Abstract
We consider a nonlinear Dirichlet problem driven by a general nonhomogeneous differential operator and with a reaction exhibiting the combined effects of a parametric singular term plus a Carathéodory perturbation f(z,x,y) which is only locally defined in x in {mathbb {R}} . Using the frozen variable method, we prove the existence of a positive smooth solution, when the parameter is small.
Highlights
Let ⊆ RN be a bounded domain with C2-boundary ∂
In this paper we study the following nonhomogeneous parametric singular Dirichlet problem with gradient dependence:
The map a : RN → RN in the differential operator is continuous and strictly monotone and satisfies certain other growth and regularity conditions listed in hypotheses H0 below
Summary
Let ⊆ RN be a bounded domain with C2-boundary ∂. – div a(Du(z)) = λu(z)–η + f (z, u(z), Du(z)) in , u|∂ = 0, u > 0, λ > 0, 0 < η < 1 In this problem, the map a : RN → RN in the differential operator is continuous and strictly monotone (maximal monotone too) and satisfies certain other growth and regularity conditions listed in hypotheses H0 below The map a : RN → RN in the differential operator is continuous and strictly monotone (maximal monotone too) and satisfies certain other growth and regularity conditions listed in hypotheses H0 below In order to use topological tools (fixed point theory), we need to find a canonical way to choose such a positive solution. Using this lemma and (1) we are led to the following growth restrictions on G(·)
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