On the Schrödinger–Poisson system with a general indefinite nonlinearity

The number of positive solutions for Schrödinger–Poisson system under the effect of eigenvalue

“…Remark In fact, our approach can be applied to System (1) when $$$ \mu V\equiv 1 $$$ and $$$ 2<p<4 $$$, that is, we can fill the gap in previous studies 7,11,12,16 by using our approach. Let us point out that the solutions obtained in Theorem 1.2 are the mountain pass type solutions. This indicates that the geometrical structure of mountain pass can be described between the parameter $$$ \lambda $$$ and the non‐local term by $$$ {\lambda}_K $$$ in the case of $$$ {\int}_{\Omega}\kern3pt g{\phi}_1^p dx>0 $$$.…”

“…In recent years, there has been much attention to SP systems on the existence and multiplicity of positive and nodal solutions under variant assumptions on $$$ V,K $$$ and $$$ h $$$ via variational methods. We refer the reader to previous studies 2–24 . For autonomous SP systems, Ruiz 2 studied the existence and multiplicity of radial solutions for system () with $$$ V(x)\equiv 1,K(x)\equiv \sqrt{\lambda }>0 $$$ and $$$ h\left(x,u\right)={\left|u\right|}^{p-2}u.$…”

On a class of nonlocal nonlinear Schrödinger equations with potential well

“…In recent years, many authors have been studying such topics (existence of two positive solutions which one of the negative energy), for example, Chen [14], Huang et al [21,22] and Shen and Han [32], consider the following Schrödinger…”

“…Actually the authors also established N ( ) may not contain any non-zero critical point of I for + √ < p ≤ . Motivated by the above works [14,21,22,30,32,36], in the present article we mainly study the existence and multiplicity of positive solutions for Eq. (P µ,λ ) can not require conditions f changes sign in R and lim |x|→∞ f (x) = f∞ < .…”

Multiple positive solutions for the Schrödinger-Poisson equation with critical growth

“…Remark 1.2. Similar condition like (g 1 ) on the sublinear term g(u) has been employed in [10]. In Theorem 1.1, the sublinear term g(u) is more extensive than the specific nonlinearity u q−1 for q ∈ (1, 2) and the condition (g 2 ) is the classical Ambrosetti-Rabinowitz condition, which plays an important role in proving the boundedness of Cerami sequence.…”