Positive Energy Solutions Research Articles

A concave–convex Kirchhoff type elliptic equation involving the fractional p-Laplacian and steep well potential

In this paper, we study the following Schrödinger–Kirchhoff type equation: , where , , N>ps, and , are three real parameters. By using the mountain pass theorem and a variant version of Ekeland's variational principle in Zhong [A generalization of Ekeland's variational principle and application to the study of relation between the weak P.S. condition and coercivity. Nonlinear Anal. 1997;29:1421–1431], we get some results. Firstly, we obtain a positive energy solution by a truncated functional and discuss their asymptotical behavior for when p<r<2p and b lies in suitable range. Secondly, for all b>0, we obtain a positive energy solution and discuss their asymptotical behavior for when , where . Moreover, we obtain other asymptotical behavior of positive energy solution when . Finally, we get a negative energy solution and the non-existence of the non-trivial solutions.

Exponential growth of positive initial energy solutions for coupled nonlinear Klein-Gordon equations with degenerate damping and source terms

In this paper we will prove that the positive initial-energy solution for coupled nonlinear Klein-Gordon equations with degenerate damping and source terms grows exponentially.

Existence of least energy positive solutions to critical Schrödinger systems in [formula omitted

In this paper, we consider the following coupled Schrödinger system with critical exponent: −Δu+λ1u=μ1|u|4u+β|v|3|u|uinΩ,−Δv+λ2v=μ2|v|4v+β|u|3|v|vinΩ,u=v=0on∂Ω.Here, Ω⊂R3 is a smooth bounded domain, −λ1(Ω)<λ1,λ2<−λ∗(Ω),μ1,μ2>0, and β>0, where λ1(Ω) is the first eigenvalue of −Δ with Dirichlet boundary condition and λ∗(Ω)∈(0,λ1(Ω)). By a variational method, we establish the existence of least energy positive solutions for positive small β.

Positive least energy solutions for k-coupled Schrödinger system with critical exponent: the higher dimension and cooperative case

In this paper, we study the following k-coupled nonlinear Schrödinger system with Sobolev critical exponent: $$\begin{aligned} \left\{ \begin{aligned} -\Delta u_i&+\lambda _iu_i =\mu _i u_i^{2^*-1}+\sum _{j=1,j\ne i}^{k} \beta _{ij} u_{i}^{\frac{2^*}{2}-1}u_{j}^{\frac{2^*}{2}} \quad \hbox {in}\;\Omega ,\\ u_i&>0 \quad \hbox {in}\; \Omega \quad \hbox {and}\quad u_i=0 \quad \hbox {on}\;\partial \Omega , \quad i=1,2,\dots , k. \end{aligned} \right. \end{aligned}$$Here \(\Omega \subset {{\mathbb {R}}}^N \) is a smooth bounded domain, \(2^{*}=\frac{2N}{N-2}\) is the Sobolev critical exponent, \(-\lambda _1(\Omega )<\lambda _i<0, \mu _i>0\) and \( \beta _{ij}=\beta _{ji}\ne 0\), where \(\lambda _1(\Omega )\) is the first eigenvalue of \(-\Delta \) with the Dirichlet boundary condition. We characterize the positive least energy solution of the k-coupled system for the purely cooperative case \(\beta _{ij}>0\), in higher dimension \(N\ge 5\). Since the k-coupled case is much more delicate, we shall introduce the idea of induction. We point out that the key idea is to give a more accurate upper bound of the least energy. It is interesting to see that the least energy of the k-coupled system decreases as k grows. Moreover, we establish the existence of positive least energy solution of the limit system in \({\mathbb {R}}^N\), as well as classification results.

On the nonlocal Schrödinger‐poisson type system in the Heisenberg group

This paper is concerned with a class of nonlocal Schrödinger‐Poisson type system with sublinear term. With the aid of the Ekeland's variational principle and the mountain pass theorem, the existence of negative energy solution, positive energy solution, and positive ground state solution is obtained, respectively. Moreover, we also obtain the multiplicity of solutions by using the Clark theorem. The novelty of this paper is that the equation has not only nonlocal term but also nonlocal coefficient.

Existence Results for p1(x,·) and p2(x,·) Fractional Choquard–Kirchhoff Type Equations with Variable s(x,·)-Order

In this article, we study a class of Choquard–Kirchhoff type equations driven by the variable s(x,·)-order fractional p1(x,·) and p2(x,·)-Laplacian. Assuming some reasonable conditions and with the help of variational methods, we reach a positive energy solution and a negative energy solution in an appropriate space of functions. The main difficulties and innovations are the Choquard nonlinearities and Kirchhoff functions with the presence of double Laplace operators involving two variable parameters.

