Positive Ground State Solution Research Articles

Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations

Abstract This article is directed to the study of the following Schrödinger-Poisson-Slater type equation: − ε 2 Δ u + V ( x ) u + ε − α ( I α ∗ ∣ u ∣ 2 ) u = λ ∣ u ∣ p − 1 u in R N , -{\varepsilon }^{2}\Delta u+V\left(x)u+{\varepsilon }^{-\alpha }\left({I}_{\alpha }\ast | u{| }^{2})u=\lambda | u{| }^{p-1}u\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where ε , λ > 0 \varepsilon ,\lambda \gt 0 are parameters, N ⩾ 2 N\geqslant 2 , ( α + 6 ) / ( α + 2 ) < p < 2 ∗ − 1 \left(\alpha +6)\hspace{0.1em}\text{/}\hspace{0.1em}\left(\alpha +2)\lt p\lt {2}^{\ast }-1 , I α {I}_{\alpha } is the Riesz potential with 0 < α < N 0\lt \alpha \lt N , and V ∈ C ( R N , R ) V\in {\mathcal{C}}\left({{\mathbb{R}}}^{N},{\mathbb{R}}) . By using variational methods, we prove that there is a positive ground state solution for the aforementioned equation concentrating at a global minimum of V V in the semi-classical limit, and then we found that this solution satisfies the property of exponential decay. Finally, the multiplicity and concentration behavior of positive solutions for the aforementioned problem is investigated by the Ljusternik-Schnirelmann theory. Our article improves and extends some existing results in several directions.

Homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign nonlinearities

In this paper, we obtain the multiplicity of homoclinic solutions for a class of asymptotically autonomous Hamiltonian systems with indefinite sign potentials. The concentration-compactness principle is applied to show the compactness. As a byproduct, we obtain the uniqueness of the positive ground state solution for a class of autonomous Hamiltonian systems and the best constant for Sobolev inequality which are of independent interests.

Nonexistence and existence of positive ground state solutions for generalized quasilinear Schrödinger equations

Nonexistence and existence of positive ground state solutions for generalized quasilinear Schrödinger equations

Positive ground state solutions for a class of fractional coupled Choquard systems

<abstract><p>In this paper, we combine the critical point theory and variational method to investigate the following a class of coupled fractional systems of Choquard type</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{array}{l} (-\Delta)^{s}u+\lambda_{1}u& = (I_{\alpha}*|u|^{p})|u|^{p-2}u+\beta v \quad &&\text{in}\\ \mathbb{R}^{N}, \ (-\Delta)^{s}v+\lambda_{2}v& = (I_{\alpha}*|v|^{p})|v|^{p-2}v+\beta u \quad &&\text{in}\ \mathbb{R}^{N}, \end{array} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>with $ s\in(0, 1), \ N\geq 3, \ \alpha\in(0, N), \ p > 1 $, $ \lambda_{i} > 0 $ are constants for $ i = 1, \ 2 $, $ \beta > 0 $ is a parameter, and $ I_{\alpha}(x) $ is the Riesz Potential. We prove the existence and asymptotic behaviour of positive ground state solutions of the systems by using constrained minimization method and Hardy-Littlewood-Sobolev inequality. Moreover, nonexistence of nontrivial solutions is also obtained.</p></abstract>

Semiclassical states for coupled nonlinear Schrödinger equations with critical frequency

In this paper, we are concerned with the coupled nonlinear Schrödinger system [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are nonnegative continuous potentials, and [Formula: see text] is a small parameter. We show the existence of positive ground state solutions for the system above and also establish the concentration behaviour as [Formula: see text], when [Formula: see text] and [Formula: see text] achieve 0 with a homogeneous behaviour or vanish in some nonempty open set with smooth boundary.

Weigthed elliptic equation of Kirchhoff type with exponential non linear growthWeigthed elliptic equation of Kirchhoff type with exponential non linear growth

"This work is concerned with the existence of a positive ground state solution for the following non local weighted problem \begin{equation*} \displaystyle \left\{ \begin{array}{rclll} L_{(\sigma,V)}u &= & \displaystyle f(x,u)& \mbox{in} \ B \\ u &>&0 &\mbox{in }B\\ u&=&0 &\mbox{on } \partial B, \end{array} \right. \end{equation*} where $$L_{(\sigma,V)}u:=g(\int_{B}(\sigma(x)|\nabla u|^{N}+V(x)|u|^{N})dx)\big[-\textmd{div} (\sigma(x)|\nabla u|^{N-2} \nabla u)+V(x)|u|^{N-2}u\big],$$ B is the unit ball of $\mathbb{R}^{N}$, $ N>2$, $\sigma(x)=\Big(\log(\frac{e}{|x|})\Big)^{\beta(N-1)}$, $\beta \in[0,1)$ the singular logarithm weight , $V(x)$ is a positif continuous potential.The Kirchhoff function $g$ is positive and continuous on $(0,+\infty)$. The nonlinearities are critical or subcritical growth in view of Trudinger-Moser inequalities of exponential type. We prove the existence of a positive ground state solution by using Mountain Pass theorem . In the critical case, the Euler-Lagrange function loses compactness except for a certain level. We dodge this problem by using adapted test functions to identify this level of compactness."

