Existence Of Least Energy Solutions Research Articles

A critical phenomenon for the least energy solutions to nonlinear scalar field equations

We study the existence of least energy solutions to the nonlinear scalar field equation: (1) − Δ u + λ u + V ( x ) u = Q ( x ) | u | p u , u ∈ H 1 ( R N ) , Where V ( x ) , Q ( x ) ∈ L ∞ ( R N ) are real functions satisfying suitable assumptions. By considering the Nehari type constraint N λ := { u ∈ H 1 ( R N ) ∖ { 0 } : ∫ R N ( | ∇ u | 2 + λ | u | 2 + V ( x ) | u | 2 ) d x = ∫ R N Q ( x ) | u | p + 2 d x } , it is shown that (1) exists at least a nontrivial least energy solutions if \\lambda _{*}:=-\\inf \\sigma (-\\Delta +V) $ ]]> λ > λ ∗ := − inf σ ( − Δ + V ) . In this argument, we point out λ ∗ is a critical value for the existence of least energy solutions restricted to N λ . That is, there exists global least energy solutions in N λ if \\lambda _{*} $ ]]> λ > λ ∗ , but not if λ < λ ∗ . Moreover, the asymptotic behavior of least energy solutions as λ → + ∞ is analyzed.

Least energy solutions to a class of nonlocal Schrödinger equations

Least energy solutions to a class of nonlocal Schrödinger equations

Least energy sign-changing solutions for discrete Kirchhoff-type problems

In this paper, by variational technique coupled with quantitative lemma, we not only investigate the existence of least energy solutions, but also point out that the energy of the least energy sign-changing solutions is strictly larger than twice that of the ground state solutions.

Critical Gagliardo–Nirenberg, Trudinger, Brezis–Gallouet–Wainger inequalities on graded groups and ground states

In this paper, we investigate critical Gagliardo–Nirenberg, Trudinger-type and Brezis–Gallouet–Wainger inequalities associated with the positive Rockland operators on graded Lie groups, which include the cases of [Formula: see text], Heisenberg, and general stratified Lie groups. As an application, using the critical Gagliardo–Nirenberg inequality, the existence of least energy solutions of nonlinear Schrödinger type equations is obtained. We also express the best constant in the critical Gagliardo–Nirenberg and Trudinger inequalities in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The obtained results are already new in the setting of general stratified Lie groups (homogeneous Carnot groups). Among new technical methods, we also extend Folland’s analysis of Hölder spaces from stratified Lie groups to general homogeneous Lie groups.

The existence of least energy and high energy solutions to the Kirchhoff type problem in high dimensions

In this paper, we study the following Kirchhoff type problem with critical growth:(0.1)−(a+b∫RN|∇u|2dx)Δu+V(x)u=βf(x)|u|r−2u−g(x)|u|q−2u+u2⁎−1inRN, where N≥4, a, b>0 are constants, β>0 is a parameter and 2<r<q<2⁎. By using variational methods, we obtain the existence of least energy solutions and high energy solutions for certain parameter ranges. Also, we obtain the existence of sign-changing solutions.

A spinorial analogue of the Brezis-Nirenberg theorem involving the critical Sobolev exponent

Let (M,g,σ) be a compact Riemannian spin manifold of dimension m≥2, let S(M) denote the spinor bundle on M, and let D be the Atiyah-Singer Dirac operator acting on spinors ψ:M→S(M). We study the existence of solutions of the nonlinear Dirac equation with critical exponent(NLD)Dψ=λψ+f(|ψ|)ψ+|ψ|2m−1ψ where λ∈R and f(|ψ|)ψ is a subcritical nonlinearity in the sense that f(s)=o(s2m−1) as s→∞. A model nonlinearity is f(s)=αsp−2 with 2<p<2mm−1, α∈R. In particular we study the nonlinear Dirac equation(BND)Dψ=λψ+|ψ|2m−1ψ,λ∈R. This equation is a spinorial analogue of the Brezis-Nirenberg problem. As corollary of our main results we obtain the existence of least energy solutions (λ,ψ) of (BND) for every λ>0, even if λ is an eigenvalue of D. For some classes of nonlinearities f we also obtain solutions of (NLD) for every λ∈R, except for non-positive eigenvalues. If m≢3 (mod 4) we obtain solutions of (BND) for every λ∈R, except for a finite number of non-positive eigenvalues. In certain parameter ranges we obtain multiple solutions of (NLD) and (BND), some near the trivial branch, others away from it.The proofs of our results are based on variational methods using the strongly indefinite energy functional associated to (NLD).

