zbMATH — the first resource for mathematics

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.

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