Ground state solutions for weighted $N$-Laplacian problem with exponential non linear growth

In this paper, we establish the existence of a positive ground state solution for a weighted problem under boundary Dirichlet condition in the unit ball of $\mathbb R^N,$ $N > 2$. The nonlinearity of the equation is critical or subcritical growth in view of Trudinger-Moser inequalities. In order to obtain our existence result we used minimax techniques combined with Trudinger-Moser inequalities. In the critical case, the associated energy does not satisfy the condition of compactness. We provide a new condition for growth and we stress its importance to check compactness levels.

Concentrating solutions for singularly perturbed double phase problems with nonlocal reaction

This paper focuses on the study of multiplicity and concentration phenomena of positive solutions for the singularly perturbed double phase problem with nonlocal Choquard reaction{−ϵpΔpu−ϵqΔqu+V(x)(|u|p−2u+|u|q−2u)=ϵμ−N(1|x|μ⁎G(u))g(u),inRN,u∈W1,p(RN)∩W1,q(RN),u>0,inRN, where 1<p<q<N, 0<μ<N, ϵ is a small positive parameter and V is the absorption potential. Combining variational and topological arguments from Nehari manifold analysis and Ljusternik-Schnirelmann category theory, we prove the existence of positive ground state solutions that concentrate around global minimum points of the potential V. In the second part of this paper, we establish the relationship between the number of positive solutions and the topology of the set where V attains its global minimum. The main results included in this paper complement several recent contributions to the study of concentration phenomena.

Chern-Simons limit of ground state solutions for the Schrödinger equations coupled with a neutral scalar field

We consider a class of the Schrödinger equation coupled with an neutral scalar field in non-radial symmetric space H1(R2). In this paper, combining the Nehari manifold, Moser iteration and some analytical skills, we get a positive ground state solution to the problem, which is classical and spherically symmetric. At the same time, this solution together with its derivatives up to order 2 has exponential decay at infinity. Moreover, the asymptotic behavior of solutions is analyzed under the sense of Chern-Simons limit. Our work can be regarded as a supplement and extension to Han et al. (2014) [12], where they rely heavily on the symmetry of work space.

Positive Ground State Solutions for Schrödinger–Poisson System Involving a Negative Nonlocal Term and Critical Exponent

Positive Ground State Solutions for Schrödinger–Poisson System Involving a Negative Nonlocal Term and Critical Exponent

Polyharmonic Kirchhoff-type problem without Ambrosetti–Rabinowitz condition

In this paper, we will study the polyharmonic Kirchhoff-type problem with singular exponential nonlinearity without the Ambrosetti–Rabinowitz condition: { − M ( ∫ Ω | ∇ m u | n m ) Δ n m m u = f ( x , u ) | x | σ in Ω , u = ∇ u = ⋯ = ∇ m − 1 u = 0 on ∂ Ω , where Ω ⊂ R n is a bounded domain with smooth boundary, 0 < σ < n , n ≥ 2 m ≥ 2 , M is a Kirchhoff function and f ( x , u ) has critical exponential growth. We use a suitable version of the Mountain Pass Theorem to prove the existence of a positive ground state solution for this problem.

Ground state solutions for critical Schrödinger equations with Hardy potential

In this paper, we investigate the following Schrödinger equation 0.1 −Δu−μ|x|2u=g(u)+|u|2*−2uinRN\\{0}, where N ⩾ 3, , is called the critical Sobolev exponent and g satisfies some appropriate subcritical conditions. For any , we prove that problem (0.1) has a positive radial ground state solution, which possesses exponential decaying property at infinity and blow-up property at origin. Moreover, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state solutions to problem (0.1) converges to a ground state solution of 0.2 −Δu=g(u)+|u|2*−2uinRN. when μ < 0, we prove that the mountain pass level of problem (0.1) in cannot be achieved. Further, we obtain a ground state radial solution of problem (0.1) whose energy is strictly greater than the mountain pass level in . Also, for any sequence {μ n } ⊂ (0, +∞) satisfying μ n → 0+, the sequence of ground state radial solutions to problem (0.1) converges to a ground state radial solution of the limiting problem as n → ∞.

Multiplicity and concentration of solutions for fractional Kirchhoff–Choquard equation with critical growth

In this paper, we are considered with class of fractional Kirchhoff–Choquard equation. Applying variational methods and topological arguments, we first investigate the existence of positive ground state solution and then consider relationship for the number of positive solutions and the topology of the set where the potential V attains its minimum. Finally, we give the concentrating behavior of solutions.

