Existence Of Ground State Solutions Research Articles

Existence and asymptotic behavior of normalized solutions for fourth‐order equations of Kirchhoff type

In this paper, we consider the following fourth‐order equation of Kirchhoff type under the normalized constraint , where are constants, and is a Lagrange multiplier. Firstly, we obtain the existence of ground state solutions for or with . Moreover, we show the nonexistence of solutions for . Finally, we investigate the asymptotic behavior of normalized solutions obtained above with respect to the parameter .

Ground state solutions for the Hamilton–Choquard elliptic system with critical exponential growth

In this paper, we study the following Hamilton–Choquard type elliptic system: − Δ u + u = ( I α ∗ F ( v ) ) f ( v ) , x ∈ R 2 , − Δ v + v = ( I β ∗ F ( u ) ) f ( u ) , x ∈ R 2 , where I α and I β are Riesz potentials, f : R → R possessing critical exponential growth at infinity and F ( t ) = ∫ 0 t f ( s ) d s. Without the classic Ambrosetti–Rabinowitz condition and strictly monotonic condition on f, we will investigate the existence of ground state solution for the above system. The strongly indefinite characteristic of the system, combined with the convolution terms and critical exponential growth, makes such problem interesting and challenging to study. With the help of a proper auxiliary system, we employ an approximation scheme and the non-Nehari manifold method to control the minimax levels by a fine threshold, and succeed in restoring the compactness for the critical problem. Existence of a ground state solution is finally established by the concentration compactness argument and some detailed estimates.

Magnetic Kirchhoff–Schrödinger equations with critical exponential growth and indefinite potential in R 2

In this paper, we are concerned with the following magnetic nonlinear equation of Kirchhoff type with critical exponential growth and an indefinite potential in R 2 m ( ∫ R 2 [ | 1 i ∇ u − A ( x ) u | 2 + V ( x ) | u | 2 ] d x ) [ ( 1 i ∇ − A ( x ) ) 2 u + V ( x ) u ] = B ( x ) f ( | u | 2 ) u , where u ∈ H 1 ( R 2 , C ), m is a Kirchhoff type function, V : R 2 → R and A : R 2 → R 2 represent locally bounded potentials, while B denotes locally bounded and f exhibits critical exponential growth. By employing variational methods and utilizing the modified Trudinger–Moser inequality, we get ground state solutions or nontrivial solutions for the above equation. Furthermore, in the special case where m is a constant equal to 1, the equation is reduced to the following magnetic nonlinear Schrödinger equation, ( 1 i ∇ − A ( x ) ) 2 u + V ( x ) u = B ( x ) f ( | u | 2 ) u in R 2 . Applying analogous methods, we can also establish the existence of ground state solutions or nontrivial solutions to this equation.

Variational problems for the system of nonlinear Schrödinger equations with derivative nonlinearities

We consider the Cauchy problem of the system of nonlinear Schrödinger equations with derivative nonlinearlity. This system was introduced by Colin and Colin (Differ Int Equ 17:297–330, 2004) as a model of laser-plasma interactions. We study existence of ground state solutions and the global well-posedness of this system by using the variational methods. We also consider the stability of traveling waves for this system. These problems are proposed by Colin–Colin as the open problems. We give a subset of the ground-states set which satisfies the condition of stability. In particular, we prove the stability of the set of traveling waves with small speed for 1-dimension.

Ground state solutions of fractional equations with Coulomb potential and critical exponent

AbstractIn this paper, we are concerned with ground state solutions to fractional equations with Coulomb potential and critical exponent. The existence of ground state solutions is established under certain conditions by proposing a new analytical method.

