## Differential and Integral Equations

### An infinite number of solutions for an elliptic problem with power nonlinearity

#### Abstract

We consider the following nonlinear elliptic equation $$\tag{0.1} A_\frac{1}{2} u = K(x) |u|^{p-1}u \hbox{ in } \Omega, \;\; u=0 \hbox{ on } \partial\Omega,$$ where $\Omega$ is a bounded domain of $\mathbb{R}^n, n\geq 1, K(x)$ is a given function, $A_\frac{1}{2}$ represents the square root of $-\Delta$ in $\Omega$ with zero Dirichlet boundary condition and $1 < p < \frac{n+1}{n-1}$, $(p > 1$ if $n=1$). We apply the Brouwer's fixed point theorem to prove that (0.1) has infinitely many distinct solutions.

#### Article information

Source
Differential Integral Equations, Volume 30, Number 1/2 (2017), 133-144.

Dates
First available in Project Euclid: 20 January 2017

https://projecteuclid.org/euclid.die/1484881223

Mathematical Reviews number (MathSciNet)
MR3599799

Zentralblatt MATH identifier
06738545

#### Citation

Sharaf, Khadijah. An infinite number of solutions for an elliptic problem with power nonlinearity. Differential Integral Equations 30 (2017), no. 1/2, 133--144. https://projecteuclid.org/euclid.die/1484881223