Nodal Solutions Research Articles

On the fractional P–Q laplace operator with weights

We exploit the existence and non-existence of positive solutions to the eigenvalue problem driven by the nonhomogeneous fractional p &q Laplacian operator with indefinite weights ( − Δ p ) α u + ( − Δ q ) β u = λ [ a | u | p − 2 u + b | u | q − 2 u ] in Ω , where Ω ⊆ R N is a smooth bounded domain that has been extended by zero. We further show the existence of a continuous family of eigenvalues in the case Ω = R N and b ≡ 0 a.e. Our approach relies strongly on variational Analysis, in which the Mountain pass theorem plays the key role. Due to the lack of spatial compactness and the embedding W α , p ( R N ) ↪ W β , q ( R N ) in R N , we employ the concentration-compactness principle of P.L. Lions [The concentration-compactness principle in the calculus of variations. The limit case. II, Rev Mat Iberoamericana. 1985;1(2):45–121]. to overcome the difficulty. Our paper can be considered as a counterpart to the important works [Alves et al. Existence, multiplicity and concentration for a class of fractional p &q Laplacian problems in R N , Commun Pure Appl Anal, 2019;18(4):2009–2045], [Benci et al. An eigenvalue problem for a quasilinear elliptic field equation. J Differ Equ, 2002;184(2):299–320], [Bobkov et al. On positive solutions for ( p , q ) -Laplace equations with two parameters, Calc Var Partial Differ Equ, 2015;54(3):3277–3301], [Colasuonno and Squassina. Eigenvalues for double phase variational integrals, Ann Mat Pura Appl (4), 2016;195(6):1917–1956], [Papageorgiou et al. Positive solutions for nonlinear Neumann problems with singular terms and convection, J Math Pures Appl (9), 2020;136:1–21], [Papageorgiou et al. Ground state and nodal solutions for a class of double phase problems, Z Angew Math Phys, 2020;71:1–15], and may have further applications to deal with other problems.

Coexistence-segregation dichotomy in the full cross-diffusion limit of the stationary SKT model

This paper studies the Lotka-Volterra competition model with cross-diffusion terms under homogeneous Dirichlet boundary conditions. We consider the asymptotic behavior of positive steady-states as equal two cross-diffusion coefficients tend to infinity (so-called the full cross-diffusion limit). The first result shows that, at the full cross-diffusion limit, the set of positive steady-state solutions can be classified into two types: the small coexistence or the complete segregation. The small coexistence can be characterized by the set of positive solutions of a nonlinear elliptic equation, while the complete segregation can be characterized by the set of nodal solutions of another nonlinear elliptic equation. The second result concerns the global bifurcation structure of the 1d model with large cross-diffusion terms to show that the branch of small coexistence bifurcates from the trivial solution and many branches of complete segregation bifurcate from solutions on the branch of small coexistence. Finally, in order to verify the theoretical results, we chase some bifurcation branches numerically via the continuation software pde2path.

Nodal method for handling irregularly deformed geometries in hexagonal lattice cores

The hexagonal nodal code RENUS has been enhanced to handle irregularly deformed hexagonal assemblies. The underlying RENUS methods involving triangle-based polynomial expansion nodal (T-PEN) and corner point balance (CPB) were extended in a way to use line and surface integrals of polynomials in a deformed hexagonal geometry. The nodal calculation is accelerated by the coarse mesh finite difference (CMFD) formulation extended to unstructured geometry. The accuracy of the unstructured nodal solution was evaluated for a group of 2D SFR core problems in which the assembly corner points are arbitrarily displaced. The RENUS results for the change in nuclear characteristics resulting from fuel deformation were compared with those of the reference McCARD Monte Carlo code. It turned out that the two solutions agree within 18 pcm in reactivity change and 0.46% in assembly power distribution change. These results demonstrate that the proposed unstructured nodal method can accurately model heterogeneous thermal expansion in hexagonal fueled cores.

Global structure of the set of 1-node solutions in a class of degenerate diffusive logistic equations

In this paper, we analyze from a numerical point of view the global structure of the set of solutions with one interior node of a degenerate diffusive logistic equation of huge interest in Population Dynamics. Our main findings reveal that the number of nodal solutions as well as the number of components in the bifurcation diagrams strongly depends on the number and position of the components where the weight function in front of the nonlinearity vanishes. No technical tool seems to be available to treat analytically these problems.