Regularity for critical fractional Choquard equation with singular potential and its applications

Abstract We study the following fractional Choquard equation ( − Δ ) s u + u ∣ x ∣ θ = ( I α * F ( u ) ) f ( u ) , x ∈ R N , {\left(-\Delta )}^{s}u+\frac{u}{{| x| }^{\theta }}=({I}_{\alpha }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ⩾ 3 N\geqslant 3 , s ∈ 1 2 , 1 s\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right) , α ∈ ( 0 , N ) \alpha \in \left(0,N) , θ ∈ ( 0 , 2 s ) \theta \in \left(0,2s) , and I α {I}_{\alpha } is the Riesz potential. The main purpose of this article is twofold. We first study the regularity of weak solutions for the aforementioned equation with critical nonlinearity, which extends the results of θ = 0 \theta =0 in Moroz-Van Schaftingen [Existence of groundstates for a class of nonlinear Choquardequations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579]. Then, as an application of the regularity results, we establish the existence of ground state solutions for above equation with the nonlinearity involving embedding top and bottom indices, which is related to the Hardy-Littlewood-Sobolev inequality and singular term 1 ∣ x ∣ θ \frac{1}{{| x| }^{\theta }} . It is worth noting that our approach is not involving the concentration-compactness principle.

Ground state solutions for generalized quasilinear Schrödinger equations

In this paper we consider the generalized quasilinear Schrödinger equations − div ( g 2 ( u ) ∇ u ) + g ( u ) g ′ ( u ) | ∇ u | 2 + V ( x ) u = h ( x , u ) , x ∈ R N , where V and h are periodic in x i , 1 ⩽ i ⩽ N. By using variational methods, we prove the existence of ground state solutions, i.e., nontrivial solutions with least possible energy.

Ground State Solutions for a Non-Local Type Problem in Fractional Orlicz Sobolev Spaces

In this paper, we study the following non-local problem in fractional Orlicz–Sobolev spaces: (−ΔΦ)su+V(x)a(|u|)u=f(x,u), x∈RN, where (−ΔΦ)s(s∈(0,1)) denotes the non-local and maybe non-homogeneous operator, the so-called fractional Φ-Laplacian. Without assuming the Ambrosetti–Rabinowitz type and the Nehari type conditions on the non-linearity f, we obtain the existence of ground state solutions for the above problem with periodic potential function V(x). The proof is based on a variant version of the mountain pass theorem and a Lions’ type result in fractional Orlicz–Sobolev spaces.

A note on a supercritical [formula omitted]-Laplacian equation with logarithmic perturbation

In this short note, we study the existence of ground state solutions for p-Laplacian elliptic problems with supercritical growth and logarithmic nonlinearity, both within the interior and on the boundary of a domain.

Multiplicity and limit of solutions for logarithmic Schrödinger equations on graphs

Let Ω be a finite connected subset of a locally finite graph G = (V, E) with the vertex set V and the edge set E. We investigate the logarithmic Schrödinger equation on Ω with the nonlinear term |u|p−2u log u2. For p > 2, through two different approaches which are the Brouwer degree theory and mountain-pass theorem, we obtain the existence of ground state solutions. We also apply the Brouwer degree theory together with the constraint variational method to prove that the equation admits a sign-changing solution which implies the multiplicity of solutions to the equation. Finally, we illustrate that as p → 2, up to a subsequence, the solutions for p > 2 shall converge to a non-trivial solution of the equation with p = 2.

Existence of Ground State Solutions for a Class of Non-Autonomous Fractional Kirchhoff Equations

We take a look at the fractional Kirchhoff problem in this paper. Using a variational approach, we show that there exists a ground state solution for this problem. Furthermore, using the approach developed by Szulkin and Weth, we also find that positive ground state solutions exist for the fractional Kirchhoff equation with p=4.

Normalized ground states for the Sobolev critical Schrödinger equation with at least mass critical growth

In the present paper, we investigate the existence of ground state solutions to the Sobolev critical nonlinear Schrödinger equation −Δu+λu=gu+|u|2∗−2u inRN,∫RN|u|2dx=m2, where N⩾3 , m > 0, 2∗:=2NN−2 , λ is an unknown parameter that will appear as a Lagrange multiplier, g is a mass critical or supercritical but Sobolev subcritical nonlinearity. With the aid of the minimization of the energy functional over a linear combination of the Nehari and Pohozaev constraints intersected with the product of the closed balls in L2(RN) of radii m and the profile decomposition, we obtain a couple of the normalized ground state solution to (Pm) that is independent of the sign of the Lagrange multiplier. This result complements and extends the paper by Bieganowski and Mederski (2021 J. Funct. Anal. 280 108989) concerning the above problem from the Sobolev subcritical setting to the Sobolev critical framework. We also answer an open problem that was proposed by Jeanjean and Lu (2020 Calc. Var. PDE 59 174). Furthermore, the asymptotic behavior of the ground state energy map is also studied.