Existence and asymptotic behavior of positive least energy solutions for coupled nonlinear Choquard equations

In this article, we study the coupled nonlinear Schrodinger equations with Choquard type nonlinearities $$\displaylines{ -\Delta u+\nu_1u=\mu_1(\frac{1}{|x|^{\alpha}} *u^2)u +\beta (\frac{1}{|x|^{\alpha}} *v^2)u \quad\hbox{in } \mathbb{R}^{N},\cr -\Delta v+\nu_2v=\mu_2(\frac{1}{|x|^{\alpha}} *v^2)v + \beta (\frac{1}{|x|^{\alpha}} *u^2)v \quad\hbox{in } \mathbb{R}^{N},\cr u,v \geq 0\quad \hbox{in } \mathbb{R}^{N}, \quad u,v \in H^{1}(\mathbb{R}^{N}), }$$ where \(\nu_1,\nu_2,\mu_1,\mu_2\) are positive constants, \(\beta&gt;0\) is a coupling constant, \(N\geq 3\), \(\alpha\in(0,N)\cap (0,4)\), and ``*'' is the convolution operator We show that the nonlocal elliptic system has a positive least energy solution for positive small \(\beta\) and positive large \(\beta\) via variational methods. For the case in which \(\nu_1=\nu_2\), \(\mu_1\neq\mu_2\), \(N=3,4,5\) and \(\alpha=N-2\), we prove the uniqueness of positive least energy solutions. Moreover, the asymptotic behaviors of the positive least energy solutions as \(\beta\to 0^{+}\) are studied. For more information see https://ejde.math.txstate.edu/Volumes/2021/47/abstr.html

Positive solutions to the sublinear Lane-Emden equation are isolated

We prove that on a smooth bounded set, the positive least energy solution of the Lane-Emden equation with sublinear power is isolated. As a corollary, we obtain that the first eigenvalue of the Dirichlet-Laplacian is not an accumulation point of the spectrum, on a smooth bounded set. Our results extend to a suitable class of Lipschitz domains, as well.

Existence of positive solution for the Choquard equation in exterior domain

In this paper, we study the nonlinear Choquard equation where is an unbounded domain, is bounded, , p = 2 if N = 3, 4, 5 or if N = 3. By global compactness analysis, we show that the problem has at least one positive high energy solution.

High energy positive solutions for a coupled Hartree system with Hardy-Littlewood-Sobolev critical exponents

We study the coupled Hartree system{−Δu+V1(x)u=α1(|x|−4⁎u2)u+β(|x|−4⁎v2)uinRN,−Δv+V2(x)v=α2(|x|−4⁎v2)v+β(|x|−4⁎u2)vinRN, where N≥5, β>max⁡{α1,α2}≥min⁡{α1,α2}>0, and V1,V2∈LN/2(RN)∩Lloc∞(RN) are nonnegative potentials. This system is critical in the sense of the Hardy-Littlewood-Sobolev inequality. For the system with V1=V2=0 we employ moving sphere arguments in integral form to classify positive solutions and to prove the uniqueness of positive solutions up to translation and dilation, which is of independent interest. Then using the uniqueness property, we establish a nonlocal version of the global compactness lemma and prove the existence of a high energy positive solution for the system assuming that |V1|LN/2(RN)+|V2|LN/2(RN)>0 is suitably small.

Homoclinic solutions for perturbed discrete Schrödinger systems in infinite lattices: Sublinear and asymptotically linear cases

When the nonlinearities are sublinear or asymptotically linear at infinity, we obtain the existence of negative energy solutions for a class of non-periodic discrete Schrödinger systems with perturbed terms. Besides, for the asymptotically linear case, we obtain the existence of positive energy solutions. We also illustrate our results by giving several examples.

Relativistic Brueckner-Hartree-Fock Theory in Infinite Nuclear Matter

On the way of a microscopic derivation of covariant density functionals, the first complete solution of the relativistic Brueckner-Hartree-Fock (RBHF) equations is presented for symmetric nuclear matter. In most of the earlier investigations, theG-matrix is calculated only in the space of positive energy solutions. On the other side, for the solution of the relativistic Hartree-Fock (RHF) equations, also the elements of this matrix connecting positive and negative energy solutions are required. So far, in the literature, these matrix elements are derived in various approximations. We discuss solutions of the Thompson equation for the full Dirac space and compare the resulting equation of state with those of earlier attempts in this direction.

