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Nodal Solutions for Supercritical Laplace Equations

, 2016
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Nodal solutions of a p-Laplacian equation

Proceedings of the London Mathematical Society, 2005
Summary: We prove that the \(p\)-Laplacian problem \(-\Delta_p u = f(x, u)\), with \(u \in W_0^{1,p}(\Omega)\) on a bounded domain \(\Omega \subset \mathbb R^N\), with \(p > 1\) arbitrary, has a nodal solution provided that \(f : \Omega\times\mathbb R \to \mathbb R\) is subcritical, and \(f(x, t) / |t|^{p-2}\) is superlinear.
Thomas Bartsch, Zhaoli Liu, Tobias Weth   +2 more
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Solution of inverse nodal problems

Inverse Problems, 1989
We show that the coefficients in a second-order differential equation can be determined from the positions of the nodes for the eigenfunctions. We prove uniqueness results, derive approximate solutions, give error bounds and present numerical experiments.
Hald, Ole H., McLaughlin, Joyce R.
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On the Geometric Measure of Nodal Sets of Solutions

Journal of Partial Differential Equations, 1994
The authors prove several results on the Hausdorff measure of nodal sets of solutions \(u\) of the equation \(Lu=0\) in the unit ball \(B\subset \mathbb{R}^ n\). Here \(L\) belongs to a special class of elliptic or parabolic operators and \(u\) is assumed to have a certain compactness property.
Han, Qing, Lin, Fanghua
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Localized Nodal Solutions for Schrödinger-Poisson Systems

Acta Mathematica Scientia, 2022
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Wang, Xing, He, Rui, Liu, Xiangqing
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Nodal Solutions for Indefinite Robin Problems

Bulletin of the Malaysian Mathematical Sciences Society, 2017
Given a bounded domain \(\Omega \subseteq \mathbb{R}^N\) with a \(C^2\)-boundary, the authors study semilinear Robin problems of the form \[ \begin{cases} -\Delta u(z)+\xi(z)u(z) = f(z,u(z)) &\text{in }\Omega,\\ \frac{\partial u}{\partial n}+\beta(z)u=0 &\text{on }\partial \Omega, \end{cases} \] where the potential function \(\xi\) belongs to \(L^s ...
Filippakis, Michael, Papageorgiou, Nikolaos S.   +1 more
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Nodal solutions of -Laplacian equations

Nonlinear Analysis: Theory, Methods & Applications, 2007
The aim of this paper is to study the existence of nodal radial solutions for the \(p(x)\)-Laplacian equation of the form \[ \begin{gathered} -\text{div}(|\nabla |^{p(x)-2}\nabla u)+ a(x)|u|^{p(x)-2} u=|u|^{q(x)- 2} u\quad\text{in }\Omega,\\ u\in W^{1,p(x)}_0(\Omega).\end{gathered}\tag{1} \] Using a variational method, the authors prove that, for any ...
Fan, Xianling, Zhao, Yuanzhang
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Nodal solutions for anisotropic \((p, q)\)-equations

Nonlinear Analysis: Real World Applications, 2022
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Zeng, Shengda, Papageorgiou, Nikolaos S.
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Nonexistence of nodal solutions of nonlinear elliptic equations

Nonlinear Analysis: Theory, Methods & Applications, 2001
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Bae, Soohyun, Pahk, Dae Hyeon
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ON NODAL SOLUTIONS OF THE NONLINEAR SCHRÖDINGER–POISSON EQUATIONS

Communications in Contemporary Mathematics, 2012
In this paper, we are interested in nodal solutions of nonlinear Schrödinger–Poisson equations. In particular, for a given natural number k we construct a radial solution changing sign exactly k-times.
Kim, Seunghyeok, Seok, Jinmyoung
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