Schrodinger Poisson System Research Articles

Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations

We study the existence and multiplicity of solutions for the Schrodinger-Bopp-Podolsky system $$\displaylines{ -\Delta u + \phi u = \omega u \quad\text{ in } \Omega \cr a^2\Delta^2\phi-\Delta \phi = u^2 \quad\text{ in } \Omega \cr u=\phi=\Delta\phi=0\quad\text{ on } \partial\Omega \cr \int_{\Omega} u^2\,dx =1 }$$ where \(\Omega\) is an open bounded and smooth domain in \(\mathbb R^{3}\), \(a>0 \) is the Bopp-Podolsky parameter. The unknowns are \(u,\phi:\Omega\to \mathbb R\) and \(\omega\in\mathbb R\). By using variational methods we show that for any \(a>0\) there are infinitely many solutions with diverging energy and divergent in norm. We show that ground states solutions converge to a ground state solution of the related classical Schrodinger-Poisson system, as \(a\to 0\). For more information see https://ejde.math.txstate.edu/Volumes/2023/66/abstr.html

Local well-posedness and standing waves with prescribed mass for Schrodinger-Poisson systems with a logarithmic potential in R^2

In this article, we consider planar Schrodinger-Poisson systems with a logarithmic external potential \(W(x)=\ln (1+|x|^2)\) and a general nonlinear term \(f\). We obtain conditions for the local well-posedness of the Cauchy problem in the energy space. By introducing some suitable assumptions on \(f\), we prove the existence of the global minimizer. In addition, with the help of the local well-posedness, we show that the set of ground state standing waves is orbitally stable.
 For more information see https://ejde.math.txstate.edu/Volumes/2023/64/abstr.html

Multiple solutions for nonhomogeneous Schrodinger-Poisson system with p-Laplacian

This article concerns the existence of solutions to the Schrodinger-Poisson system $$\displaylines{ -\Delta_p u+|u|^{p-2}u+\lambda\phi u=|u|^{q-2}u+h(x) \quad \hbox{in }\mathbb{R}^3,\\ -\Delta \phi=u^2 \quad \hbox{in }\mathbb{R}^3, }$$ where \( 4/3 < p < 12/5 \), \( p < q < p^{*}=3p/(3-p) \), \(\Delta_p u =\hbox{div}(|\nabla u|^{p-2}\nabla u)\), \(\lambda >0\), and \(h \not= 0\). The multiplicity results are obtained by using Ekeland's variational principle and the mountain pass theorem.

Multiplicity of high energy solutions for fractional Schrodinger-Poisson systems with critical frequency

In this article we study the fractional Schrodinger-Poisson system $$\displaylines{ \epsilon^{2s}(-\Delta)^s u+V(x)u=\phi |u|^{2^*_s-3}u,\quad x\in \mathbb{R}^3, \cr (-\Delta)^s\phi=|u|^{2^*_s-1}, \quad x\in \mathbb{R}^3, }$$ where \(s\in(1/2,1)\), \(\epsilon>0\) is a parameter, \(2^*_s=6/(3-2s)\) is the critical Sobolev exponent, \(V\in L^{\frac{3} {2s}}(\mathbb{R}^3)\) is a nonnegative function which may be zero in some region of \(\mathbb{R}^3\). By means of variational methods, we present the number of high energy bound states with the topology of the zero set of V for small \(\epsilon\).

Logarithmic estimates for mean-field models in dimension two and the Schrödinger–Poisson system

In dimension two, we investigate a free energy and the ground state energy of the Schrodinger-Poisson system coupled with a logarithmic nonlinearity in terms of underlying functional inequalities which take into account the scaling invariances of the problem. Such a system can be considered as a nonlinear Schrodinger equation with a cubic but nonlocal Poisson nonlinearity, and a local logarithmic nonlinearity. Both cases of repulsive and attractive forces are considered. We also assume that there is an external potential with minimal growth at infinity, which turns out to have a logarithmic growth. Our estimates rely on new logarithmic interpolation inequalities which combine logarithmic Hardy-Littlewood-Sobolev and logarithmic Sobolev inequalities. The two-dimensional model appears as a limit case of more classical problems in higher dimensions.

Ground state solutions for a class of fractional Schrodinger-Poisson system with critical growth and vanishing potentials

AbstractIn this paper, we study the fractional Schrödinger-Poisson system(−Δ)su+V(x)u+K(x)ϕ|u|q−2u=h(x)f(u)+|u|2s∗−2u,in R3,(−Δ)tϕ=K(x)|u|q,in R3,$$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} (-{\it\Delta})^{s}u+V(x)u+ K(x) \phi|u|^{q-2}u=h(x)f(u)+|u|^{2^{\ast}_{s}-2}u,&\mbox{in}~ {\mathbb R^{3}},\\ (-{\it\Delta})^{t}\phi=K(x)|u|^{q},&\mbox{in}~ {\mathbb R^{3}}, \end{array}\right. \end{array}$$wheres,t∈ (0, 1), 3 < 4s< 3 + 2t,q∈ (1,2s∗$\begin{array}{} \displaystyle 2^*_s \end{array}$/2) are real numbers, (−Δ)sstands for the fractional Laplacian operator,2s∗:=63−2s$\begin{array}{} \displaystyle 2^{*}_{s}:=\frac{6}{3-2s} \end{array}$is the fractional critical Sobolev exponent,K,Vandhare non-negative potentials andV,hmay be vanish at infinity.fis aC1-function satisfying suitable growth assumptions. We show that the above fractional Schrödinger-Poisson system has a positive and a sign-changing least energy solution via variational methods.

