Differential and Integral Equations

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

Khadijah Sharaf

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We consider the following nonlinear elliptic equation \begin{equation} \tag{0.1} A_\frac{1}{2} u = K(x) |u|^{p-1}u \hbox{ in } \Omega, \;\; u=0 \hbox{ on } \partial\Omega, \end{equation} 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

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

First available in Project Euclid: 20 January 2017

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations 35R11: Fractional partial differential equations


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

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