The existence of ground state normalized solution for Kirchhoff equation

In this paper, we study the existence of ground state solution for the following Kirchhoff equation with combined nonlinearities − ( a + b ∫ R N | ∇u | 2 ) Δu = λu + | u | p − 2 u + μ | u | q − 2 u in R N , 1 ≤ N ≤ 3 with prescribed mass ∫ R N u 2 = c 2 , where a > 0 , b > 0 , c > 0 , μ < 0 , 2 < q ≤ 2 + 8 N < p < 2 ∗ . Under certain assumptions on the parameter μ, we obtain the existence of normalized solution ( u ~ , λ c ) ∈ S c × R , where S c = { u ∈ H 1 ( R N ) : ∫ R N u 2 = c 2 } .

Ground state solutions for a kind of superlinear elliptic equations with variable exponent

AbstractIn this paper, we focus on the existence of ground state solutions for the $p(x)$ p ( x ) -Laplacian equation $$ \textstyle\begin{cases} -\Delta _{p(x)}u+\lambda \vert u \vert ^{p(x)-2}u=f(x,u)+h(x) \quad \text{in } \Omega , \\ u=0,\quad \text{on }\partial \Omega . \end{cases} $$ { − Δ p ( x ) u + λ | u | p ( x ) − 2 u = f ( x , u ) + h ( x ) in Ω , u = 0 , on ∂ Ω . Using the constraint variational method, quantitative deformation lemma, and strong maximum principle, we proved that the above problem admits three ground state solutions, especially speaking, one solution is sign-changing, one is positive, and one is negative. Our results improve on those existing in the literature.

Properties of the least action level and the existence of ground state solution to fractional elliptic equation with harmonic potential

In this article we consider the following fractional semilinear elliptic equation \[(-\Delta)^su+|x|^2u =\omega u+|u|^{2\sigma}u \quad \text{ in } \mathbb{R}^N,\] where \(s\in (0,1)\), \(N\gt 2s\), \(\sigma\in (0,\frac{2s}{N-2s})\) and \(\omega\in (0, \lambda_1)\). By using variational methods we show the existence of a symmetric decreasing ground state solution of this equation. Moreover, we study some continuity and differentiability properties of the ground state level. Finally, we consider a bifurcation type result.

Existence of Ground State Solutions for the Schrödinger–Poisson System in $\mathbb{R}^{2}$

Existence of Ground State Solutions for the Schrödinger–Poisson System in $\mathbb{R}^{2}$

Ground state solutions of a magnetic nonlinear Choquard equation with lower critical exponent

In this article, we establish the existence of ground state solutions for a magnetic nonlinear Choquard equation with lower critical exponent by variational methods.

Existence of Ground State Solutions for Choquard Equation with the Upper Critical Exponent

In this article, we investigate the existence of a nontrivial solution for the nonlinear Choquard equation with upper critical exponent see Equation (6). The Riesz potential in this case has never been studied. We establish the existence of the ground state solution within bounded domains Ω⊂RN. Variational methods are used for this purpose. This method proved to be instrumental in our research, enabling us to address the problem effectively. The study of the existence of ground state solutions for the Choquard equation with a critical exponent has applications and relevance in various fields, primarily in theoretical physics and mathematical analysis.

The Existence of Ground State Solutions for a Schrödinger–Bopp–Podolsky System with Convolution Nonlinearity

The Existence of Ground State Solutions for a Schrödinger–Bopp–Podolsky System with Convolution Nonlinearity

Ground state solutions for a class of generalized quasilinear Schrödinger‐Maxwell system with critical growth

In this paper, we consider the following generalized quasilinear Schrödinger‐Maxwell system Under suitable conditions on , , and , we obtain the existence of ground state solutions with critical growth. To some extent, our results improve and supplement some existing relevant results.