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Note: Please see pdf for full abstract with equations. In this paper, we study the fractional Schrödinger–Poisson system with general nonlinearity as follows: (−∆)su + u + l(x)φu = ƒ(u) in R2, (−∆)tφ = l(x)u2 in R2, where 1/2 < t ≤ s < 1, the potential l ∈ C(R2, R+) and the nonlinearity ƒ ∈ C(R, R) does not require the classical (AR)-condition. When l(x) ≡ µ > 0 is a parameter, by establishing new estimates for the fractional Laplacian, we find two positive solutions, depending on the range of µ. As a result, a positive ground state solution with negative energy exists for the non-autonomous system without any symmetry on l(x). When l(x) is radially symmetric, we show that the symmetry breaking phenomenon can occur, and that a non-radial ground state solution with negative energy exists. Furthermore, under additional assumptions on l(x), three positive solutions are found. The intrinsic differences between the planar SP system and the planar fSP system are analyzed.

Note: Please see pdf for full abstract with equations. In this paper, we study the fractional Schrödinger–Poisson system with general nonlinearity as follows: (−∆)su + u + l(x)φu = ƒ(u) in R2, (−∆) tφ = l(x)u2 in R2, where 1/2 < t ≤ s < 1, the potential l ∈ C(R2, R+) and the nonlinearity ƒ ∈ C(R,R) does not require the classical (AR)-condition. When l(x) ≡ µ > 0 is a parameter, by establishing new estimates for the fractional Laplacian, we find two positive solutions, depending on the range of µ. As a result, a positive ground state solution with negative energy exists for the non-autonomous system without any symmetry on l(x). When l(x) is radially symmetric, we show that the symmetry breaking phenomenon can occur, and that a non-radial ground state solution with negative energy exists. Furthermore, under additional assumptions on l(x), three positive solutions are found. The intrinsic differences between the planar SP system and the planar fSP system are analyzed. 2020 Mathematics Subject Classification. Primary: 35B38; Secondary: 35J60

We study the following Kirchhoff type equation:−a+b∫RN|∇u|2dxΔu+u=k(x)|u|p−2u+m(x)|u|q−2u in RN,$$\begin{equation*}\begin{array}{ll} -\left(a+b\int\limits_{\mathbb{R}^{N}}|\nabla u|^{2}\mathrm{d}x\right)\Delta u+u =k(x)|u|^{p-2}u+m(x)|u|^{q-2}u~~\text{in}~~\mathbb{R}^{N}, \end{array} \end{equation*}$$whereN=3,a,b>0$ a,b \gt 0 $,1

In the paper, we study a class of semilinear fractional semilinear elliptic equations involving concave-convex nonlinearities: \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} (-\Delta)^{\alpha} u+V_{\lambda }\left( x\right) u = f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, \\ u\in H^{\alpha}(\mathbb{R}^{N}), & \end{array}\right. \end{equation*} $\end{document} where \begin{document}$ \alpha\in (0,1] $\end{document} , \begin{document}$ 1 2\alpha\right), $\end{document} the potential \begin{document}$ V_{\lambda }(x) = \lambda a(x)-b(x) $\end{document} and the parameter \begin{document}$ \lambda >0. $\end{document} Under some suitable assumptions on \begin{document}$ a,b $\end{document} and the weight functions \begin{document}$ f,g $\end{document} , we obtain the existence and multiplicity of non-trivial (positive) solutions for \begin{document}$ \lambda $\end{document} large enough. An interesting phenomenon is that we do not need the condition that weight functions \begin{document}$ f, g $\end{document} are integrable or bounded on whole space \begin{document}$ \mathbb{R}^{N}. $\end{document}

We study the non-existence, existence and multiplicity of positive solutions to the following nonlinear Kirchhoff equation:% \begin{equation*} \left\{ \begin{array}{l} -M\left( \int_{\mathbb{R}^{3}}\left\vert \nabla u\right\vert ^{2}dx\right) \Delta u+\mu V\left( x\right) u=Q(x)\left\vert u\right\vert ^{p-2}u+\lambda f\left( x\right) u\text{ in }\mathbb{R}^{N}, \\ u\in H^{1}\left( \mathbb{R}^{N}\right) ,% \end{array}% \right. \end{equation*}% where $N\geq 3,20\right) ,$ the potential $V$ is a nonnegative function in $\mathbb{R}% ^{N}$ and the weight function $Q\in L^{\infty }\left( \mathbb{R}^{N}\right) $ with changes sign in $\overline{\Omega }:=\left\{ V=0\right\} .$ We mainly prove the existence of at least two positive solutions in the cases that $% \left( i\right) $ $20$ sufficiently large, where $\lambda _{1}\left( f_{\Omega }\right) $ is the first eigenvalue of $-\Delta $ in $% H_{0}^{1}\left( \Omega \right) $ with weight function $f_{\Omega }:=f|_{% \overline{\Omega }},$ whose corresponding positive principal eigenfunction is denoted by $\phi _{1}.$ Furthermore, we also investigated the non-existence and existence of positive solutions if $a,\lambda $ belongs to different intervals., Comment: 34 pages

