We consider the positive solutions of a quasilinear elliptic equation with $p$-Laplacian, logistic-type growth rate function, and a constant yield harvesting. We use sub-super-solution methods to prove the existence of a maximal positive solution when the harvesting rate is under a certain positive constant.

This paper presents a functional approach to a nonlinear model describing the complete physical process of water infiltration into an unsaturated soil, including the saturation occurrence and the advance of the wetting front. The model introduced in this paper involves a multivalued operator covering the simultaneous saturated and unsaturated flow behaviors and enhances the study of the displacement of the free boundary between these two flow regimes. The model resides in Richards' equation written in pressure form with an initial condition and boundary conditions which in this work express the inflow due to the rain on the soil surface on the one hand, and characterize a certain permeability corresponding to the underground boundary, on the other hand. Existence, uniqueness, and regularity results for the transformed model in diffusive form, that is, for the moisture of the soil, and the existence of the weak solution for the pressure form are proved in the 3D case. The main part of the paper focuses on the existence of the free boundary between the saturated and unsaturated parts of the soil, and this is proved, in the 1D case, for certain stronger assumptions on the initial data and boundary conditions.

We prove that critical values set of a differentiable map lies on a line of certain smoothness class.

We are interested in a nonlinear boundary value problem for ${\left({\left|u^{″}\right|}^{p-2}{u}^{″}\right)}^{\prime \text{}\prime }=\lambda {\left|u\right|}^{p-2}u$ in $\left[0,1\right]$, $p>1$, with Dirichlet and Neumann boundary conditions. We prove that eigenvalues of the Dirichlet problem are positive, simple, and isolated, and form an increasing unbounded sequence. An eigenfunction, corresponding to the $n$th eigenvalue, has precisely $n-1$ zero points in $\left(0,1\right)$. Eigenvalues of the Neumann problem are nonnegative and isolated, $0$ is an eigenvalue which is not simple, and the positive eigenvalues are simple and they form an increasing unbounded sequence. An eigenfunction, corresponding to the $n$th positive eigenvalue, has precisely $n+1$ zero points in $\left(0,1\right)$.

We prove the existence, uniqueness, and continuous dependence of a generalized solution of a nonlinear reaction-diffusion system with only integral terms in the boundaries. We first solve a particular case of the problem by using the energy-integral method. Next, via an iteration procedure, we derive the obtained results to study the solvability of the stated problem.