Abstract
We consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter lambda . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.
Highlights
Let ⊆ RN be a bounded domain with a C2-boundary ∂
We study the following singular eigenvalue problem for the Dirichlet ( p, q)-Laplacian pu(z) −
For r ∈ (1, ∞), by r we denote the r -Laplace differential operator defined by r u = div(|Du|r−2 Du) for all u ∈ W 1,r ( )
Summary
Let ⊆ RN be a bounded domain with a C2-boundary ∂. Proposition 4 For every λ > 0, problem (Qλ) admits a unique positive solution uλ ∈ intC+ and uλ → 0 in C01( ) as λ → 0+. Proof First we prove the existence of a positive solution To this end, let ψλ : W01,p( ) → R be the C1-functional defined by ψλ(u). We have 0 ≤ λc2 [urλ−τ − urλ−τ ](uτλ − uτλ) d z ⇒ uλ = uλ This proves the uniqueness of the positive solution uλ ∈ intC+ of problem (Qλ). The nonlinear regularity theorem of Lieberman [22] and the compact embedding of C01,α( ) := C1,α( ) C01( ) (0 < α < 1) into C01( ), imply that uλ → 0 in C01( ) as λ → 0+ This completes the proof of the proposition.
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