Differential and Integral Equations

Existence of solution for a anisotropic equation with critical exponent

Claudianor Oliveira Alves and Abdallah El Hamidi

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Using variational methods we establish existence of nontrivial solutions for the following class of anisotropic critical problem $$ {(P_{\lambda})} \qquad \left \{ \begin{array}{l} - \displaystyle \sum_{i=1}^{N} \frac{\partial}{\partial x_{i}} \Big ( \Big | \frac{\partial u}{\partial x_{i}} \Big |^{p_{i}-2}\frac{\partial u}{\partial x_{i}} \Big )= \lambda f(u) + g(u) , \quad \mbox{in} \quad \Omega \\ u \geq 0, \quad \mbox{in} \quad \Omega \\ u=0, \quad \mbox{on} \quad \partial \Omega , \end{array} \right. $$ where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N$, $\lambda$ is a positive parameter, $g(u)$ behaves like $|u|^{p^*-2}u$, $p^{*}$ is the critical exponent for this class of problem and $f$ is a continuous function verifying some adequate assumptions.

Article information

Differential Integral Equations, Volume 21, Number 1-2 (2008), 25-40.

First available in Project Euclid: 20 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B33: Critical exponents 35J20: Variational methods for second-order elliptic equations 47J30: Variational methods [See also 58Exx] 58E05: Abstract critical point theory (Morse theory, Ljusternik-Schnirelman (Lyusternik-Shnirel m an) theory, etc.)


Alves, Claudianor Oliveira; El Hamidi, Abdallah. Existence of solution for a anisotropic equation with critical exponent. Differential Integral Equations 21 (2008), no. 1-2, 25--40. https://projecteuclid.org/euclid.die/1356039057

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