In general, the existence of nodal solution for Schrodinger–Poisson systems with the nonlinearity f ( x ) | u | p − 2 u ( 4 ≤ p 6 ) in R 3 can be established by using the nodal Nehari manifold method. However, for the case where 2 p 4 , such an approach is not applicable because Palais-Smale sequences restricted on the nodal Nehari manifold can be not bounded. In this paper, we introduce a novel constraint method to prove the existence of nodal solution to a class of non-autonomous Schrodinger–Poisson systems in the case where 2 p 4 . We conclude that such solution changes sign exactly once in R 3 and is bounded in H 1 ( R 3 ) × D 1 , 2 ( R 3 ) . Moreover, the existence of least energy nodal solution is obtained in the case where 1 + 73 3 p 4 , which remains unsolved in the existing literature.

In this paper, we study saturable nonlinear Schrodinger equations with nonzero intensity function which makes the nonlinearity become not superlinear near zero. Using the Nehari manifold and the Lusternik-Schnirelman category, we prove the existence of multiple positive solutions for saturable nonlinear Schrodinger equations with nonzero intensity function which satisfies suitable conditions. The ideas contained here might be useful to obtain multiple positive solutions of the other non-homogeneous nonlinear elliptic equations.

In this paper, we are concerned with the following two-component system of Schrodinger equations with Hartree nonlinearity: \begin{document}$ \begin{equation*} \begin{cases} -\varepsilon ^{2}\Delta u+V_{1}\left( x\right) u+\lambda _{1}\left(I_{\varepsilon }\ast |u|^{p+1}\right)|u|^{p-1}u\\ \qquad\quad\; = \left(\mu _{1}|u|^{2p}+\beta(x) |u|^{q-1}|v|^{q+1}\right)u, & \text{in }\mathbb{R}^{N}, \\ -\varepsilon ^{2}\Delta v+V_{2}\left( x\right) v+\lambda _{2}\left(I_{\varepsilon }\ast |v|^{p+1}\right)|v|^{p-1}v\\ \qquad\quad\; = \left(\mu _{2}|v|^{2p}+\beta(x) |v|^{q-1}|u|^{q+1}\right)v, & \text{in }\mathbb{R}^{N}\,, \\ u,v\in H^{1}(\mathbb{R}^{N}),\quad u,v>0, \end{cases} \end{equation*} $\end{document} where \begin{document}$ 0 is a small parameter, \begin{document}$ 0 , \begin{document}$ I_{\varepsilon}(x) = \frac{\Gamma((N-\alpha)/2)} {\Gamma(\alpha/2)\pi^{\frac{N}{2}}2^{\alpha}\varepsilon^{\alpha}}\frac{1}{|x|^{N-\alpha}}, \; x\in\mathbb{R}^{N}\setminus\{0\} $\end{document} , \begin{document}$ \alpha\in(0,N),\; N = 3,4,5 $\end{document} and \begin{document}$ \lambda _{l}\geq0,\; \mu _{l}>0,\; l = 1,2, $\end{document} are constants. Under some suitable assumptions on the potentials \begin{document}$ V_{l}(x),l = 1,2, $\end{document} and the coupled function \begin{document}$ \beta(x) $\end{document} , we prove the existence and multiplicity of positive solutions for the above system by using energy estimates, the Nehari manifold technique and the Lusternik-Schnirelmann theory. Furthermore, the existence and nonexistence of the least energy positive solutions are also explored.

In this paper, we consider the following Schrodinger-Poisson system with Hartree-type nonlinearity −Δu+u+λφu=(Iα∗|u|p)|u|p−2u,in R3,−Δφ=u2,in R3, where λ>0, 0

In this paper we investigate the existence, multiplicity and asymptotic behavior of positive solution for the nonlocal nonlinear Schrödinger equations. We exploiting the relationship between the Nehari manifold and eigenvalue problems to discuss how the Nehari manifold changes as parameters μ, λ changes and show how existence, multiplicity and asymptotic results for positive solutions of the equation are linked to properties of the manifold.

