Nodal Solutions Research Articles

The nodal solution for a problem involving the logarithmic and exponential nonlinearities

ABSTRACT In this work, we study the existence of the least energy sign-changing solution for the following degenerate Kirchhoff-type problem involving the fractional N/s-Laplacian with logarithmic and both subcritical and critical exponential nonlinearities: { M ( ∫ R 2 N | u ( x ) − u ( y ) | N s | x − y | 2 N d x d y ) ( − Δ ) N / s s u = | u | q − 2 uln ⁡ | u | 2 + μf ( u ) , in Ω , u = 0 , in R N ∖ Ω , where Ω ⊂ R N is a bounded domain. The proof is based on constrained variational method, fractional Trudinger–Moser inequality, quantitative deformation lemma and Brouwer's degree theory in Nehari sets. To be more precise, the least energy sign-changing solution is obtained by minimizing the energy functional on the sign-changing Nehari manifold.

Positive and nodal solutions for parametric superlinear weighted [formula omitted]-equations

We consider a nonlinear eigenvalue problem for equations driven by a weighted (p,q)-Laplacian and a superlinear reaction. We prove a global (with respect to the parameter λ>0) existence and multiplicity result. We also generate nodal (sign-changing) solutions. Finally, we determine the topological properties of the solution set and prove the continuity properties of the solution multifunction.

A Multigroup Homogeneous Flux Reconstruction Method Based on the ANOVA-HDMR Decomposition

In this paper, we formulate a new flux reconstruction method based on the High Dimensional Model Representation (HDMR) of the nodal flux shape. The method requires the conventional nodal parameters obtained as a result of the transverse-integration nodal solution procedure as well as corner values calculated by a combination of finite-volume and finite-difference approximations. The proposed flux reconstruction procedure is applicable to rectangular nodes and more than two energy groups. The flux reconstruction method is applied to a number of homogeneous and heterogeneous test problems in order to evaluate its accuracy.

An abstract sub-supersolution method in partially ordered spaces

We present a general framework in which sub-supersolution techniques can be applied. Specifically, we consider a normed vector space endowed with a partial order. This allows us to introduce a notion of sub- and supersolutions relative to an equation driven by operators defined from the space to its topological dual. We provide sufficient conditions to guarantee that the interval determined by an ordered pair of sub-supersolutions contains a solution of the initial problem. We also study existence of extremal positive or negative solutions and location of nodal solutions. Examples of applications to quasilinear boundary value problems are given.

Ground state and nodal solutions for fractional Orlicz problems with lack of regularity and without the Ambrosetti-Rabinowitz condition

We consider a non-local Shrödinger problem driven by the fractional Orlicz g-Laplace operator as follows(P)(−△g)αu+g(u)=K(x)f(x,u),inRd, where d≥3,(−△g)α is the fractional Orlicz g-Laplace operator, f:Rd×R→R is a measurable function and K is a positive continuous function. Employing the Nehari manifold method and without assuming the well-known Ambrosetti-Rabinowitz and differentiability conditions on the non-linear term f, we prove that the problem (P) has a ground state of fixed sign and a nodal (or sign-changing) solutions.

Bifurcation of nodal solutions for the Moore–Nehari differential equation

Bifurcation of nodal solutions for the Moore–Nehari differential equation

Fractional delay integrodifferential equations of nonsingular kernels: Existence, uniqueness, and numerical solutions using Galerkin algorithm based on shifted Legendre polynomials

This paper considers linear and nonlinear fractional delay Volterra integrodifferential equation of order [Formula: see text] in the Atangana–Beleanu–Caputo (ABC) sense. We used continuous Laplace transform (CLT) to find equivalent Volterra integral equations that have been used together with the Arzela–Ascoli theorem and Schauder’s fixed point theorem to prove the local existence solution. Moreover, the obtained Volterra integral equations and the contraction mapping theorem have been successfully applied to construct and prove the global existence and uniqueness of the solution for the considered fractional delay integrodifferential equation (FDIDE). The Galerkin algorithm instituted within shifted Legendre polynomials (SLPs) is applied in the approximation procedure for the corresponding delay equation. Indeed, by this algorithm, we get algebraic system models and by solving this system we gained the approximated nodal solution. The reliability of the method and reduction in the size of the computational work give the algorithm wider applicability. Linear and nonlinear examples are included with some tables and figures to show the effectiveness of the method in comparison with the exact solutions. Finally, some valuable notes and details extracted from the presented results were presented in the last part, with the sign to some of our future works.

