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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.

35J60 Nonlinear elliptic equations
35J20 Variational methods for second-order elliptic equations
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