Positive least energy solutions for k-coupled critical systems involving fractional Laplacian

In this paper, we study the following \begin{document}$ k $\end{document} -coupled critical system: \begin{document}$ \begin{equation*} \left\{ \begin{aligned} (-\Delta)^s u_i +\lambda_iu_i & = \mu_i u_i^{2^*-1}+\sum\limits_{j = 1,j\ne i}^{k} \beta_{ij} u_{i}^{\frac{2^*}{2}-1}u_{j}^{\frac{2^*}{2}} \quad \hbox{in}\;\Omega, u_i & = 0 \quad \hbox{in}\;\; \; \mathbb R^N\backslash\Omega, \quad i = 1,2,\cdots, k. \end{aligned} \right. \end{equation*} $\end{document} Here \begin{document}$ (-\Delta)^s $\end{document} is the fractional Laplacian operator, \begin{document}$ 0 , \begin{document}$ 2^{*} = \frac{2N}{N-2s} $\end{document} is a fractional Sobolev critical exponent, \begin{document}$ N>2s $\end{document} , \begin{document}$ - \lambda_s( \Omega) 0 $\end{document} , \begin{document}$ \beta_{ij} = \beta_{ji}\ne 0 $\end{document} and \begin{document}$ \Omega\subset {\mathbb R}^N $\end{document} is a smooth bounded domain, where \begin{document}$ \lambda_s( \Omega) $\end{document} is the first eigenvalue of \begin{document}$ (-\Delta)^{s} $\end{document} with the homogeneous Dirichlet boundary datum. We characterize the positive least energy solution of the \begin{document}$ k $\end{document} -coupled fractional critical system for the purely cooperative case \begin{document}$ \beta_{ij}>0 $\end{document} with \begin{document}$ N> 4s $\end{document} . We shall introduce the idea of induction to prove our results. We point out that the key idea is to give a more accurate upper bound of the least energy. It's interesting to see that the least energy of the \begin{document}$ k $\end{document} -coupled system decreases as \begin{document}$ k $\end{document} grows. Moreover, we establish the existence of positive least energy solution of the limit system in \begin{document}$ \mathbb R^N $\end{document} , as well as classification results. Meanwhile, we also construct a positive solution for a more general system involving subcritical items. Besides, we investigated in the asymptotic behaviour of the positive least energy solutions of the critical system. We point out that the results of the fractional critical systems have some coincidences with those of the critical Schrodinger systems.

Positive vortex solutions and phase separation for coupled Schrodinger system with singular potential

We consider the existence of rotating solitary waves (vortices) for a coupled Schrodinger equations by finding solutions to the singular system $$\displaylines{ -\Delta u+\lambda_1 u+\frac{u}{|x|^2}=\mu_1 u^3+\beta u v^2, \quad x\in\mathbb{R}^2, \cr -\Delta v+\lambda_2 v+\frac{v}{|x|^2}=\mu_2 v^3+\beta u^2 v, \quad x\in\mathbb{R}^2, \cr u,v \geq 0,\quad x\in\mathbb{R}^2, }$$ where \(\lambda_1,\lambda_2,\mu_1, \mu_2\) are positive parameters, \(\beta\neq 0\). We show that this system has a positive least energy solution for the cases When either \(\beta\) is negative or \(\beta\) is positive and small or large. Moreover, if \(\lambda_1=\lambda_2\), then the solution is unique. We also study the limiting behavior of the least energy solutions in the repulsive case for \(\beta\to-\infty\), and phase separation. For more information see https://ejde.math.txstate.edu/Volumes/2020/108/abstr.html

Fast decaying ground states for elliptic equations with exponential nonlinearity

We show the existence of positive minimal energy solutions for the equation −div(K(x)∇u)=K(x)f(u) in R2, where K(x)=exp(|x|2∕4) and f:R→R is a superlinear continuous function with exponential subcritical or exponential critical growth. We use as a main tool the Nehari manifold method.

Nonlinear flux “concave–convex” problems: a fibering method approach

This paper studies the nonlinear flux problem: where $$\varDelta _p$$ stands for the p-Laplacian operator, $$\varOmega \subset {\mathbb {R}}^N$$ is a bounded smooth domain, $$\lambda$$ is a positive parameter and $$\nu$$ stands for the outer unit normal at $$\partial \varOmega$$ . The exponents q, r are assumed to vary in the concave convex regime $$1< q< p < r$$ while $$1< p < N$$ and r is subcritical $$r < p^*$$ . Our objective here is showing the existence, for every $$0< \lambda < {{\bar{\lambda }}}$$ , of two different sets of infinitely many solutions of (P). The energy functional associated to the problem exhibits a different sign on each of these sets. The analysis of positive energy solutions involves the so-called fibering method (Drabek and Pohozaev in Proc R Soc Edinb Sect A 127(4):703–726, 1997). Our results have been inspired by similar ones in Garcia-Azorero et al. (J Differ Equ 198(1):91–128, 2004), Garcia-Azorero and Peral (Trans Am Math Soc 323(2):877–895, 1991) and El Hamidi (Commun Pure Appl Anal 3(2):253–265, 2004). This work can be considered as a natural continuation of Sabina de Lis (Differ Equ Appl 3(4):469–486, 2011), Sabina de Lis and Segura de Leon (Adv Nonlinear Stud 15(1):61–90, 2015) and Sabina de Lis and Segura de Leon (Nonlinear Anal 113:283–297, 2015). The main achievement of the latter of these works consisted in showing a global existence result of positive solutions to (P).