Nodal solutions for an asymptotically linear Kirchhoff-type problem in ℝ N

ABSTRACT In this paper, we are concerned with nodal solutions for a class of Kirchhoff-type problems where , a, b>0, f satisfies some asymptotically linear growth conditions. First of all, when N = 3, b>0 and , b>0 sufficiently small, we attain infinitely many nodal solutions by an equivalent transformation. Based on this, via a new and direct approach, the least energy sign-changing radial solutions can be obtained for the above problem. What's more, we also establish non-existence results of nodal solutions for and b large enough.

An inequality of Bessel functions and applications to transcritical bifurcation problems of nonlinear elliptic equations

We prove a weighted integral inequality of the Bessel function of zero order, which implies the positivity of the integral of the cube of nodal radial eigenfunctions of the Laplacian operator in the planar disk. This solves an open problem in Bonheure et al. (2016) and Corsato et al. (2020). The positivity plays a crucial role in transcritical bifurcation problems of nodal radial solutions of nonlinear elliptic equations.

Nodal Solutions for Quasilinear Schrödinger Equations with Asymptotically 3-Linear Nonlinearity

In this paper, we are concerned with the quasilinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u-u\Delta(u^2)=g(u),\ \ x\in \mathbb{R}^{N}, \end{equation*} where $N\geq3$, $V$ is radially symmetric and nonnegative, and $g$ is asymptotically 3-linear at infinity. In the case of $\inf_{\mathbb{R}^N}V>0$, we show the existence of a least energy sign-changing solution with exactly one node, and for any integer $k>0$, there are a pair of sign-changing solutions with $k$ nodes. Moreover, in the case of $\inf_{\mathbb{R}^N}V=0$, the problem above admits a least energy sign-changing solution with exactly one node. The proof is based on variational methods. In particular, some new tricks and the method of sign-changing Nehari manifold depending on a suitable restricted set are introduced to overcome the difficulty resulting from the appearance of asymptotically 3-linear nonlinearities.

A new type of nodal solutions to singularly perturbed elliptic equations with supercritical growth

In this paper, we aim to investigate the following class of singularly perturbed elliptic problem{−ε2Δu+|x|ηu=|x|ηf(u)inA,u=0on∂A, where ε>0, η∈R, A={x∈R2N:0<a<|x|<b}, N≥2 and f is a nonlinearity of C1 class with supercritical growth. By a reduction argument, we show that there exists a nodal solution uε with exactly two positive and two negative peaks, which concentrate on two different orthogonal spheres of dimension N−1 as ε→0. In particular, we establish different concentration phenomena of four peaks when the parameter η>2, η=2 and η<2.

Pin power reconstruction for hexagonal geometry in nodal neutronics program Ants

The pin power reconstruction methodology for hexagonal geometry in the multigroup nodal neutronics program Ants is described and verified against full core Serpent Monte Carlo solutions in four VVER benchmarks. In order to provide the best possible reference solution, Serpent uses the same nuclear data libraries and simulation options both to generate the required group constants for Ants and to solve the full core reference solution. The root mean square of the relative differences in 2D pin powers between Ants and Serpent was between 0.83% and 1.11% in the four benchmarks, while the root mean square of the relative differences in the 2D intra-assembly peaking factor (Kk) was between 0.54% and 0.71%. Increasing the number of energy groups in the nodal solution tended improve the accuracy of the nodal solution as did the use of critical spectrum condensed group constant data instead of infinite spectrum group constants.

Crack analysis using a hybrid numerical manifold method with node-based strain smoothing and double-interpolation

In this paper, a hybrid numerical manifold method with node-based strain smoothing and double-interpolation is proposed for crack analysis and propagation. By introducing an allocation scheme, the empty strain smoothing domain issue encountered in the numerical manifold method framework is avoided, while the flexibility in smooth domain formulation is preserved. Our method’s hybrid approach stems from the double-interpolation locally implemented on the tip domain, where strain smoothing becomes cumbersome and costly. High accuracy and continuous nodal stress solutions were achieved in whole domains. The proposed model is validated through several comparisons of analytical and experimental results, and the results obtained for crack analysis and growths are shown to be accurate and efficient.

Localized nodal solutions for semiclassical nonlinear Kirchhoff equations

Localized nodal solutions for semiclassical nonlinear Kirchhoff equations

Revisiting viscous transonic decretion disks of Be stars

Aims. In the context of Be stars, we re-studied the viscous transonic decretion disk model of these stars. This model is driven by a radiative force due to an ensemble of optically thin lines and viscosity considering the Shakura–Sunyaev prescription. Methods. The nonlinear equation of motion presents a singularity (sonic point) and an eigenvalue, which is also the initial condition at the stellar surface. Then, to obtain this eigenvalue, we set it as a radial quantity and performed a detailed topological analysis. We describe a numerical method for solving either nodal or saddle transonic solutions. Results. The value of the viscosity α barely determines the location of the sonic point, but it determines the topology of the solution. We found two nodal solutions, which are almost indistinguishable. Saddle solutions were found for lower values of α than required for the nodal solutions. In addition, rotational velocity does not play a determinate role in the velocity (and density) profile, because the viscosity effects collapse all the solutions to almost a unique one in a small region above the stellar surface. Conclusions. A suitable combination of line-force parameters and/or disk temperature gives the location of the sonic point lower than 50 stellar radii, describing a truncated disk. This could explain the SED turn-down observed in Be stars without needing a binary companion.

