Nonlinear Scalar Field Equations Research Articles

The existence and multiplicity of L 2-normalized solutions to nonlinear Schrödinger equations with variable coefficients

Abstract The existence of L 2–normalized solutions is studied for the equation − Δ u + μ u = f ( x , u ) in R N , ∫ R N u 2 d x = m . $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R}}^{N},\quad {\int }_{{\mathbf{R}}^{N}}{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L 2-subcritical growth and (μ, u) ∈ R × H 1(R N ) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L 2 constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.

An upper bound for the least energy of a sign-changing solution to a zero mass problem

Abstract We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation − Δ u = f ( u ) , u ∈ D 1,2 ( R N ) , $-{\Delta}u=f\left(u\right), u\in {D}^{1,2}\left({\mathrm{R}}^{N}\right),$ where N ≥ 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing solution whose energy is smaller that 12c 0 if N = 5, 6 and smaller than 10c 0 if N ≥ 7, where c 0 is the ground state energy.

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.

Normalized solutions of the Schrödinger equation with potential

AbstractIn this paper, for dimension and prescribed mass , we consider the following nonlinear scalar field equation with constraint: where is a Lagrange multiplier, . In particular, satisfies mass supercritical and Sobolev subcritical growth. We prove the existence results of the normalized solution and infinitely many normalized solutions to the above system under some proper assumptions on the functions by the mountain pass argument.

Nondegeneracy of ground states for nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies in three and four dimensions

We consider nonlinear scalar field equations involving the Sobolev-critical exponent at high frequencies ω. Since the limiting profile of the ground states as ω→∞ is the Aubin–Talenti function and degenerate in a certain sense, from the point of view of perturbation methods, the nondegeneracy problem for the ground states at high frequencies is subtle. In addition, since the limiting profile (Aubin–Talenti function) fails to lie in L2(Rd) for d=3,4, the nondegeneracy problem for d=3,4 is more difficult than that for d≥5 and an applicable methodology is not known. In this paper, we solve the nondegeneracy problem for d=3,4 by modifying the arguments in Akahori et al. (2020) and Akahori et al. (2019). We also show that the linearized operator around the ground state has exactly one negative eigenvalue.

Normalized solutions to nonlinear scalar field equations with doubly nonlocal terms and critical exponent

In this paper, we study the existence and regularity of normalized solutions to the following doubly nonlocal equation−Δu=λu+μ1(Iα⁎|u|p)|u|p−2u+μ2(Iβ⁎|u|p)|u|p−2uinR3, having a prescribed mass ∫R3u2=a>0, where λ∈R will arise as a Lagrange multiplier, α,β∈(0,3) with α⩽β, p∈[2⁎β,2α⁎], Iα and Iβ are Riesz potentials, μ1,μ2>0 are constants. In particular, we estimate the energy level ingeniously and consider the existence of normalized solutions to the above equation with Hardy–Littlewood–Sobolev lower critical exponent p=2⁎β or upper critical exponent p=2α⁎. The results are supplements to the works of Cao et al. (2021) [5].

Kirchhoff–Boussinesq-type problems with positive and zero mass

Using variational methods we show the existence of solutions for the following class of elliptic Kirchhoff–Boussinesq-type problems given by Δ 2 u − Δ p u + u = h ( u ) , in R N and Δ 2 u − Δ p u = f ( u ) , in R N , where 2 < p ≤ 2 N N − 2 for N ≥ 3 and 2 ∗ ∗ = ∞ for N = 3, N = 4, 2 ∗ ∗ = 2 N N − 4 for N ≥ 5 and h and f are continuous functions that satisfy hypotheses considered by Berestycki and Lions [Nonlinear scalar field. Arch Rational Mech Anal. 1983;82:313–345]. More precisely, the problem with the nonlinearity h is related to the Positive mass case and the problem with the nonlinearity f is related to the Zero mass case. The main argument is to find a Palais–Smale sequence satisfying a property related to Pohozaev identity, as in Hirata et al. [Nonlinear scalar field equations in RN: mountain pass and symmetric mountain pass approaches. Topol Methods Nonlinear Anal. 2010;35:253–276], which was used for the first time by Jeanjean [On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer- type problem set on R N . Proc R Soc Edinb Sect A. 1999;129:787–809].

Biharmonic Nonlinear Scalar Field Equations

Abstract We prove a Brezis–Kato-type regularity result for weak solutions to the biharmonic nonlinear equation $$ \begin{align*} &amp; \Delta^2 u = g(x,u)\qquad\text{in }\mathbb{R}^N\end{align*}$$with a Carathéodory function $g:\mathbb {R}^N\times \mathbb {R}\to \mathbb {R}$, $N\geq 5$. The regularity results give rise to the existence of ground state solutions provided that $g$ has a general subcritical growth at infinity. We also conceive a new biharmonic logarithmic Sobolev inequality $$ \begin{align*} &amp; \int_{\mathbb{R}^N}|u|^2\log |u|\, \text{d}x\leq\frac{N}{8}\log \Big(C\int_{\mathbb{R}^N}|\Delta u|^2\, \text{d}x \Big), \quad\text{for } u \in H^2(\mathbb{R}^N), \; \int_{\mathbb{R}^N}u^2\, \text{d}x = 1, \end{align*}$$for a constant $0&amp;lt;C&amp;lt; \big (\frac {2}{\pi e N}\big )^2$ and we characterize its minimizers.

Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies in three and four dimensions

Uniqueness of ground states for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent at high frequencies in three and four dimensions

Global behavior of the ground state energy of the nonlinear scalar field equation

We study the global behavior of the ground state energy for the nonlinear scalar field equation (including the $p$-Laplacian version), and give a complete description of this energy in terms of nonlinearity power. More precisely,let$\mathcal{E}_{\lambda,q}(u)=\frac{1}{p}(\|\nabla~u\|_p^p+\lambda\|u\|_p^p)-\frac{1}{q}\|u\|_q^q$ bethe energy functional in $~W^{1,p}(\mathbb{R}^N)$,where $\lambda&gt;0$, $1&lt;p&lt;N$, $p&lt;q&lt;p^*:=\frac{pN}{N-p}$, and $p^*$ is the critical Sobolev exponent. It is known that $\mathcal{E}_{\lambda,q}(u)$ has a unique radially symmetric mountain-pass critical point $U_{\lambda,q}$, which is called the ground state 解 to the corresponding nonlinear scalar field equation and whose energy $\mathcal{E}_{\lambda,q}(U_{\lambda,q})$ is called the ground state energy $\mathfrak{m}(\lambda,q)$.We show that there is a decreasing function $\lambda_0(q)$ defined in $q\in~[p,p^*]$ with $\lambda_0(p)=1~$ and $\lambda_0(p^*)=0$such that $\mathfrak{m}(\lambda,r)$ is strictly increasing for $r\in(p,~q)$ with $\lambda\in(0,~\lambda_0(q))$ being fixed, $\mathfrak{m}(\lambda,r)$ is strictly decreasing for $r\in(q,~p^*)$ with $\lambda\in(\lambda_0(q),~1)$ being fixed, and $\mathfrak{m}(\lambda,r)$ is strictly decreasing for all $r\in(p,~p^*)$ with $\lambda\in[1,~+\infty)$ being fixed.We also deduce the precise asymptotic behavior of $\lambda_0(q)$ as $q\to~p$ and $q\to~p^*$. This is done by establishing a relation between power-law scalar fieldequations and logarithmic-law scalar field equations,which is of the independent interest.

Non-existence of ground states and gap of variational values for 3D Sobolev critical nonlinear scalar field equations

In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists (see Theorems 1.1 and 1.2). Furthermore, we prove the difference between two typical variational problems used to characterize the ground states (see Theorem 1.3).

An inhomogeneous perturbation for a class of nonlinear scalar field equations

This paper deals with a class of nonlinear scalar field equations with an inhomogeneous perturbation. Two positive solutions were obtained using the variational methods.

Soliton-like solutions of nonlinear scalar and electromagnetic field equations in gravitational theory

In this work, exact analytical static spherical symmetric solutions to the nonlinear interacting electromagnetic and massless scalar field equations have been determined taking into account the proper gravitational field of elementary particles. The obtained results prove that all solutions of the Einstein equation and those of the interacting fields are regular with localized energy density. All non-zero components of the 4-vector potential are solutions to the inverted Painlevé-Gambier XI equation. Moreover, the total energy of interacting fields is limited and the total charge of elementary particles is finite. These solutions obtained are soliton-like and can be used as a model to describe the internal structure of elementary particles.

Exact Static Plane Symmetric Soliton-Like Solutions to the Nonlinear Interacting Electromagnetic and Scalar Field Equations in General Relativity

This research work is related to soliton solutions considered as models that can describe the complex configuration of elementary particles from the study of the interactions of their fields. It is interested in the interaction of fields between two different elementary particles by expressing their physical properties. For that, we have obtained, exact static plane symmetric soliton-like solutions to the nonlinear equations of interacting electromagnetic and scalar fields taking into account the own gravitational field of elementary particles using the calibrated invariance function P(I). It has been proved that all solutions of the Einstein, nonlinear electromagnetic and scalar field equations are regular with the localized energy density. Moreover, the total charge of particles is finite and the total energy of the interaction fields is bounded. It have been emphasized the importance to the own gravitational field of elementary particles and the role of the nonlinearity of fields in the determination of these solutions. In flat space-time, soliton-like solutions exist but the total energy of the interaction fields is equal to zero. We have also shown that in the linear case, soliton-like solutions are absent.

