This work establishes the existence of weak solutions for p-Laplacian-like equations in weighted Sobolev spaces under Dirichlet boundary conditions, assuming the data is in the weighted Lebesgue space.

We prove in this paper some existence and uniqueness results of weak solutions for some nonlinear degenerate parabolic problems arising from capillarity phenomena with the Dirichlet type boundary condition.

We investigate invariant finite-dimensional linear spaces for a class of nonlinear evolution equations with second-order time dependence of the type ${\partial }_{tt}u=-{\partial }_x\left(A(x,u)u\right)+{\partial }_{xx}\left(B(x,u)u\right)+f\left(u\right),$ where the coefficients may depend nonlinearly on the solution. Using direct algebraic verification and invariant subspace reductions, we derive several exact solutions and reduce the governing equations to finite-dimensional systems of ordinary differential equations. Various examples are presented to illustrate the applicability of the method to nonlinear transport, diffusion, and reaction models.

This paper investigates a class of generalized nonlinear variable-coefficient beam-type evolution equations motivated by the modeling of nonhomogeneous elastic structures with nonlinear effects. The invariant subspace method is employed to construct exact solutions by identifying appropriate finite-dimensional invariant subspaces associated with the nonlinear operators. This approach reduces the original partial differential equations to systems of nonlinear ordinary differential equations. Several representative nonlinear beam-type equations are studied, leading to polynomial, trigonometric, hyperbolic, and negative-power invariant subspaces together with their corresponding exact solutions. The results demonstrate the effectiveness of the invariant subspace method for constructing explicit exact solutions of nonlinear variable-coefficient fourth-order beam-type equations and highlight its applicability to a broad class of nonlinear beam models.