On p(x)-Laplacian equations in ℝ^N with nonlinearity sublinear at zero

2024/09/23 by Shibo Liu, Liu, Shibo, Chunshan Zhao +1
Mathematics · #advanced mathematical theories #Nonlinear Partial Differential Equations #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2409.15540

Abstract

Let p,q be functions on ℝ^N satisfying 1≪ q≪ p≪ N, we consider p(x)-Laplacian problems of the form \ % -Δ_p(x)u+V(x)\vert u\vert ^p(x)-2u=λ\vert u\vert ^q(x)-2u+g(x,u),
u∈ W^1,p(x)(ℝ^N).% . To apply variational methods, we introduce a subspace X of W^1,p(x)(ℝ^N) as our working space. Compact embedding from X into L^q(x)(ℝ^N) is proved, this enable us to get nontrivial solution of the problem; and two sequences of solutions going to ∞ and 0 respectively, when g(x,⋅) is odd.

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