Existence of least energy positive solutions to Schrödinger systems with mixed competition and cooperation terms: the critical case

In this paper we investigate the existence of solutions to the following Schrodinger system in the critical case $$\begin{aligned} -\Delta u_{i}+\lambda _{i}u_{i}=u_{i}\sum _{j = 1}^{d}\beta _{ij}u_{j}^{2} \text { in } \Omega , \quad u_i=0 \text { on } \partial \Omega , \qquad i=1,\ldots ,d. \end{aligned}$$Here, $$\Omega \subset {\mathbb {R}}^{4}$$ is a smooth bounded domain, $$d\ge 2$$, $$-\lambda _{1}(\Omega ) 0$$ for every i, $$\beta _{ij}=\beta _{ji}$$ for $$i\ne j$$, where $$\lambda _{1}(\Omega )$$ is the first eigenvalue of $$-\Delta $$ with Dirichlet boundary conditions. Under the assumption that the components are divided into m groups, and that $$\beta _{ij}\ge 0$$ (cooperation) whenever components i and j belong to the same group, while $$\beta _{ij}<0$$ or $$\beta _{ij}$$ is positive and small (competition or weak cooperation) for components i and j belonging to different groups, we establish the existence of nonnegative solutions with m nontrivial components, as well as classification results. Moreover, under additional assumptions on $$\beta _{ij}$$, we establish existence of least energy positive solutions in the case of mixed cooperation and competition. The proof is done by induction on the number of groups, and requires new estimates comparing energy levels of the system with those of appropriate sub-systems. In the case $$\Omega ={\mathbb {R}}^4$$ and $$\lambda _1=\cdots =\lambda _m=0$$, we present new nonexistence results. This paper can be seen as the counterpart of Soave and Tavares (J Differ Equ 261:505–537, 2016) in the critical case, while extending and improving some results from Chen and Zou (Arch Ration Mech Anal 205:515–551, 2012) and Guo et al. (Nonlinearity 31:314–339, 2018).

Global existence and blow up of positive initial energy solution of a quasilinear wave equation with nonlinear damping and source terms

In this paper, we consider a quasilinearwave equation having nonlinear damping and source terms and obtained global existence and blow up results under certain polynomial growth conditions on the nonlinear functions σi, βi, (i = 1, 2, ..., n), f and g. We obtain global existence result for positive initial energy solution using Galerkin approximation procedure and nonexistence (blow up) result using the technique introduced by Georgiev and Todorova (1994) with little modification for our problem.

Global existence and blow up of positive initial energy solution of a quasilinear wave equation with nonlinear damping and source terms

In this paper, we consider a quasilinearwave equation having nonlinear damping and source terms utt−Δut−∑i=1n∂∂xiσi(x, uxi)+β(x, utxi)+f(x, ut)=g(x, u) and obtained global existence and blow up results under certain polynomial growth conditions on the nonlinear functions σi, βi, (i = 1, 2, ..., n), f and g. We obtain global existence result for positive initial energy solution using Galerkin approximation procedure and nonexistence (blow up) result using the technique introduced by Georgiev and Todorova (1994) with little modification for our problem.

Helicity rotation induced by Lorentz boosts

In this paper, we calculate the helicity rotation angle induced by Lorentz boosts. This is relevant for the study of Lorentz boost effects on quantum entanglement encoded in pairs of massive fermions, which are described in terms of positive energy solutions of the Dirac equation with definite helicity. A Lorentz boost describing the change to an inertial frame moving at uniform speed will in general rotate the particle’s helicity. We obtain the coefficients of the helicity superposition in the boosted frame and specialize our results for a perpendicular boost geometry. We verify that the helicity rotation angle can be obtained in terms of the Wigner rotation angle for spin [Formula: see text] states, bridging the framework considered in our previous works to the one of the Wigner rotations. Finally, we calculate the boost-induced spin-parity entanglement for a single particle.