Constant Sign and Sign Changing Solutions for Systems of Quasilinear Elliptic Equations

2010 ◽  
Vol 19 (2) ◽  
pp. 255-269 ◽  
Author(s):  
Dumitru Motreanu ◽  
Zhitao Zhang
Keyword(s):  
Elliptic Equations ◽  
Quasilinear Elliptic Equations ◽  
Constant Sign ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic

Existence of nodal solutions for quasilinear elliptic problems in ℝN

2015 ◽  
Vol 145 (5) ◽  
pp. 937-957
Author(s):  
Ann Derlet ◽  
François Genoud
Keyword(s):  
Elliptic Equations ◽  
Weighted Sobolev Space ◽  
Point Theory ◽  
Quasilinear Elliptic Equations ◽  
Nodal Solutions ◽  
Laplacian Equation ◽  
Quasilinear Elliptic Problems ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic ◽  
Constraint Minimization

We prove the existence of one positive, one negative and one sign-changing solution of a p-Laplacian equation on ℝN with a p-superlinear subcritical term. Sign-changing solutions of quasilinear elliptic equations set on the whole of ℝN have scarcely been investigated in the literature. Our assumptions here are similar to those previously used by some authors in bounded domains, and our proof uses fairly elementary critical point theory, based on constraint minimization on the nodal Nehari set. The lack of compactness due to the unbounded domain is overcome by working in a suitable weighted Sobolev space.

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

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.

Quasilinear Equations via Elliptic Regularization Method

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Jia-Quan Liu ◽  
Xiang-Qing Liu ◽  
Zhi-Qiang Wang
Keyword(s):  
Elliptic Equations ◽  
Regularization Method ◽  
Quasilinear Elliptic Equations ◽  
Quasilinear Equations ◽  
Elliptic Regularization ◽  
Nodal Property ◽  
Quasilinear Problems ◽  
Sign Changing Solutions ◽  
Quasilinear Elliptic

AbstractIn this paper we study a class of quasilinear problems, in particular we deal with multiple sign-changing solutions of quasilinear elliptic equations. We further develop an approach used in our earlier work by exploring elliptic regularization. The method works well in studying multiplicity and nodal property of solutions.

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