Existence and concentration results for fractional Schrodinger-Poisson system via penalization method

This article concerns the positive solutions for the fractional Schrodinger-Poisson system $$\displaylines{ \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u)\quad \text{in }\mathbb{R}^3,\cr \varepsilon^{2t}(-\Delta)^t\phi=u^2\quad \text{in }\mathbb{R}^3, }$$ where \(\varepsilon>0\) is a small parameter, \((-\Delta)^\alpha\) denotes the fractional Laplacian of orders \(\alpha=s,t\in(3/4,1)\), \(V\in C(\mathbb{R}^3,\mathbb{R})\) is the potential function and \(f:\mathbb{R}\to\mathbb{R}\) is continuous and subcritical. Under a local condition imposed on the potential function, we relate the number of positive solutions with the topology of the set where the potential attains its minimum values. Moreover, we considered some properties of these positive solutions, such as concentration behavior and decay estimate. In the proofs we apply variational methods, penalization techniques and Ljusternik-Schnirelmann theory.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/14/abstr.html

Ground states for asymptotically linear fractional Schrödinger–Poisson systems

In this paper we consider the following fractional Schrodinger–Poisson system $$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^s u+u+K(x)\phi (x)u=g(x, u), \quad x\in \mathbb {R}^{3}, \\ (-\Delta )^s \phi =K(x)u^2, \quad x\in \mathbb {R}^{3}, \end{array}\right. } \end{aligned}$$ where $$s\in (\frac{1}{2},1)$$ and g(x, u) is asymptotically linear at infinity. Under certain assumptions on K(x) and g(x, u), we prove the existence of ground state solutions by variational methods.

Ground state solutions for the fractional Schrödinger-Poisson system with critical growth

Abstract We are interested in the existence of a ground state solution for the fractional Schrodinger-Poisson system with critical growth { ( − Δ ) s u + ϕ u = g ( u ) + | u | 2 s * − 2 u , in R 3 , ( − Δ ) t ϕ = u 2 , in R 3 , where 2 s * = 6 3 − 2 s , s , t ∈ ( 0 , 1 ) and 2 s + 2 t > 3 . Under some mild assumptions on g ( u ) , using the compactness lemma of Strauss, a positive ground state solution is obtained by approximating the Pohozaev-Nehari type ground state solutions of the subcritical problems.

Ground state and multiple solutions for critical fractional Schrodinger-Poisson equations with perturbation terms

In this article, we study a class of critical fractional Schrodinger-Poisson system with two perturbation terms. By using variational methods and Lusternik-Schnirelman category theory, the existence of ground state and two nontrivial solutions are established.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/07/abstr.html

Multiple positive solutions for nonhomogeneous Schrodinger-Poisson systems with Berestycki-Lions type conditions

In this article, we consider the multiplicity of solutions for nonhomogeneous Schrodinger-Poisson systems under the Berestycki-Lions type conditions. With the aid of Ekeland's variational principle, the mountain pass theorem and a Pohozaev type identity, we prove that the system has at least two positive solutions.
 For more information see https://ejde.math.txstate.edu/Volumes/2021/01/abstr.html

The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function

In this paper, we study the multiplicity of two spikes nodal solutions for a nonautonomous Schrodinger–Poisson system with the nonlinearity \begin{document}$ f(x)\vert u\vert ^{p-2}u(2 in \begin{document}$ \mathbb{R}^{3} $\end{document} . By assuming that the weight function \begin{document}$ f\in C(\mathbb{R}^{3},\mathbb{R}^{+}) $\end{document} has \begin{document}$ m $\end{document} maximum points in \begin{document}$ \mathbb{R}^{3} $\end{document} , we conclude that such system admits \begin{document}$ m^{2} $\end{document} distinct nodal solutions, each of which has exactly two nodal domains. The proof is based on a natural constraint approach developed by us as well as the generalized barycenter map.