Multiple and Nodal Solutions for Parametric Dirichlet Equations Driven by the Double Phase Differential Operator

We consider a nonlinear parametric Dirichlet problem driven by the double phase differential operator. Using variational tools combined with critical groups, we show that for all small values of the parameter, the problem has at least three nontrivial bounded solutions which are ordered and we provide the sign information for all of them. Two solutions are of constant sign and the third one is nodal. Finally, we determine the asymptotic behavior of the nodal solution as the parameter converges to zero.

On the Bahri–Lions conjecture for elliptic equations with non-symmetric nonlinearities

We prove the existence of infinitely many solutions for a class of elliptic Dirichlet problems with non-symmetric nonlinearities. In particular, in suitable domains of {mathbb {R}}^n with nge 3, this result gives a positive answer to a well known conjecture formulated by A. Bahri and P.L. Lions. The proof is based on a minimization method which does not require the use of techniques of deformation from the symmetry. This method allows us to piece together solutions of Dirichlet problems in suitable subdomains, so we obtain infinitely many nodal solutions with a prescribed nodal structure.

Equivalently analytical solution for the large deformation of slender beams under follower loads: a second-order ANCF approach

Follower loads interact with the object's configuration, and the two are unknown variables in geometrically nonlinear analysis. This work aims to solve the large deformation of slender beams under follower loads via the Bernoulli–Euler beam element of absolute nodal coordinate formulation. The second-order approximate function of the beam centerline is utilized to model the beam configuration and the Frenet frame. The elastic force vector and its tangent stiffness matrix are formulated following the decoupling elastic line approach. The constraint of adjacent elements is imposed using Lagrange multipliers. The exact kinematics characteristics of typical follower loads are demonstrated; sequentially, the external force vectors and the load stiffness matrices are rigorously derived based on the configuration and the Frenet frame modelled by the second-order approximate function. Benchmark problems of straight and curved beams under different follower loads are solved, wherein the accuracy and efficiency of the proposed approach are validated as compared to relevant analytical and numerical solutions. It is found that the proposed element yields equivalently analytical nodal solutions, including position, rotation and curvature; considerably fewer proposed elements are used as compared with other finite element methods; the load stiffness matrix of the bending moment raises computational efficiency of the proposed element; the obtained solutions are more accurate than that of the B22 element of ABAQUS when solving pressure loading cases. By further extending the proposed approach, it can enable the unveiling of the spatio-temporal evolution characteristic of slender structures and follower loads from the equivalently analytical perspective.

Existence of constant sign and nodal solutions for a class of (p,q)-Laplacian-Kirchhoff problems

Existence of constant sign and nodal solutions for a class of (p,q)-Laplacian-Kirchhoff problems

Positive and nodal ground state solutions for a critical Schrödinger-Poisson system with indefinite potentials

We consider the Schrödinger-Poisson system{−Δu+V(x)u+K(x)ϕu=a(x)|u|p−2u+|u|4u,x∈R3,−Δϕ=K(x)u2,x∈R3, where 4<p<6 and the potentials V, a are allowed to change their signs. Under some reasonable assumptions on V, K and a, we apply the constraint minimization argument to establish the existence of positive ground state solutions and ground state nodal solutions.

Nodal solution for critical Kirchhoff-type equation with fast increasing weight in mathbb{R}^{2}

In this paper, we investigate the existence of a least-energy sign-changing solutions for the following Kirchhoff-type equation: −(1+b∫R2K(x)|∇u|2dx)div(K(x)∇u)=K(x)f(u),x∈R2,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ - \\biggl(1+b \\int _{\\mathbb{R}^{2}} K(x) \\vert \ abla u \\vert ^{2}\\,dx \\biggr) \\operatorname{div} \\bigl(K(x)\ abla u \\bigr)=K(x)f(u),\\quad x\\in \\mathbb{R}^{2}, $$\\end{document} where f has exponential subcritical or exponential critical growth in the sense of the Trudinger–Moser inequality. By using the constrained variational methods, combining the deformation lemma and Miranda’s theorem, we prove the existence of a least-energy sign-changing solution. Moreover, we also prove that this sign-changing solution has exactly two nodal domains.