Normalized ground states of the nonlinear Schrödinger equation with at least mass critical growth

We propose a simple minimization method to show the existence of least energy solutions to the normalized problem{−Δu+λu=g(u)inRN,N≥3,u∈H1(RN),∫RN|u|2dx=ρ>0, where ρ is prescribed and (λ,u)∈R×H1(RN) is to be determined. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed ball in L2(RN) of radius ρ is demonstrated, which allows to provide general growth assumptions imposed on g. We cover the most known physical examples and nonlinearities with growth considered in the literature so far as well as we admit the mass critical growth at 0.

A remark on Kirchhoff-type equations in ℝ4 involving critical growth

ABSTRACT We consider the following Kirchhoff type equation: where a, b are positive constants and . Under the critical growth assumptions on g, we obtain the existence of least energy solutions by studying the associated minimization problem with a new constraint. The mountain pass characterization of least energy solutions is also investigated without the monotonicity assumption on . Our results present some new observations.

Least energy solutions for fractional Kirchhoff problems with logarithmic nonlinearity

In this paper, we study the existence of least energy solutions to the following fractional Kirchhoff problem with logarithmic nonlinearity M([u]s,pp)(−Δ)psu=h(x)|u|θp−2uln|u|+λ|u|q−2ux∈Ω,u=0x∈RN∖Ω,where s∈(0,1), 1<p<N∕s, Ω⊂RN is a bounded domain with Lipschitz boundary, M([u]s,pp)=[u]s,p(θ−1)p with θ≥1 and [u]s,p is the Gagliardo seminorm of u, h∈C(Ω¯) may change sign, λ>0 is a parameter, q∈(1,ps∗) and (−Δ)ps is the fractional p−Laplacian. When θp<q<ps∗ and h is a positive function on Ω, the existence of least energy solutions is obtained by restricting the discussion on Nehari manifold. When 1<q<θp and h is a sign-changing function on Ω, two local least energy solutions are obtained by using the Nehari manifold approach.

On the existence of least energy solution for Kirchhoff equation in R3

On the existence of least energy solution for Kirchhoff equation in R3

Multiplicity of radial and nonradial solutions to equations with fractional operators

<p style='text-indent:20px;'>In this paper, we study the existence of radial and nonradial solutions to the scalar field equations with fractional operators. For radial solutions, we prove the existence of infinitely many solutions under <inline-formula><tex-math id="M1">\begin{document}$ N \geq 2 $\end{document}</tex-math></inline-formula>. We also show the existence of least energy solution (with the Pohozaev identity) and its mountain pass characterization. For nonradial solutions, we prove the existence of at least one nonradial solution under <inline-formula><tex-math id="M2">\begin{document}$ N \geq 4 $\end{document}</tex-math></inline-formula> and infinitely many nonradial solutions under either <inline-formula><tex-math id="M3">\begin{document}$ N = 4 $\end{document}</tex-math></inline-formula> or <inline-formula><tex-math id="M4">\begin{document}$ N \geq 6 $\end{document}</tex-math></inline-formula>. We treat both of the zero mass and the positive mass cases.

Limiting cases of Sobolev inequalities on stratified groups

In this paper we present critical Gagliardo-Nirenberg, Trudinger-type and Brezis-Gallouet-Wainger inequalities concerning the limiting cases of the embedding theorems for Sobolev spaces on stratified groups. Moreover, using the critical Gagliardo-Nirenberg inequality the existence of least energy solutions of the nonlinear Schrodinger type equations can be obtained. We also express the best constant in the critical Gagliardo-Nirenberg inequality in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations.

On the existence of least energy solutions to a Kirchhoff-type equation in [formula omitted

This paper concerns the following Kirchhoff-type equation −(α+β∫R3|∇u(x)|2dx)Δu(x)+ωu(x)=|u(x)|p−1u(x)inR3,where α,β,ω>0 are constants. The existence of positive least energy solutions to this equation for all p∈(1,5) has been established. Our result extends and improves the recent results in the literature.

Vector solutions for two coupled Schrödinger equations on Riemannian manifolds

We are interested in two coupled nonlinear Schrödinger equations on a smooth compact boundaryless Riemannian manifold, which arises from Bose-Einstein condensates and nonlinear optics theory. The existence of least energy solutions, multiplicity of positive solutions, as well as synchronized vector solutions is obtained.