Saddle solutions for the Choquard equation with a general nonlinearity

In the spirit of Berestycki and Lions (Arch. Rational Mech. Anal., 82: 313–345, 1983), we prove the existence of saddle-type nodal solutions for the Choquard equation $$\begin{aligned} -\Delta u + u= \big (I_\alpha *F(u)\big )F'(u)\qquad \text { in }\;\mathbb {R}^N \end{aligned}$$ where $$N\ge 2$$ and $$I_\alpha$$ is the Riesz potential of order $$\alpha \in (0,N)$$ . Given a finite Coxeter group G with rank $$k\le N$$ , we construct a G-groundstate uniformly with lowest energy amongst G-saddle solutions for the Choquard equation in a noncompact setting. Moreover, if $$F'$$ is odd and has constant sign on $$(0,+\infty )$$ , then every G-groundstate maintains signed on the fundamental domain of the corresponding Coxeter group and receives opposite signs on any two adjacent regions so that nodal domains of G-groundstate are of cone shapes demonstrating Coxeter’s symmetric configurations in $$\mathbb {R}^N$$ . These results further complete the variational framework in constructing sign-changing solutions for the Choquard equation but still require a quadratic or super-quadratic growth on F near the origin.

Global Structure of a Nodal Solutions Set of Mean Curvature Equation in Static Spacetime

Global Structure of a Nodal Solutions Set of Mean Curvature Equation in Static Spacetime

An adaptive tailored finite point method for the generalized Burgers’ equations

A tailored finite point method, using a minimal machinery algorithm yet utilizing the initial conditions and local properties of the solution to the hilt, is proposed to serve as a global platform to solve the generalized Burgers’ equation. On an explicit centred 4-point stencil, the nodal solutions at the advanced time level are written as a linear combination of the nodal solutions at the preceding level. The scalars in this linear combination are determined using a set of basis functions. The extraction of these basis functions is through the fundamental solutions derived via the method of separation of variables. This, in turn, brings in the influence of the local properties of the general solution. The nodal maximum of the initial conditions is chosen as the separation constant to actuate continuous dependence on the initial conditions. Withal this separation constant works well for the generalized Burgers’ equation, thereby instituting a common platform to solve these class of equations. The non-linearity is taken care of through an iterative technique where the non-linear term is replaced by an anterior temporal level iterated value. Conditional stability is established through the von-Neumann stability analysis. The method is consistent with second-order convergence in spatial variables and first-order convergence in the temporal variable. Numerical experiments are conducted on multifarious examples, and the obtained results are very much in accordance with the available exact solutions. The numerical results of examples with no useable closed-form representation of the exact solution are vindicated through the double meshing principle. Also, the error analysis establishes that the method works pretty well on coarse meshes, wherefore cutting the computational cost and increasing rapidity. Despite being a simple and straightforward algorithm with no usage of elite techniques, the method stands on par with quite a few methods in the literature.

Arbitrarily small nodal solutions for nonhomogeneous Robin problems

In the present paper, we consider a nonlinear Robin problem driven by a nonhomogeneous differential operator and with a reaction which is only locally defined. Using cut-off techniques and variational tools, we show that the problem has a sequence of nodal solutions converging to zero in C 1 ( Ω ‾ ).

Multiple nodal and semi-nodal solutions to a nonlinear Choquard-type system

We study the nonlinearly coupled Choquard-type system(0.1){−Δu1+μ1u1=a1(Iα⁎|u1|p)|u1|p−2u1+β(Iα⁎|u2|p)|u1|p−2u1,x∈Ω,−Δu2+μ2u2=a2(Iα⁎|u2|p)|u2|p−2u2+β(Iα⁎|u1|p)|u2|p−2u2,x∈Ω,u1=u2=0on∂Ω, where Ω is a bounded smooth domain in RN with N≥3, p∈(N+αN,N+αN−2), α∈(0,N), Iα is the Riesz potential, and μ1,μ2,a1,a2,β are positive constants. For every k∈N, we prove that there exists βk>0 such that system (0.1) possesses k nodal solutions and k semi-nodal solutions for β∈(0,βk) and p>2. Additionally, the existence of least energy nodal solutions is also obtained.