Infinitely many smooth nodal solutions for Orlicz Robin problems

In this note, we study a Robin problem driven by the Orlicz g-Laplace operator. In particular, by using a regularity result and Kajikiya’s theorem, we prove that the problem has a whole sequence of distinct smooth nodal solutions converging to the trivial one. The analysis is developed in the most general abstract setting that corresponds to Orlicz–Sobolev function spaces.

Global multiplicity for parametric anisotropic Neumann (p,q)-equations

We consider a Neumann boundary value problem driven by the anisotropic $(p,q)$-Laplacian plus a parametric potential term. The reaction is ``superlinear". We prove a global (with respect to the parameter) multiplicity result for positive solutions. Also, we show the existence of a minimal positive solution and finally, we produce a nodal solution.

Localized nodal solutions for system of critical Choquard equations

In this paper, we study the system of critical Choquard equations ɛ2Δvj−a(x)vj+∑i=1kβijɛ−α(∫RN|vi(y)|2α∗dy|x−y|N−α)|vj|2α∗−2vj+λj|vj|q−2vj=0,x∈RN,vj(x)→0as|x|→∞,j=1,…,k,where N≥3, λj>0,j=1,…,k,(N−4)+<α<N,2α∗=N+αN−2, max{2,2∗−1}<q<2∗=2NN−2, ɛ>0 is a small parameter and the potential function a is bounded and positive. By the truncation method and the method of invariant sets of descending flow, we establish for small ɛ the existence of a sequence of localized nodal solutions concentrating near a given local minimum point of the potential function a.

Nodal solutions for logarithmic weighted N-laplacian problem with exponential nonlinearities

In this article, we study the following problem $$-{\rm div} (\omega(x)|\nabla u|^{N-2} \nabla u) = \lambda\ f(x,u) \quad\mbox{ in }\quad B, \quad u=0 \quad\mbox{ on } \quad\partial B,$$ where $B$ is the unit ball of $\mathbb{R^{N}}$, $N\geq2$ and $ w(x)$ a singular weight of logarithm type. The reaction source $f(x,u)$ is a radial function with respect to $x$ and is subcritical or critical with respect to a maximal growth of exponential type. By using the constrained minimization in Nehari set coupled with the quantitative deformation lemma and degree theory, we prove the existence of nodal solutions.

Least-energy nodal solutions of nonlinear equations with fractional Orlicz–Sobolev spaces

In this paper, we study the existence of least-energy nodal (sign-changing) weak solutions for a class of fractional Orlicz equations given by ( − △ g ) α u + g ( u ) = K ( x ) f ( u ) , in R N , where N ⩾ 3 , ( − △ g ) α is the fractional Orlicz g-Laplace operator, while f ∈ C 1 ( R ) and K is a positive and continuous function. Under a suitable conditions on f and K, we prove a compact embeddings result for weighted fractional Orlicz–Sobolev spaces. Next, by a minimization argument on Nehari manifold and a quantitative deformation lemma, we show the existence of at least one nodal (sign-changing) weak solution.

Nodal solutions for a supercritical problem with variable exponent and logarithmic nonlinearity

Nodal solutions for a supercritical problem with variable exponent and logarithmic nonlinearity

Nodal solutions for critical Robin double phase problems with variable exponent

Nodal solutions for critical Robin double phase problems with variable exponent

Nodal solutions for singular semilinear elliptic systems

In this paper, we prove existence of nodal solutions for singular semilinear elliptic systems without variational structure where its both components are of sign changing. Our approach is based on sub-supersolutions method combined with perturbation arguments involving singular terms.

Rich dynamics in planar systems with heterogeneous nonnegative weights

This paper studies the global structure of the set of nodal solutions of a generalized Sturm–Liouville boundary value problem associated to the quasilinear equation$ -(\phi(u'))' = \lambda u + a(t)g(u), \quad \lambda\in {\mathbb R}, $where $ a(t) $ is non-negative with some positive humps separated away by intervals of degeneracy where $ a\equiv 0 $. When $ \phi(s) = s $ this equation includes a generalized prototype of a classical model going back to Moore and Nehari [35], 1959. This is the first paper where the general case when $ {\lambda}\in\mathbb{R} $ has been addressed when $ a\gneq 0 $. The semilinear case with $ a\lneq 0 $ has been recently treated by López-Gómez and Rabinowitz [28,29,30].

Ground state and nodal solutions for a class of fractional Schrödinger equations involving exponential growth

Ground state and nodal solutions for a class of fractional Schrödinger equations involving exponential growth