Nonlinear scalar field equation with competing nonlocal terms * *d’Avenia and Pomponio are members of GNAMPA of INdAM and are partially supported by PRIN 2017JPCAPN Qualitative and quantitative aspects of nonlinear PDEs. d’Avenia is also supported by FRA2019 of Politecnico di Bari. Mederski was partially supported by the National Science Centre, Poland, Grant No. 2017/26/E/ST1/00817 and by the Deutsche Forschungsgemeinschaft

We find radial and nonradial solutions to the following nonlocal problem under general assumptions, in the spirit of Berestycki and Lions, imposed on f and g, where N ⩾ 3, 0 ⩽ β ⩽ α < N, ω ⩾ 0, are continuous functions with corresponding primitives F, G, and I α , I β are the Riesz potentials. If β > 0, then we deal with two competing nonlocal terms modelling attractive and repulsive interaction potentials.

Normalized solutions for fractional nonlinear scalar field equations via Lagrangian formulation

We study existence of solutions for the fractional problem where N ⩾ 2, s ∈ (0, 1), m > 0, μ is an unknown Lagrange multiplier and satisfies Berestycki–Lions type conditions. Using a Lagrangian formulation of the problem (P m ), we prove the existence of a weak solution with prescribed mass when g has L 2 subcritical growth. The approach relies on the construction of a minimax structure, by means of a Pohozaev’s mountain in a product space and some deformation arguments under a new version of the Palais–Smale condition introduced in Hirata and Tanaka (2019 Adv. Nonlinear Stud. 19 263–90); Ikoma and Tanaka (2019 Adv. Differ. Equ. 24 609–46). A multiplicity result of infinitely many normalized solutions is also obtained if g is odd.

Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities

We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp decay estimates and asymptotics in the presence of a critically dispersing constant coefficient cubic nonlinearity. The Klein-Gordon models considered in this paper are motivated by the study of the asymptotic stability of kink solutions to classical nonlinear scalar field equations on the real line.

General class of optimal Sobolev inequalities and nonlinear scalar field equations

We find a class of optimal Sobolev inequalities(∫RN|∇u|2dx)NN−2≥CN,G∫RNG(u)dx,u∈D1,2(RN),N≥3, where the nonlinear function G:R→R of class C1 satisfies general growth assumptions in the spirit of the fundamental works of Berestycki and Lions. We admit, however, a wider class of problems involving zero, positive and infinite mass cases as well as G need not be even. We show that any minimizer is radial up to a translation. Moreover, up to a dilation, it is a least energy solution of the nonlinear scalar field equation−Δu=g(u)in RN,with g=G′. In particular, if G(u)=u2log⁡|u|, then the sharp constant is CN,G:=2⁎(N2)2N−2e2(N−1)N−2(π)NN−2 and uλ(x)=eN−12−λ22|x|2 with λ>0 constitutes the whole family of minimizers up to translations. The optimal inequality provides a new proof of the classical logarithmic Sobolev inequality based on a Pohozaev manifold approach. Moreover, if N≥4, then there is at least one nonradial solution and if, in addition, N≠5, then there are infinitely many nonradial solutions of the nonlinear scalar field equation. The energy functional associated with the problem may be infinite on D1,2(RN) and is not Fréchet differentiable in its domain. We present a variational approach to this problem based on a new variant of Lions' lemma in D1,2(RN).

Nonradial solutions of nonlinear scalar field equations

We prove new results concerning the nonlinear scalar field equation with a nonlinearity g satisfying the general assumptions due to Berestycki and Lions. In particular, we find at least one nonradial solution for any N ⩾ 4 minimizing the energy functional on the Pohozaev constraint in a subspace of consisting of nonradial functions. If in addition N ≠ 5, then there are infinitely many nonradial solutions. These solutions are sign-changing. The results give a positive answer to a question posed by Berestycki and Lions in [, ]. Moreover, we build a critical point theory on a topological manifold, which enables us to solve the above equation as well as to treat new elliptic problems.

A mass supercritical problem revisited

In any dimension $N\geq1$ and for given mass $m>0$, we revisit the nonlinear scalar field equation with an $L^2$ constraint: $$ -\Delta u=f(u)-\mu u, \quad u \in H^1(\mathbb{R}^N) \quad \text{with} \quad \|u\|^2_{L^2(\mathbb{R}^N)}=m. $$ where $\mu\in\mathbb{R}$ will arise as a Lagrange multiplier. Assuming only that the nonlinearity $f$ is continuous and satisfies weak mass supercritical conditions, we show the existence of ground states and reveal the basic behavior of the ground state energy $E_m$ as $m>0$ varies. In particular, to overcome the compactness issue when looking for ground states, we develop robust arguments which we believe will allow treating other $L^2$ constrained problems in general mass supercritical settings. Under the same assumptions, we also obtain infinitely many radial solutions for any $N\geq2$ and establish the existence and multiplicity of nonradial sign-changing solutions when $N\geq4$. Finally we propose two open problems.