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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