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

Abstract 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 = (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.