Schrodinger-Poisson systems with singular potential and critical exponent

In this article we study the Schrodinger-Poisson system $$\displaylines{ -\Delta u +V(|x|)u+\lambda\phi u = f(u), \quad x\in\mathbb{R}^3, \cr -\Delta \phi =u^2, \quad x\in\mathbb{R}^3, }$$ where V is a singular potential with the parameter \(\alpha\) and the nonlinearity f satisfies critical growth. By applying a generalized version of Lions-type theorem and the Nehari manifold theory, we establish the existence of the nonnegative ground state solution when \(\lambda=0\). By the perturbation method, we obtain a nontrivial solution to above system when \(\lambda\neq 0\).
 For more information see https://ejde.math.txstate.edu/Volumes/2020/130/abstr.html

Ground-state solutions to a Kirchhoff-type transmission problem

In this paper, we consider the existence of ground-state solutions to nonlinear Kirchhoff-type transmission problems by using the methods from (Silvia Cingolani and Tobias Weth, On the planar Schrodinger-Poisson system, Ann. Inst. H. Poincare Anal. Non Lineaire 33 (2016), no. 1, 169-197). Here, we avoid the conditions under which the nonlinear terms of $f$ and $g$ are forms of $C^1$. In particular, when $N=2$, the existence of ground-state solutions is established to the Kirchhoff-type transmission problem with exponent-type nonlinearity.

Existence and asymptotic behaviour of solutions for a quasi-linear Schrödinger–Poisson system with a critical nonlinearity

In this paper we consider the following quasilinear Schrodinger–Poisson system $$\begin{aligned} \left\{ \begin{array}[c]{ll} - \Delta u +u+\phi u = \lambda f(x,u)+|u|^{4}u &{}\ \text{ in } {\mathbb {R}}^{3} \\ -\Delta \phi -\varepsilon ^{4} \Delta _4 \phi = u^{2} &{} \ \text{ in } {\mathbb {R}}^{3}, \end{array} \right. \end{aligned}$$ depending on the two parameters $$\lambda ,\varepsilon >0$$ . We first prove that, for $$\lambda $$ larger than a certain $$\lambda ^{*}>0$$ , there exists a solution for every $$\varepsilon >0$$ . Later, we study the asymptotic behaviour of these solutions whenever $$\varepsilon $$ tends to zero, and we prove that they converge to the solution of the Schrodinger–Poisson system associated.

Ground state solutions for a Schrödinger-Poisson system with unconventional potential

We consider the Schrodinger-Poisson system with nonlinear term Q(x)|u|p−1u, where the value of $$\mathop {\lim}\limits_{\left| x \right| \to \infty} \,\,Q\left(x \right)$$Q(x) may not exist and Q may change sign. This means that the problem may have no limit problem. The existence of nonnegative ground state solutions is established. Our method relies upon the variational method and some analysis tricks.

Multiple Positive Solutions for the Fractional Schrödinger–Poisson Systems Involving Singular Terms

In this paper, we study the existence of multiple positive solutions for a class of fractional Schrodinger–Poisson systems involving singular terms and critical nonlinearities. Using the Nehari manifold method, we obtain a multiplicity result for them.

Positive solutions of Schrodinger-Poisson systems with Hardy potential and indefinite nonlinearity

In this article, we study the nonlinear Schrodinger-Poisson system $$\displaylines{ -\Delta u+u-\mu\frac{u}{|x|^2}+l(x) \phi u=k(x)|u|^{p-2}u \quad x\in\mathbb{R}^3, \cr -\Delta\phi=l(x)u^2 \quad x\in\mathbb{R}^3, }$$ where \(k\in C(\mathbb{R}^3)\) and 4<p<6, k changes sign in \(\mathbb{R}^3\) and \(\limsup_{|x|\to\infty}k(x)=k_{\infty}<0\). We prove that Schrodinger-Poisson systems with Hardy potential and indefinite nonlinearity have at least one positive solution, using variational methods. For more information see https://ejde.math.txstate.edu/Volumes/2020/47/abstr.html

Multiple positive solutions for nonhomogeneous Schrödinger–Poisson system in ℝ3*

In this paper, we study the multiplicity of solutions for a class of nonhomogeneous Schrodinger–Poisson systems under general superlinear conditions at infinity. With the aid of Ekeland’s variational principle, Jeanjean’s monotone method, Pohožaev’s identity, and the mountain pass theorem, we prove that a Schrodinger–Poisson system has at least two positive solutions, which generalizes and improves some recent results in the literature.

Fractional Schrodinger-Poisson systems with weighted Hardy potential and critical exponent

In this article we consider the fractional Schrodinger-Poisson system $$\displaylines{ (-\Delta)^{s} u - \mu \frac{\Phi(x/|x|)}{|x|^{2s}} u +\lambda \phi u = |u|^{2^*_s-2}u,\quad \text{in } \mathbb{R}^3,\cr (-\Delta)^t \phi = u^2, \quad \text{in } \mathbb{R}^3, }$$ where \(s\in(0,3/4)\), \(t\in(0,1)\), \(2t+4s=3\), \(\lambda>0\) and \(2^*_s=6/(3-2s)\) is the Sobolev critical exponent. By using perturbation method, we establish the existence of a solution for \(\lambda\) small enough. For more information see https://ejde.math.txstate.edu/Volumes/2020/01/abstr.html