Abstract
We study the following -Laplacian problem with singular term: , , , , where is a bounded domain, . We obtain the existence of solutions in .
Highlights
After Kovacik and Rakosnık first discussed the Lp x Ω spaces and Wk,p x Ω spaces in 1, a lot of research has been done concerning these kinds of variable exponent spaces, for example, see 2–5 for the properties of such spaces and 6–9 for the applications of variable exponent spaces on partial differential equations
We study the existence of the weak solutions for the following p x Laplacian problem:
Let P Ω be the set of all Lebesgue measurable functions p : Ω → 1, ∞
Summary
After Kovacik and Rakosnık first discussed the Lp x Ω spaces and Wk,p x Ω spaces in 1 , a lot of research has been done concerning these kinds of variable exponent spaces, for example, see 2–5 for the properties of such spaces and 6–9 for the applications of variable exponent spaces on partial differential equations. In W1,p x Ω spaces, there are a lot of studies on p x -Laplacian problems; see 8, 9. We study the existence of the weak solutions for the following p x Laplacian problem:. For all p x ∈ P Ω , we denote p supx∈Ωp x , p− infx∈Ωp x , and denote by p1 x fact that inf{p2 x − p1 x } > 0. A typical example of 1.1 is the following problem involving subcritical SobolevHardy exponents of the form. When b x 1, the solution of the p-Laplacian equations without singularity has been studied by many researchers. The study of problem 1.1 with variable exponents is a new topic.
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