Mountain pass solutions for the fractional Berestycki-Lions problem

We investigate the existence of least energy solutions and infinitely many solutions for the following nonlinear fractional equation \begin{align*} (-\Delta)^{s} u = g(u) \mbox{ in } \mathbb R^{N}, \end{align*} where $s\in (0,1)$, $N\geq 2$, $(-\Delta)^{s}$ is the fractional Laplacian and $g: \mathbb R \rightarrow \mathbb R $ is an odd $\mathcal{C}^{1, \alpha}$ function satisfying Berestycki-Lions type assumptions. The proof is based on the symmetric mountain pass approach developed by Hirata, Ikoma and Tanaka in [33]. Moreover, by combining the mountain pass approach and an approximation argument, we also prove the existence of a positive radially symmetric solution for the above problem when $g$ satisfies suitable growth conditions which make our problem fall in the so called “zero mass” case.

On concentration of least energy solutions for magnetic critical Choquard equations

In the present paper, we consider the following magnetic nonlinear Choquard equation{(−i∇+A(x))2u+μg(x)u=λu+(|x|−α⁎|u|2⁎α)|u|2α⁎−2uinRn,u∈H1(Rn,C) where n≥4, 2α⁎=2n−αn−2, α∈(0,n), μ>0, λ>0 is a parameter, A(x):Rn→Rn is a magnetic vector potential and g(x) is a real valued potential function on Rn. Using variational methods, we establish the existence of least energy solution under some suitable conditions. Moreover, the concentration behavior of solutions is also studied as μ→+∞.

Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well

In this paper, we consider the following nonlinear Choquard equation driven by fractional Laplacian urn:x-wiley:mma:media:mma4653:mma4653-math-0001 where V(x) is a nonnegative continuous potential function, 0&lt;s&lt;1, N≥2, (N−4s)+&lt;α&lt;N, and λ is a positive parameter. By variational methods, we prove the existence of least energy solution which localizes near the bottom of potential well int(V−1(0)) as λ large enough.

A critical problem on the Hardy–Sobolev inequality in boundary singularity case

We study a Neumann problem with the Hardy–Sobolev nonlinearity. In boundary singularity case, the impact of the mean curvature at singularity on existence of least-energy solution is well known. Existence and nonexistence of least-energy solution is studied by Hashizume (2017) except for lower dimension case. In this paper, we improve this previous work. More precisely, we study four dimensional case and show existence of minimizer in critical case in some sense.

Least energy solutions of nonlinear Schrödinger equations involving the fractional Laplacian and potential wells

We are concerned with the existence of least energy solutions of nonlinear Schrodinger equations involving the fractional Laplacian $$ (-\Delta)^{s}u(x)+\lambda V(x)u(x)=u(x)^{p-1},\quad u(x)\geq0,\quad x\in \rz^N, $$ for sufficiently large $\la$, $2＜p＜\frac{2N}{N-2s}$ for $N \geq 2$. $V(x)$ is a real continuous function on $\rz^N$. Using variational methods we prove the existence of least energy solution $u_\la(x)$ which localizes near the potential well int $V^{-1}(0)$ for $\la$ large. Moreover, if the zero sets int $V^{-1}(0)$ of $V(x)$ include more than one isolated component, then $u_\la(x)$ will be trapped around all the isolated components. However, in Laplacian case $s=1$, when the parameter $\la$ is large, the corresponding least energy solution will be trapped around only one isolated component and become arbitrarily small in other components of int $V^{-1}(0)$. This is the essential difference with the Laplacian problems since the operator $(-\Delta)^{s}$ is nonlocal.

Nonlocal Schrödinger-Kirchhoff equations with external magnetic field

The paper deals with existence and multiplicity of solutions of the fractional Schr\"{o}dinger--Kirchhoff equation involving an external magnetic potential. As a consequence, the results can be applied to the special case \begin{equation*} (a+b[u]_{s,A}^{2\theta-2})(-\Delta)_A^su+V(x)u=f(x,|u|)u\,\, \quad \text{in $\mathbb{R}^N$}, \end{equation*} where $s\in (0,1)$, $N>2s$, $a\in \mathbb{R}^+_0$, $b\in \mathbb{R}^+_0$, $\theta\in[1,N/(N-2s))$, $A:\mathbb{R}^N\rightarrow\mathbb{R}^N$ is a magnetic potential, $V:\mathbb{R}^N\rightarrow \mathbb{R}^+$ is an electric potential, $(-\Delta )_A^s$ is the fractional magnetic operator. In the super- and sub-linear cases, the existence of least energy solutions for the above problem is obtained by the mountain pass theorem, combined with the Nehari method, and by the direct methods respectively. In the superlinear-sublinear case, the existence of infinitely many solutions is investigated by the symmetric mountain pass theorem.