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Existence and multiplicity of positive solutions for singular quasilinear problems. (English) Zbl 1168.35358
Summary: We combine perturbation arguments and variational methods to study the existence and multiplicity of positive solutions for a class of singular \(p\)-Laplacian problems \[ \begin{cases} -\Delta_ pu=a(x)u^ {-\gamma}+\lambda f(x,u)\quad &\text{ in }\Omega,\\ \quad u\in W_ {0}^ {1,p}(\Omega),\;u>0\quad &\text{ in }\Omega, \end{cases}\tag{1} \] where \(\Omega \subset \mathbb{R}^ {N}\), \(N\geq 1\), is a \(C^ {2}\) bounded domain, \(1<p<\infty\), \(\gamma >0\) is a constant, \(\lambda >0\) is a parameter, \(f\) is a Carathéodory function and \(a\geq 0\) is measurable and satisfies \(\varphi^ {-\gamma}a\in L^ q(\Omega)\) for some \(q>N\) and \(\varphi\in C_ 0^ 1(\overline{\Omega})\), \(\varphi \geq 0\).
In the first two theorems we prove the existence of solutions in the sense of distributions. By strengthening the hypotheses, in the third and last result, we establish the existence of two ordered positive weak solutions.

MSC:
35J62 Quasilinear elliptic equations
35B32 Bifurcations in context of PDEs
35B09 Positive solutions to PDEs
35D30 Weak solutions to PDEs
35J20 Variational methods for second-order elliptic equations
47J30 Variational methods involving nonlinear operators
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