Ground state for critical elliptic systems with perturbation term of superlinear type

This work deals with the existence of at least one positive ground state solution for a stationary perturbed critical elliptic system with superlinear potential. Our problems involve the critical Sobolev constants which generate the lack of compactness in unbounded domains; we overcome such difficulty by using the concept of the Palais–Smale convergence. We make recourse to the Ekeland variational principle to show that our problem has a positive time-independent solution with positive energy as the total energy of the system.

Existence and concentration of positive solutions to a fractional system with saturable term

In this paper, we consider a fractional Laplacian system with saturable nonlinearity. Under some assumptions on the parameters and potential functions, we obtain the existence and concentration behavior of the positive ground state solution by variational methods.

Ground state for relativistic nonlinear Schrödinger equations involving general nonlinear term

In this paper, the existence and nonexistence of positive solutions for a relativistic nonlinear Schrödinger equation with vanish potential and general nonlinear term were obtained. First, we reduce this equation to a semilinear elliptic equation by a change. Then, using mountain pass theorem and Strauss compactness lemma, we get the existence of positive radial ground state solutions for this equation. Moreover, the nonexistence of nontrivial solutions for this equation was gotten by Pohozaev identity.

Positive Ground State Solutions for a Critical Nonlocal Problem in Dimension Three

Positive Ground State Solutions for a Critical Nonlocal Problem in Dimension Three

Ground states for critical fractional Schrödinger‐Poisson systems with vanishing potentials

This paper deals with a class of fractional Schrödinger‐Poisson system with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where is the fractional critical exponent. The problem is set on the whole space, and compactness issues have to be tackled. By employing the mountain pass theorem, concentration‐compactness principle, and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on , , , and .

The Hardy–Schrödinger operator on the Poincaré ball: Compactness, multiplicity, and stability of the Pohozaev obstruction

Let Ω be a smooth relatively compact domain containing zero in the Poincaré ball model of the Hyperbolic space Bn (n≥3) and let −ΔBn be the Laplace-Beltrami operator on Bn. We consider problems of non-existence, existence, and multiplicity of variational solutions for the borderline Dirichlet problem,(1){−ΔBnu−γV2u−λu=V2⋆(s)|u|2⋆(s)−2u in Ωu=0 on ∂Ω, where γ<(n−2)24, 0<s<2, 2⋆(s):=2(n−s)n−2 being the corresponding critical Sobolev exponent, while V2 (resp., V2⋆(s)) is a Hardy-type potential (resp., Hardy-Sobolev weight) that is invariant under hyperbolic scaling and which behaves like 1r2 (resp. 1rs) at the origin. The bulk of this paper is a sharp blow-up analysis that we perform on certain approximate solutions of (1) with bounded but arbitrary high energies. When these approximate solutions are positive, our analysis leads to improvements of results in [5] regarding positive ground state solutions for (1), as we show that they exist whenever n≥4, 0≤γ≤(n−2)24−1 and λ>0. The latter result also holds true for n≥3 and γ>(n−2)24−1 provided the domain has a positive “hyperbolic mass”. On the other hand, the same analysis yields that if γ>(n−2)24−1 and the mass is non vanishing, then there is a surprising stability of regimes where no variational positive solution exists. As for higher energy solutions to (1), we show that there are infinitely many of them provided n≥5, γ<(n−2)24−4 and λ>n−2n−4(n(n−4)4−γ).

On fractional logarithmic Schrödinger equations

Abstract We study the following fractional logarithmic Schrödinger equation: ( − Δ ) s u + V ( x ) u = u log u 2 , x ∈ R N , {\left(-\Delta )}^{s}u+V\left(x)u=u\log {u}^{2},\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ≥ 1 N\ge 1 , ( − Δ ) s {\left(-\Delta )}^{s} denotes the fractional Laplace operator, 0 &lt; s &lt; 1 0\lt s\lt 1 and V ( x ) ∈ C ( R N ) V\left(x)\in {\mathcal{C}}\left({{\mathbb{R}}}^{N}) . Under different assumptions on the potential V ( x ) V\left(x) , we prove the existence of positive ground state solution and least energy sign-changing solution for the equation. It is known that the corresponding variational functional is not well defined in H s ( R N ) {H}^{s}\left({{\mathbb{R}}}^{N}) , and inspired by Cazenave (Stable solutions of the logarithmic Schrödinger equation, Nonlinear Anal. 7 (1983), 1127–1140), we first prove that the variational functional is well defined in a subspace of H s ( R N ) {H}^{s}\left({{\mathbb{R}}}^{N}) . Then, by using minimization method and Lions’ concentration-compactness principle, we prove that the existence results.