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Three solutions for singular $$p$$-Laplacian type equations. (English) Zbl 1177.35082
Summary: We consider the singular $$p$$-Laplacian type equation
\begin{aligned} -\text{div}(|x|^{-\beta} a(x,\nabla u))= \lambda f(x,u) &\quad \text{in }\Omega,\\ u=0 &\quad \text{on }\partial\Omega, \end{aligned}
where $$0\leq\beta<N-p$$, $$\Omega$$ is a smooth bounded domain in $$\mathbb R^N$$ containing the origin, $$f$$ satisfies some growth and singularity conditions. Under some mild assumptions on $$a$$, applying the three critical points theorem developed by Bonanno, we establish the existence of at least three distinct weak solutions to the above problem if $$f$$ admits some hypotheses on the behavior at $$u=0$$ or perturbation property.

##### MSC:
 35J60 Nonlinear elliptic equations 35J20 Variational methods for second-order elliptic equations
##### Keywords:
$$p$$-Laplacian operator; singularity; multiple solutions
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