Narrow Results

In this paper, we establish several results involving the minimum and maximum principles and the comparison principles for elliptic equations and parabolic equations on finite graphs. The results are then used to prove the monotonicity and asymptotic properties of solutions for parabolic equations whose initial values are given by the equation $\mathrm{\Delta }\psi +f(\psi )=0$ with Dirichlet boundary conditions. Finally, an illustration with numerical experiments is provided to demonstrate our main results.

We prove some sharp estimates on the summability properties of the second derivatives of solutions to the equation –Δ_pu = f(x), under suitable assumptions on the source term. As an application, we deduce some strong comparison principles for the p-Laplacian, in the case of vanishing source terms.

Given $\mathrm{\Omega }$ a bounded open subset of ${ℝ}^N$, we consider non-negative solutions to the singular semilinear elliptic equation $-\mathrm{\Delta }u=\frac{f}{u^{\beta }}$ in $H_{\mathrm{\text{loc}}}^1(\mathrm{\Omega })$, under zero Dirichlet boundary conditions. For $\beta >0$ and $f\in L^1(\mathrm{\Omega })$, we prove that the solution is unique.

In this paper, we investigate various comparison principles for quasilinear elliptic equations of $p$-Laplace type with lower-order terms that depend on the solution and its gradient. More specifically, we study comparison principles for equations of the following form:

Our purpose is to generalize some recent comparison principles for operators driven by $p$-Laplacian to a wide class of quasilinear equations including $(p,q)$-Laplacian. It turns out, in particular, that adding a $q$-Laplacian to $p$-Laplacian allows to weaken the assumptions needed on the Hamiltonian of lower order terms. The results are specialized in the case that the Hamiltonian has at most polynomial growth in the gradient with coefficients depending on $x$ and $u$.