Lower and upper solutions for elliptic problems in nonsmooth domains

Abstract

In this paper we prove some existence results of semilinear Dirichlet problems in nonsmooth domains in presence of lower and upper solutions well-ordered or not. We first prove existence results in an abstract setting using degree theory. We secondly apply them for domains with conical points.

Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions

“…In addition, following the ideas of [5,16] we achieve multiplicity results for problem (1) in presence of more than a pair of lower and upper solutions. Now we present some preliminaries concerning the new degree theory for a class of discontinuous operators that we need for our results (for further details see [6]).…”

“…We say that the admissible discontinuity curve γ is inviable for the differential equation if it satisfies (4) or (5).…”

“…Multiplicity results. In [5,10,16,18] conditions which guarantee the existence of several solutions for second-order problems were given. They are based on degree theory and lower and upper solutions technique.…”

Monotone iterative sequences for non-local elliptic problems

“…The maximum principle does not hold if g(u) is increasing, but the constructing ordered upper and lower solutions guarantee the existence of a solution for the problem (1.3)-(1.4), without the condition on g to be decreasing [10] (Theorem 3.1). If g(u) satisfies the one-sided Lipschitz condition, then we can define two convergent monotone sequences of upper and lower solutions.…”

Analytical sequences of upper and lower solutions for a class of elliptic equations

“…Similar arguments have been applied for a class of nonlocal problems in [3], and analytical bounds have been obtained for the critical (blow up) parameter. In a recent work, De Coster et al [9] have proved the existence of solutions for the problem in nonsmooth domains in the presence of upper and lower solutions without ordering. If the upper and lower solutions are close to each other then they give us good information about the solution.…”