We consider the nonlinear Dirichlet problem and develop conditions for the function such that the considered problem has a positive classical solution. Moreover, we present some results showing that is a bifurcation point in and in .
In the paper, we obtain the existence of symmetric or monotone positive solutions and establish a corresponding iterative scheme for the equation , , where , , subject to nonlinear boundary condition. The main tool is the monotone iterative technique. Here, the coefficient may be singular at .
The main goal in this paper is to prove the existence of radial positive solutions of the quasilinear elliptic system⎧ -Δpu = f(x,u,v) in Ω,⎨ -Δqv = g(x,u,v) in Ω,⎩ u = v = 0 on ∂Ω,where Ω is a ball in RN and f, g are positive continuous functions satisfying f(x, 0, 0) = g(x, 0, 0) = 0 and some growth conditions which correspond, roughly speaking, to superlinear problems. Two different sets of conditions, called strongly and weakly coupled, are given in order to obtain existence. We use...
We consider the boundary value problem involving the one dimensional -Laplacian, and establish the precise intervals of the parameter for the existence and non-existence of solutions with prescribed numbers of zeros. Our argument is based on the shooting method together with the qualitative theory for half-linear differential equations.
We provide existence and stability results for semilinear Dirichlet problems with nonlinearities satisfying some general local growth conditions. We derive a general abstract result which we then apply to prove the existence of solutions, their stability and continuous dependence on parameters for a sixth order ODE with Dirichlet type boundary data.
We solve the problem of the existence and uniqueness of coexistence states for the classical predator-prey model of Lotka-Volterra with diffusion in the scalar case.