A note on the boundary regularity of solutions to quasilinear elliptic equations

2018 ◽  
Vol 24 (2) ◽  
pp. 849-858 ◽  
Author(s):  
Giuseppe Riey ◽  
Berardino Sciunzi
Keyword(s):  
Laplace Operator ◽  
Critical Points ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Boundary Regularity ◽  
Regularity Of Solutions ◽  
Quasilinear Elliptic Equations ◽  
Second Derivatives ◽  
Quasilinear Elliptic ◽  
Derivatives Of

We study the summability up to the boundary of the second derivatives of solutions to a class of Dirichlet boundary value problems involving the p-Laplace operator. Our results are meaningful for the cases when the Hopf’s Lemma cannot be applied to ensure that there are no critical points of the solution on the boundary of the domain.

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.

scholarly journals Solvability of quasilinear elliptic equations with strong dependence on the gradient

2000 ◽  
Vol 5 (3) ◽  
pp. 159-173 ◽  
Author(s):  
Darko Žubrinić
Keyword(s):  
Elliptic Equations ◽  
Strong Dependence ◽  
Strong Solutions ◽  
Outer Radius ◽  
Regularity Of Solutions ◽  
Quasilinear Elliptic Equations ◽  
A Posteriori ◽  
The Right ◽  
Equation Solvability ◽  
Quasilinear Elliptic

We study the problem of existence of positive, spherically symmetric strong solutions of quasilinear elliptic equations involvingp-Laplacian in the ball. We allow simultaneous strong dependence of the right-hand side on both the unknown function and its gradient. The elliptic problem is studied by relating it to the corresponding singular ordinary integro-differential equation. Solvability range is obtained in the form of simple inequalities involving the coefficients describing the problem. We also study a posteriori regularity of solutions. An existence result is formulated for elliptic equations on arbitrary bounded domains in dependence of outer radius of domain.

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