Existence of a Least Energy Nodal Solution for a Class of Quasilinear Elliptic Equations with Exponential Growth

AbstractIn this paper we prove an existence result for a least energy nodal (or sign-changing) solution for the class of p&q problems given bywhere Ω is a smooth bounded domain in ℝ

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Nodal Solutions for a Quasilinear Elliptic Equation Involving the p-Laplacian and Critical Exponents

Advanced Nonlinear Studies ◽

10.1515/ans-2017-6022 ◽

2018 ◽

Vol 18 (1) ◽

pp. 17-40

Author(s):

Yinbin Deng ◽

Shuangjie Peng ◽

Jixiu Wang

Keyword(s):

Elliptic Equations ◽

Order Term ◽

Quasilinear Elliptic Equations ◽

Nodal Solution ◽

Nodal Solutions ◽

Content Type ◽

Nodal Domains ◽

Change Of Variables ◽

Graphic Content ◽

Quasilinear Elliptic

AbstractThis paper is concerned with the following type of quasilinear elliptic equations in{\mathbb{R}^{N}}involving thep-Laplacian and critical growth:-\Delta_{p}u+V(|x|)|u|^{p-2}u-\Delta_{p}(|u|^{2})u=\lambda|u|^{q-2}u+|u|^{2p^{% *}-2}u,which arises as a model in mathematical physics, where{2<p<N},{p^{*}=\frac{Np}{N-p}}. For any given integer{k\geq 0}, by using change of variables and minimization arguments, we obtain, under some additional assumptions onpandq, a radial sign-changing nodal solution with{k+1}nodal domains. Since the critical exponent appears and the lower order term (obtained by a transformation) may change sign, we shall use delicate arguments.

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Quasilinear elliptic equations involving the -Laplacian with critical exponential growth in

Nonlinear Analysis ◽

10.1016/j.na.2009.06.006 ◽

2009 ◽

Vol 71 (12) ◽

pp. 6157-6169 ◽

Cited By ~ 15

Author(s):

Youjun Wang ◽

Jun Yang ◽

Yimin Zhang

Keyword(s):

Exponential Growth ◽

Elliptic Equations ◽

Quasilinear Elliptic Equations ◽

Critical Exponential Growth ◽

Quasilinear Elliptic

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

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/prm.2018.88 ◽

2019 ◽

Vol 149 (5) ◽

pp. 1163-1173

Author(s):

Vladimir Bobkov ◽

Sergey Kolonitskii

Keyword(s):

Boundary Conditions ◽

Elliptic Equations ◽

Dirichlet Boundary ◽

Dirichlet Boundary Conditions ◽

Quasilinear Elliptic Equations ◽

Nodal Set ◽

Sign Changing Solutions ◽

Quasilinear Elliptic ◽

Least Energy

AbstractIn 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 nodal set of any least energy sign-changing solution intersects the boundary of Ω. The proof is based on a moving polarization argument.

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