Existence of solutions of semilinear systems with gradient dependence via eigenvalue criteria

Abstract

In this paper new criteria are established for the existence of positive radial solutions of a semilinear elliptic system depending on the gradient. These criteria are determined by some relationships between the upper and lower bounds on suitable stripes of R n of the nonlinearities of the system and the principal characteristic values of some associated linear Hammerstein integral operators. Moreover, using smoothing tools, the totality of the involved cone is established.2010 Mathematics Subject Classification. Primary 45G15, 35J57, secondary 35B07, 47H30.

New results on the sign of the Green function of a two-point n-th order linear boundary value problem

“…Incidentally, the calculation of the smallest eigenvalue of (4) with μ = 0 is also relevant to prove the existence of solutions of nonlinear boundary value problems of the type Ly + p(x)g(y) = 0, in particular, by comparing that eigenvalue with the quotient g (y) y for different values of y, especially when y → 0 + and when y → +∞. This approach was started by Erbe [11] for symmetric kernels and extended by Webb and Lan [12][13][14] and many others, [15] being a recent example.…”

Coupled elliptic systems depending on the gradient with nonlocal BCs in exterior domains

“…Cianciaruso and co-authors [8,9,12], De Figueiredo and co-authors [13,14] and Faria and co-authors [22].…”