Existence of a continuum of solutions for a quasilinear elliptic singular problem

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文摘

In this paper we study the existence of positive solution

u \in H_{0}^{1} false">(Ω false">)

for some quasilinear elliptic equations, having lower order terms with quadratic growth in the gradient and singularities, whose model is

- Δ u + μ false">(x false">) \frac{{false">| \nabla u false">|}^{2}}{u^{γ} + u^{β}} = λ u^{p} + f_{0} false">(x false">), x \in Ω, 0 < γ \leq β, 0 < p .

Using topological methods we obtain the existence of an unbounded continuum of solutions. In the case μ(x)

μ false">(x false">)

is constant we derive the existence of solution for every fa1bdf593afffe711f1df5922d034" title="Click to view the MathML source">λ>0

λ > 0

if faeeb30694e688c68644f3c72e3314" title="Click to view the MathML source">1<γ<2

1 < γ < 2

for any β and p<1

p < 1

. Even more for μ∈L^∞(Ω)

μ \in L^{\infty} false">(Ω false">)

we prove this result if β≤1

β \leq 1

and p<2−β

p < 2 - β

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