# On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations

```@article{Bobkov2019OnAP,
title={On a property of the nodal set of least energy sign-changing solutions for quasilinear elliptic equations},
journal={Proceedings of the Royal Society of Edinburgh: Section A Mathematics},
year={2019},
volume={149},
pages={1163 - 1173}
}```
• Published 10 July 2017
• Mathematics
• Proceedings of the Royal Society of Edinburgh: Section A Mathematics
Abstract In this note, we prove the Payne-type conjecture about the behaviour of the nodal set of least energy sign-changing solutions for the equation \$-\Delta _p u = f(u)\$ in bounded Steiner symmetric domains \$ \Omega \subset {{\open R}^N} \$ under the zero Dirichlet boundary conditions. The nonlinearity f is assumed to be either superlinear or resonant. In the latter case, least energy sign-changing solutions are second eigenfunctions of the zero Dirichlet p-Laplacian in Ω. We show that the…
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## References

SHOWING 1-10 OF 32 REFERENCES
Qualitative properties of nodal solutions of semilinear elliptic equations in radially symmetric domains
• Mathematics
• 2004
Abstract We study the qualitative properties of sign changing solutions of the Dirichlet problem Δ u + f ( u ) = 0 in Ω, u = 0 on ∂Ω, where Ω is a ball or an annulus and f is a C 1 function with f (
On qualitative properties of solutions for elliptic problems with the p-Laplacian through domain perturbations
• Mathematics, Physics
Communications in Partial Differential Equations
• 2019
Abstract We study the dependence of least nontrivial critical levels of the energy functional corresponding to the zero Dirichlet problem in a bounded domain upon domain perturbations. Assuming that
Asymptotics and symmetries of least energy nodal solutions of Lane-Emden problems with slow growth
• Mathematics
• 2008
In this paper, we consider the Lane-Emden problem [GRAPHICS] where is a bounded domain in R-N and p > 2. First, we prove that, for p close to 2, the solution is unique once we. x the projection on
ON EXISTENCE OF NODAL SOLUTION TO ELLIPTIC EQUATIONS WITH CONVEX-CONCAVE NONLINEARITIES
In a bounded connected domainR N , N > 1, with a smooth boundary, we consider the Dirichlet boundary value problem for elliptic equation with a convex-concave nonlinearity ( −�u = �juj q−2 u + juj −2
On the nodal set of the second eigenfunction of the laplacian in symmetric domains in RN
— We present a simple proof of the fact that if Ω is a bounded domain in R , N ≥ 2, which is convex and symmetric with respect to k orthogonal directions, 1 ≤ k ≤ N , then the nodal sets of the
On the structure of the second eigenfunctions of the -Laplacian on a ball
• Mathematics
• 2015
In this paper, we prove that the second eigenfunctions of the \$p\$-Laplacian, \$p>1\$, are not radial on the unit ball in \$\mathbb{R}^N,\$ for any \$N\ge 2.\$ Our proof relies on the variational
A SIGN-CHANGING SOLUTION FOR A SUPERLINEAR DIRICHLET PROBLEM
• Mathematics
• 1997
In previous work by Castro, Cossio, and Neuberger [2], it was shown that a superlinear Dirichlet problem has at least three nontrivial solutions when the derivative of the nonlinearity at zero is
Lane–Emden problems: Asymptotic behavior of low energy nodal solutions
• Mathematics
• 2013
Abstract We study the nodal solutions of the Lane–Emden–Dirichlet problem { − Δ u = | u | p − 1 u , in Ω , u = 0 , on ∂ Ω , where Ω is a smooth bounded domain in R 2 and p > 1 . We consider solutions
Nodal line structure of least energy nodal solutions for Lane–Emden problems
• Mathematics
• 2009
In this Note, we consider the Lane–Emden problem −Δu=λ2|u|p−2u with Dirichlet boundary conditions, where the domain Ω is an open bounded subset of R2, λ2 is the second eigenvalue of −Δ, and p>2. We
A Strong Maximum Principle for some quasilinear elliptic equations
In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain Ω ⊂ ℝn,n ⩾ 1, is in fact positive everywhere. Here we prove that the same