Multiple existence and qualitative property of nodal solutions for coupled elliptic equations

Haoyu Li; Zhi-Qiang Wang

Home/Archive/arXiv

arXiv · 2504.02209

Multiple existence and qualitative property of nodal solutions for coupled elliptic equations

Haoyu Li·Zhi-Qiang Wang

View original source ↗Read source document ↗Export metadata ↓

Abstract

The paper studies nodal solutions having prescribed componentwise nodal data for the following coupled nonlinear elliptic equations \begin{equation} \left\{ \begin{array}{lr} -Δu_{j}+ u_{j}= u^{3}_{j}+β\sum_{i=1, i\neq j}^N u_{j}u_{i}^{2} \,\,\,\,\,\,\, \mbox{in}\ Ω,\nonumber u_{j}\in H_{0,r}^{1}(Ω), \,\,\,\,\,\,\,\,j=1,\dots,N.\nonumber \end{array} \right. \end{equation} Here, $Ω\subset\mathbb{R}^n$ is a bounded and radial domain with $n=2,3$. The coupling constant $β\leq-1$ is in the repulsive regime. We investigate the solution structure for both positive and nodal solutions, proving multiple existence of solutions with prescribed nodal data and providing qualitative estimates for the nodal numbers of the inter-componentwise differences of solutions with both upper and lower bounds. Our general framework is for nodal solutions though our results are new also for positive solutions.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Haoyu Li, Zhi-Qiang Wang. 2025-04-03. Multiple existence and qualitative property of nodal solutions for coupled elliptic equations. https://arxiv.org/abs/2504.02209

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

De Giorgi-Nash-Moser theory for kinetic equations with nonlocal diffusions

We extend the De Giorgi--Nash--Moser theory to nonlocal hypoelliptic equations arising in kinetic theory as linearized models for the non-cutoff Boltzmann equation. Assuming that the nonlocal tail in velocity of weak solutions belongs locally to $L^p_{t,x}$ for some $p>N_{d}/(2s)$, where $N_{d}$ is the drift homogeneous dimension and $2s\in(0,2)$ is the order of diffusion, we prove a local $L^2$-$L^\infty$ estimate, from which we also deduce a nonlocal strong Harnack inequality. The tail summability threshold for the local estimate is optimal: for every $1\leq p \leq N_{d}/(2s)$, we construct a family of nonnegative, time-dependent solutions that rules out this estimate even for the pure fractional Kolmogorov equation. The local boundedness estimate also allows for possibly unbounded source terms.

math.AP↗

arXiv2026

Multiple positive solutions to a perturbed Gelfand problem involving mixed local-nonlocal operators and singular nonlinearity

We investigate a perturbed Gelfand problem involving a mixed local-nonlocal $p$-Laplacian operator with singular nonlinearity: \begin{equation*} \begin{aligned} -Δ_p u + (-Δ_p)^s u = λ\frac{f(u)}{u^β}\ \text{in} \ Ω\newline u >0\ \text{in} \ Ω,\ u =0\ \text{in} \ \mathbb{R}^N \setminus Ω\end{aligned} \end{equation*} where $Ω\subset \mathbb{R}^N$ is a smooth bounded domain, $λ> 0 $ is a parameter, $0\leq β<1$ and $f$ is a non-decreasing $C^1$-function with $f(0)>0$. Using the method of sub- and supersolutions, we present a novel multiplicity result and, in specific cases, we also prove a three-solution theorem using Amann's fixed point theorem. Our construction of sub-supersolutions avoids the conventional reliance on ODE techniques and Green's function estimates, thereby making it more adaptable to the nonlinear and nonlocal framework. Additionally, we establish a Hopf-type Strong Comparison Principle for the linear operator with singular nonlinearity, marking the first result of its kind for mixed local-nonlocal operators. This result is crucial in deriving a third solution and holds broader mathematical significance.

math.AP↗

arXiv2026

The Obstacle Problem Arising from the American Chooser Option

We study the finite-horizon obstacle problem associated with the American chooser option. Its obstacle is the maximum of two independently determined American call and put values and has a spatial kink where these values cross. Using truncation, penalization, and comparison, we construct a local strong Sobolev solution and prove uniqueness within an exponential-growth class. We show that contact occurs only on the smooth exercise-payoff branches and that both contact regions have nonempty sections at every time. We then establish strict monotonicity, terminal limits, and local Lipschitz continuity of the two free boundaries. Localized one-phase equations, quantitative nondegeneracy, and parabolic boundary Harnack estimates yield $C^\infty$ regularity on the open time interval.

math.AP↗