Differential and Integral Equations

On a class of nonlinear elliptic equations with lower order terms

A. Alvino, M.F. Betta, A. Mercaldo, and R. Volpicelli

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In this paper, we prove an existence result for weak solutions to a class of Dirichlet boundary value problems whose prototype is \begin{equation*} \label{pa} \left\{ \begin{array}{lll} -\Delta_p u =\beta |\nabla u|^{q} +c(x)|u|^{p-2}u +f & & \text{in}\ \Omega \\ u=0 & & \text{on}\ \partial \Omega , \end{array} \right. \end{equation*} where $\Omega $ is a bounded open subset of $\mathbb R^N$, $N\geq 2$, $\Delta_p u={\rm div} \left(|\nabla u|^{p-2}\nabla u\right)$, $1 < p < N$, $ p-1 < q\le p-1+\frac p N$, $\beta $ is a positive constant, $c\in L^{\frac N p}(\Omega)$ with $c\ge 0$, $c\neq 0$ and $f\in L^{(p^*)'}(\Omega).$ We further assume smallness assumptions on $c$ and $f$. Our approach is based on Schauder's fixed point theorem.

Article information

Differential Integral Equations, Volume 32, Number 3/4 (2019), 223-232.

First available in Project Euclid: 23 January 2019

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations 35J25: Boundary value problems for second-order elliptic equations


Alvino, A.; Betta, M.F.; Mercaldo, A.; Volpicelli, R. On a class of nonlinear elliptic equations with lower order terms. Differential Integral Equations 32 (2019), no. 3/4, 223--232. https://projecteuclid.org/euclid.die/1548212430

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