We study the following non-autonomous Schrodinger–Poisson systems: $$\begin{aligned} \left\{ \begin{array}{ll} -\triangle u+u+K(x)\phi u=a(x)|u|^{p-2}u &{} \quad \text {in}\ {\mathbb {R}}^{3}\\ -\triangle \phi =K(x)u^{2} &{}\quad \text {in}\ {\mathbb {R}}^{3} \end{array} \right. \end{aligned}$$ with $$2

We study the Schrodinger–Poisson system: $$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u+u+\lambda \phi u=a\left( x\right) \left| u\right| ^{p-2}u &{} \text { in }{{\mathbb {R}}}^{3}, \\ -\Delta \phi =u^{2} &{} \ \text {in }{{\mathbb {R}}}^{3}, \end{array} \right. \end{aligned}$$ where parameter $$\lambda >0$$ , $$20$$ and other suitable conditions, we explore the energy functional corresponding to the system which is bounded below on $$ H^{1}\left( {{\mathbb {R}}}^{3}\right) $$ and the existence and multiplicity of positive (ground state) solutions for $$\left[ \frac{A\left( p\right) }{p} a_{\infty }\right] ^{2/\left( p-2\right) }

The existence and multiplicity of solutions for a class of non-local elliptic boundary value problems with superlinear source functions are investigated in this paper. Using variational methods, we examine the changes arise in the solution behaviours as a result of the non-local effect. Comparisons are made of the results here with those of the elliptic boundary value problem in the absence of the non-local term under the same prescribed conditions to highlight this effect of non-locality on the solution behaviours. Our results here demonstrate that the complexity of the solution structures is significantly increased in the presence of the non-local effect with the possibility ranging from no permissible positive solution to three positive solutions and, contrary to those obtained in the absence of the non-local term, the solution profiles also vary depending on the superlinearity of the source functions., 35 pages, 6 figures

We are concerned with a class of nonlocal elliptic equations as follows: \begin{document}$ \left\{ \begin{align} & -M\left( \int_{{{\mathbb{R}}^{N}}}{|}\nabla u{{|}^{2}}dx \right)\Delta u+\lambda V\left( x \right)u=f(x,u)\ \ \ \ \text{in }{{\mathbb{R}}^{N}}, \\ & u\in {{H}^{1}}({{\mathbb{R}}^{N}}), \\ \end{align} \right. $\end{document} where \begin{document}$ N\geq 1, $\end{document} \begin{document}$ \lambda>0 $\end{document} is a parameter, \begin{document}$ M(t) = am(t)+b $\end{document} with \begin{document}$ a, b>0 $\end{document} and \begin{document}$ m\in C(\mathbb{R}^{+}, \mathbb{R}^{+}) $\end{document} , \begin{document}$ V\in C(\mathbb{R}^{N}, \mathbb{R}^{+}) $\end{document} and \begin{document}$ f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R}) $\end{document} satisfying \begin{document}$ \lim_{|u|\rightarrow \infty }f(x, u) /|u|^{k-1} = q(x) $\end{document} uniformly in \begin{document}$ x\in \mathbb{R}^{N} $\end{document} for any \begin{document}$ 2 ( \begin{document}$ 2^{\ast} = \infty $\end{document} for \begin{document}$ N = 1, 2 $\end{document} and \begin{document}$ 2^{\ast} = 2N/(N-2) $\end{document} for \begin{document}$ N\geq 3 $\end{document} ). Unlike most other papers on this problem, we are more interested in the effects of the functions \begin{document}$ m $\end{document} and \begin{document}$ q $\end{document} on the number and behavior of solutions. By using minimax method as well as Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity of positive solutions for the above problem.

In this work, we study the quantum Zakharov system which describes the nonlinear interaction between high-frequency quantum Langmuir waves and low-frequency quantum ion-acoustic waves. We show the existence and the stability of the standing waves of quantum Zakharov system for the spacial dimensions d = 1 , 2 , 3 .

We study the existence of positive solutions for the non-autonomous Schrodinger-Poisson system: \begin{document}$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + u + \lambda K\left( x \right)\phi u = a\left( x \right){{\left| u \right|}^{p - 2}}u}&{{\text{in }}{\mathbb{R}^3},} \\ { - \Delta \phi = K\left( x \right){u^2}}&{{\text{in }}{\mathbb{R}^3},} \end{array}} \right.$ \end{document} where \begin{document} $\lambda >0$\end{document} , \begin{document} $2 and both \begin{document} $K\left( x\right) $\end{document} and \begin{document} $a\left( x\right) $\end{document} are nonnegative functions in \begin{document} $\mathbb{R}^{3}$\end{document} , which satisfy the given conditions, but not require any symmetry property. Assuming that \begin{document} $% \lim_{\left\vert x\right\vert \rightarrow \infty }K\left( x\right) = K_{\infty }\geq 0$\end{document} and \begin{document} $\lim_{\left\vert x\right\vert \rightarrow \infty }a\left( x\right) = a_{\infty }>0$\end{document} , we explore the existence of positive solutions, depending on the parameters \begin{document} $\lambda$\end{document} and \begin{document} $p$\end{document} . More importantly, we establish the existence of ground state solutions in the case of \begin{document} $3.18 \approx \frac{{1 + \sqrt {73} }}{3} .

