Multiple solutions for a modified quasilinear Schrödinger elliptic equation with a nonsquare diffusion term

This paper is an attempt to establish the existence and multiplicity results of nontrivial solutions to singular systems with sign-changing weight, nonlinear singularities, and critical exponent. By using variational methods, the Nehari manifold, and under sufficient conditions on the parameter η which represent some physical meanings, we prove some existing results by researching the critical points as the minimizers of the energy functional associated with the proposed problem (2) on the constraint defined by the Nehari manifold, which are solutions of our system, under some sufficient conditions on the parameters α, β, μ, and η. To the best of our knowledge, this paper is one of the first contributions to the study of singular systems with sign-changing weight, nonlinear singularities, and critical exponent.

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Group invariant solutions for the planar Schrödinger-Poisson equations

Zhou

2023

era

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<abstract>This paper is concerned with the following planar Schrödinger-Poisson equations <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} -\Delta{u}+V(x)u+\left(\ln{|\cdot|}\ast |u|^p\right)|u|^{p-2}u = f(x,u),\; \; \; x\in\mathbb{R}^{2}, \end{equation*} $\end{document} </tex-math></disp-formula> where $ p\geq2 $ is a constant, and $ V(x) $ and $ f(x, u) $ are continuous, mirror symmetric or rotationally periodic functions. The nonlinear term $ f(x, u) $ satisfies a certain monotonicity condition and has critical exponential growth in the Trudinger-Moser sense. We adopted a version of mountain pass theorem by constructing a Cerami sequence, which in turn leads to a ground state solution. Our method has two new insights. First, we observed that the integral $ \int_{\mathbb{R}^2}\int_{\mathbb{R}^2}\ln{(|x-y|)}|u(x)|^{p}|u(y)|^pdxdy $ is always negative if $ u $ belongs to a suitable space. Second, we built a new Moser type function to ensure the boundedness of the Cerami sequence, which further guarantees its compactness. In particular, by replacing the monotonicity condition with the Ambrosetti–Rabinowitz condition, our approach works also for the subcritical growth case.</abstract>

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Multiple solutions for a modified quasilinear Schrödinger elliptic equation with a nonsquare diffusion term

Abstract: In this paper, we establish the results of multiple solutions for a class of modified nonlinear Schrödinger equation involving the p-Laplacian. The main tools used for analysis are the critical points theorems by Ricceri and the dual approach.

Cited by 3 publications

References 26 publications

A necessary and sufficient condition for the existence of entire large solutions to a k-Hessian system

A necessary and sufficient condition for the existence of entire large solutions to a k-Hessian system

Multiple Solutions for Singular Systems with Sign-Changing Weight, Nonlinear Singularities and Critical Exponent

Group invariant solutions for the planar Schrödinger-Poisson equations

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