A Geometric B-Spline Approach to Mountain-Pass Type Solutions of Nonlinear Dirichlet Problems

We study the numerical computation of nontrivial critical points of variational functionals associated with nonlinear Dirichlet problems involving the p -Laplacian. Previous numerical mountain pass approaches typically relied on finite element discretizations of the underlying function space. In contrast, we employ a discretization based on a geometric B-spline representation of the solution. The function space is approximated by smooth spline curves parameterized by control points, yielding a finite-dimensional geometric representation of the variational problem. Within this discrete space we apply a mountain-pass type up–down method. This allows the search for saddle-type critical points to be carried out directly in the space of spline control points. The descent direction is obtained through an auxiliary Poisson equation, providing a Sobolev gradient that stabilizes the iteration. Convergence of the numerical procedure is monitored via the Euler–Lagrange residual, ensuring that the computed spline approximation satisfies the variational problem up to a prescribed tolerance. Numerical experiments for the model case $$p=2$$ p = 2 with nonlinearity f(u)=u³ f ( u ) = u 3 on Ω =(0,1) Ω = ( 0 , 1 ) show that the method computes nontrivial solutions, including sign-changing profiles depending on the initialization. The results demonstrate the successful application of B-splines in the numerical solution of nonlinear variational problems.