In this paper, we study a class of linearly coupled Schrodinger systems with steep potential wells, which arises from Bose–Einstein condensates. The existence of positive ground states is investigated by exploiting the relation between the Nehari manifold and fiberring maps. Some interesting phenomena are that we do not need the weight functions in the nonlinear terms to be integrable or bounded and we can relax the upper control condition of the coupling function. Moreover, the decay rate and concentration phenomenon of positive ground states are also studied.

We investigate a class of nonlinear biharmonic equations with p-Laplacian { Δ 2 u − β Δ p u + λ V ( x ) u = f ( x , u ) in R N , u ∈ H 2 ( R N ) , where N ≥ 1 , β ∈ R , λ > 0 is a parameter and Δ p u = div ( | ∇ u | p − 2 ∇ u ) with p ≥ 2 . Unlike most other papers on this problem, we replace Laplacian with p-Laplacian and allow β to be negative. Under some suitable assumptions on V ( x ) and f ( x , u ) , we obtain the existence and multiplicity of nontrivial solutions for λ large enough. The proof is based on variational methods as well as Gagliardo–Nirenberg inequality.

In this paper, we study the existence, multiplicity and concentration of positive solutions for the following indefinite semilinear elliptic equations involving concave-convex nonlinearities: \begin{eqnarray} \left\{\begin{array}{l} -\Delta u+V_{\lambda }\left( x\right) u=f\left( x\right) \left\vert u\right\vert ^{q-2}u+g\left( x\right) \left\vert u\right\vert ^{p-2}u & \text{in }\mathbb{R}^{N}, \\ u\geq 0, & \text{in }\mathbb{R}^{N}, \end{array}\right. \end{eqnarray} where$ 1 < q < 2 < p < 2^{\ast} (2^{\ast } = \frac{2N}{N-2}$ for $N \geq 3) $ the potential $V_{\lambda }(x)=\lambda V^{+}(x)-V^{-}(x)$ with $ V^{\pm }=\max \left\{ \pm V,0\right\} $ and the parameter $\lambda >0.$ We assume that the functions $f,g$ and $V$ satisfy suitable conditions with the potential $V$ and the weight function $g$ without the assumptions of infinite limits.

In this article, we study a Kirchhoff type problem with a positive parameter $\lambda$, $$\displaylines{ -K\Big( \int_{\Omega }|\nabla u|^{2}dx\Big) \Delta u=\lambda f(x,u) , \quad \text{in } \Omega , \cr u=0, \quad \text{on } \partial \Omega , }$$ where $K:[0,+\infty )\to \mathbb{R} $ is a continuous function and $f:\Omega \times \mathbb{R}\to \mathbb{R}$ is a $L^{1}$-Caratheodory function. Under suitable assumptions on K(t) and f(x,u), we obtain the existence of infinitely many solutions depending on the real parameter $\lambda$. Unlike most other papers, we do not require any symmetric condition on the nonlinear term $f(x,u)$. Our proof is based on variational methods.

We consider a class of indefinite Kirchhoff-type equations:where , , and . We require that f(x, u) is “local" sublinear at the origin and “local" linear at infinity on u, respectively. Using mountain pass theorem and Ekeland variational principle, we obtain the existence and multiplicity of nontrivial solutions. In particular, the existence criterion of three nontrivial solutions is established. Furthermore, the nonexistence of nontrivial solutions is explored as well.

We study the indefinite Kirchhoff-type problem where Ω is a smooth bounded domain in and . We require that f is sublinear at the origin and superlinear at infinity. Using the mountain pass theorem and Ekeland variational principle, we obtain the multiplicity of non-trivial non-negative solutions. We improve and extend some recent results in the literature.

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.

In this paper, we investigate the multiplicity of positive solutions for a class of Schrödinger–Poisson systems with concave and convex nonlinearities as follows: [Formula: see text] where [Formula: see text] are two parameters, [Formula: see text], [Formula: see text] is a potential well, [Formula: see text] and [Formula: see text]. Such problem cannot be studied by applying variational methods in a standard way, since the (PS) condition is still unsolved on [Formula: see text] due to [Formula: see text]. By developing a novel constraint approach, we prove that the above problem admits at least two positive solutions.