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Nontrivial solutions for a class of one-parameter problems with singular \(\phi\)-Laplacian. (English) Zbl 1274.35078
Summary: We study the mixed boundary value problem with singular \(\phi\)-Laplacian \[ [r^{N-1} \phi(u')]'=r^{N-1}[\alpha(r)u^{q-1}-\lambda p(r,u)]{\text{ in }}[0,R],\;u'(0)=0= u(R), \] where \(\lambda>0\) is a parameter, \(q>1, \alpha:[0,R]\to\mathbb R\) is positive on \((0,R)\) and the function \(p:[0,R]\times [0,A]\to\mathbb R\) is positive on \((0,R)\times (0,A)\), with \(p(r,0)=0= p(r,A)\) for all \(r\in [0,R]\). Using a variational approach, we provide sufficient conditions ensuring the existence of at least one or at least two nontrivial solutions, for large enough values of the parameter.

MSC:
35J20 Variational methods for second-order elliptic equations
35J60 Nonlinear elliptic equations
35J93 Quasilinear elliptic equations with mean curvature operator
35J87 Unilateral problems for nonlinear elliptic equations and variational inequalities with nonlinear elliptic operators
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