We consider a second-order nonlinear degenerate anisotropic parabolic equation in the case when flux vector is only continuous and nonnegative diffusion matrix bounded measurable. The concepts of entropy sub- supersolution Cauchy problem are introduced, so that solution this problem, understood sense Chen-Perthame, both an supersolution. It established maximum subsolutions also subsolution prob...

In this article, using the sub-supersolution method and Rabinowitztype global bifurcation theory, we prove some results on existence, uniqueness and multiplicity of positive solutions for some singular nonlocal elliptic problems.

We show a comparison principle for viscosity superand subsolutions to Hamilton-Jacobi equations with discontinuous Hamiltonians. The key point is that the Hamiltonian depends upon u and has a special structure. The supersolution must enjoy some additional regularity.

Assuming that a system of quasilinear equations of gradient type admits a strict supersolution and a strict subsolution, we show that it also admits a positive solution.

Lu = aij(x)Diju for u ∈ C(Ω). Suppose u ∈ C(Ω) is a supersolution in Ω, i.e. Lu ≤ 0. Then if φ ∈ C(Ω) satisfies Lφ > 0, we get L(u− φ) < 0 in Ω, hence by the maximum principle, u− φ does not have interior local minima in Ω. Put differently, if φ ∈ C(Ω) is such that u− φ has a local minimum at x0 ∈ Ω, then necessarily Lφ(x0) ≤ 0. Geometrically, u− φ having a local minimum at x0 means that the gr...

Let E be a complete, separable metric space and A be an operator on Cb(E). We give an abstract definition of viscosity sub/supersolution of the resolvent equation λu − Au = h and show that, if the comparison principle holds, then the martingale problem for A has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in par...

We establish the existence of multiple solutions for a nonvariational elliptic systems involving p(x)-Laplacian operator. The approach combines methods sub–supersolution and Leray–Schauder topological degree.

This paper studies the H 1 control problem for a nonlinear, unbounded, innnite dimensional system with state constraints. We characterize the solvability of the problem by means of a Hamilton-Jacobi-Isaacs equation, proving that the H 1 problem can be solved if and only if the HJI equation has a nonnegative viscosity supersolution, vanishing at the origin.

In this paper we investigate numerically positive solutions of the equation −Δu = λu+u with Dirichlet boundary condition in a boundary domain Ω for λ > 0 and 0 < q < 1 < p < 2∗, we will compute and visualize the range of λ, this problem achieves a numerical solution. Keywords—positive solutions; concave-convex; sub-supersolution method; pseudo arclength method.