Taiwanese Journal of Mathematics

EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR NONLOCAL $\overrightarrow{p}(x)$-LAPLACIAN PROBLEM

G. A. Afrouzi and M. Mirzapour

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Abstract

In this paper, we study the nonlocal anisotropic $\overrightarrow{p}(x)$-Laplacian problem of the following form \begin{gather*} - \sum_{i=1}^N M_{i} \Big( \int_{\Omega} \frac{|\partial_{x_i} u|^{p_i(x)}}{p_i(x)} dx \Big) \partial_{x_i} \Big( |\partial_{x_i} u|^{p_i(x)-2} \partial_{x_i} u \Big) = f(x,u) \quad \text{in } \Omega, \\ u=0 \quad \text{on } \partial \Omega. \end{gather*} By means of a direct variational approach and the theory of the anisotropic variable exponent Sobolev space, we obtain the existence and multiplicity of weak energy solution. Moreover, we get much better results with $f$ in a special form.

Article information

Source
Taiwanese J. Math., Volume 18, Number 1 (2014), 219-236.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706339

Digital Object Identifier
doi:10.11650/tjm.18.2014.2596

Mathematical Reviews number (MathSciNet)
MR3162121

Zentralblatt MATH identifier
1357.35141

Subjects
Primary: 35J62: Quasilinear elliptic equations 35J70: Degenerate elliptic equations 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Keywords
anisotropic Sobolev spaces variable exponent mountain pass theorem Fountain theorem dual Fountain theorem

Citation

Afrouzi, G. A.; Mirzapour, M. EXISTENCE AND MULTIPLICITY OF SOLUTIONS FOR NONLOCAL $\overrightarrow{p}(x)$-LAPLACIAN PROBLEM. Taiwanese J. Math. 18 (2014), no. 1, 219--236. doi:10.11650/tjm.18.2014.2596. https://projecteuclid.org/euclid.twjm/1499706339


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