| {\nabla }_{H}u{| }^{2}{\rm{d}}\xi \right){\Delta }_{H}u+\mu \phi u=\lambda | u{| }^{q-2}u\mathrm{ln}| u{| }^{2}+| u{| }^{2}u& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ -{\Delta }_{H}\phi ={u}^{2}& \hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}\Omega ,\\ u=\phi =0& \hspace{0.1em}\text{on}\hspace{0.1em}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ H {\Delta }_{H} is the Kohn-Laplacian operator in the first Heisenberg group H 1 {{\mathbb{H}}}^{1} , Ω \Omega is a smooth bounded domain of H 1 {{\mathbb{H}}}^{1} , q ∈ ( 2 θ , 4 ) q\in \left(2\theta ,4) , μ ∈ R \mu \in {\mathbb{R}} , and λ > 0 \lambda \gt 0 are some real parameters. Under suitable assumptions on the Kirchhoff function M M , which cover the degenerate case, we prove the existence of nontrivial solutions for the above problem when λ > 0 \lambda \gt 0 is sufficiently large. Moreover, our results are new even in the Euclidean case.
Positive solution for a nonlocal problem with strong singular nonlinearity
In this article, we consider a nonlocal problem with a strong singular term and a general weight function. By using Ekeland’s variational principle, we prove a necessary and sufficient condition for the existence of a positive solution. Moreover, a method of algebraic analysis is used to deal with the multiplicity of solutions. Compared with the existing literature, our problems and results are novel.
This paper establishes the existence of unique and multiple solutions to two nonlocal equations with fractional operators. The main results are obtained using the variational method and algebraic analysis. The conclusion is that there exists a constant λ*>0 such that the equations have only three, two, and one solution, respectively, for λ∈(0,λ*), λ=λ*, and λ>λ*. The main conclusions fill the gap in the knowledge of this kind of fractional-order problem.
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