This paper is concerned with the following Schrödinger-Kirchhoff-Poisson system:where a ≥ 0, b > 0 and η, λ > 0, ⊂ R 3 is a bounded smooth domain. With the help of the variational methods, the existence of a non-trivial solution is obtained. MSC: 35B38; 35G99
In this paper, we study the following fractional Schrödinger equations:> 0, and are real parameter. 2 * is the critical Sobolev exponent. We prove a fractional Sobolev-Hardy inequality and use it together with concentration compact theory to get a ground state solution. Moreover, concentration behaviors of nontrivial solutions are obtained when the coefficient of the potential function tends to infinity.
In this study, we consider the following quasilinear Choquard equation with singularity − Δ u + V ( x ) u − u Δ u 2 + λ ( I α ∗ ∣ u ∣ p ) ∣ u ∣ p − 2 u = K ( x ) u − γ , x ∈ R N , u > 0 , x ∈ R N , \left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right. where I α {I}_{\alpha } is a Riesz potential, 0 < α < N 0\lt \alpha \lt N , and N + α N < p < N + α N − 2 \displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2} , with λ > 0 \lambda \gt 0 . Under suitable assumption on V V and K K , we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as λ → 0 \lambda \to 0 .
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