Abstract
In this paper, the nonlinear quasilinear elliptic problem with the logarithmic nonlinearity−div∇up−2∇u=axφpulogu+hxψpuinΩ⊂Rnwas studied. By means of a double perturbation argument and Nehari manifold, the authors obtain the existence results.
Highlights
In this paper, we consider the existence of solution to the following problem −div Àj∇ujp−2 Á ∇u =aðxÞφpðuÞ log juj + hðxÞψpðuÞ, inΩ, ð1Þ where Ω ⊂ Rn, φpðzÞ = jzjp−2z, ψpðzÞ = jzjp−1z, p > 2, and n ≥ 1
We always suppose that aðxÞ is a sign-changing function; hðxÞ ≥ 0 is a ∈C1 function
Equations of the above form are mathematical models occurring in studies of the p-Laplace equation, generalized reaction-diffusion theory [1], non-Newtonian fluid theory [2, 3], non-Newtonian filtration theory [4, 5], and the turbulent flow of a gas in porous medium [6]
Summary
We consider the existence of solution to the following problem where Ω ⊂ Rn, φpðzÞ = jzjp−2z, ψpðzÞ = jzjp−1z, p > 2, and n ≥ 1. We consider the existence of positive solution for problem ð1Þ with Neumann boundary conditions. Following [15, 16] (see [17]), for each 0 < ε < 1, we consider Journal of Function Spaces a family of approximate problems
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