Existence Of Ground State Solutions Research Articles

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 ) Δ u = λ u + | u | p − 2 u + μ | u | q − 2 u in R N , 1 ≤ N ≤ 3 with prescribed mass , where , , . Under certain assumptions on the parameter μ, we obtain the existence of normalized solution , where .

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.

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.

A Study of Biharmonic Equation Involving Nonlocal Terms and Critical Sobolev Exponent

AbstractIn this paper, we investigate the existence of ground state solutions and non-existence of non-trivial weak solution of the equation $$\begin{aligned} \Delta ^{2} u= \Big (|x|^{-\theta }*|u|^{p_{\theta }}\Big )|u|^{p_{\theta }-2}u +\alpha \Big (|x|^{-\gamma }*|u|^{p}\Big )|u|^{p-2}u \quad \text{ in } {{\mathbb {R}}}^{N}, \end{aligned}$$ Δ 2 u = ( | x | - θ ∗ | u | p θ ) | u | p θ - 2 u + α ( | x | - γ ∗ | u | p ) | u | p - 2 u in R N , where $$0<p\le p_{\gamma }^{*}$$ 0 < p ≤ p γ ∗ , $$\alpha >0$$ α > 0 , $$\theta , \gamma \in (0,N)$$ θ , γ ∈ ( 0 , N ) , $$p_{\theta }=\frac{2(N-\theta )}{N-4}$$ p θ = 2 ( N - θ ) N - 4 and $$N\ge 5$$ N ≥ 5 . Firstly, we prove the non-existence by establishing Pohozaev type of identity. Next, we study the existence of ground state solutions by using the minimization method on the associated Nehari manifold.

Existence of Ground State Solutions for Kirchhoff Problems with Hardy Potential

Existence of Ground State Solutions for Kirchhoff Problems with Hardy Potential

Ground State Solutions of Fractional Choquard Problems with Critical Growth

In this article, we investigate a class of fractional Choquard equation with critical Sobolev exponent. By exploiting a monotonicity technique and global compactness lemma, the existence of ground state solutions for this equation is obtained. In addition, we demonstrate the existence of ground state solutions for the corresponding limit problem.

Ground state solutions for Choquard equations with Bessel potential

Firstly, we study the existence, regularity and decay of positive solutions for the elliptic system where , g satisfies Berestycki-Lions type conditions. In Sobolev space, the system can be transformed into the Choquard equations with Bessel potential where is the fundamental solution of the linear operator on . Moreover, by assuming that g is nondecreasing, we obtain the radial symmetry and monotonic property of positive solutions by moving plane method, and the existence of ground state solutions by non-Nehari manifold method.

Existence of ground state solutions for weighted biharmonic problem involving non linear exponential growth

In this article, we study the following problem $$\begin{aligned} \Delta (w_{\beta }(x)\Delta u) = \ f(x,u) \quad \text{ in } \quad B, \quad u=\frac{\partial u}{\partial n}=0 \quad \text{ on } \quad \partial B, \end{aligned}$$ where B is the unit ball of $${\mathbb {R}}^{4}$$ and $$ w_{\beta }(x)$$ a singular weight of logarithm type. The reaction source f(x, u) is a radial function with respect to x and it is critical in view of exponential inequality of Adams’ type. The existence result is proved by using the constrained minimization in Nehari set coupled with the quantitative deformation lemma and degree theory results.

Existence of ground state solutions for a Choquard double phase problem

In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form −Lp,qa(u)+|u|p−2u+a(x)|u|q−2u=∫RNF(y,u)|x−y|μdyf(x,u)inRN,where Lp,qa is the double phase operator given by Lp,qa(u)≔div(|∇u|p−2∇u+a(x)|∇u|q−2∇u),u∈W1,H(RN),0<μ<N, 1<p<N, p<q<p+αpN, 0≤a(⋅)∈C0,α(RN) with α∈(0,1] and f:RN×R→R is a continuous function that satisfies a subcritical growth. Based on the Hardy–Littlewood–Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.

On nonlinear fractional Schrödinger equations with indefinite and Hardy potentials

This paper is concerned with a class of fractional Schrödinger equation with Hardy potential ( − Δ ) s u + V ( x ) u − κ | x | 2 s u = f ( x , u ) , x ∈ R N , where s ∈ ( 0 , 1 ) and κ ⩾ 0 is a parameter. Under some suitable conditions on the potential V and the nonlinearity f, we prove the existence of ground state solutions when the parameter κ lies in a given range by using the non-Nehari manifold method. Moreover, we investigate the continuous dependence of ground state energy about κ. Finally, we are able to explore the asymptotic behavior of ground state solutions when κ tends to 0.