Multiplicity Results for Two Classes of Boundary-Value Problems

Abstract

Multiplicity results are provided for two classes of boundary-value problems with cubic nonlinearities, depending on a parameter A. In particular, it is proved that for sufficiently large A, there are exactly two solutions, and that all solutions lie on a single smooth solution curve. The last fact allows one to use continuation techniques to compute all solutions.

Exact multiplicity for some nonlinear elliptic equations in balls

“…If the corresponding linearized equation (2.2) has only the trivial solution w = 0, then by the implicit function theorem we can solve (1.1) for λ < λ 1 and λ close to λ 1 , obtaining a continuous curve in λ of solutions u(x, λ). As in Korman and Ouyang [8], this process of decreasing λ cannot be continued indefinitely, since for sufficiently small λ > 0, the problem (1.1) has no solution. Let λ 0 be the infimum of λ for which we can continue the curve of solutions to the left.…”

“…While our idea is in the same line with [8], there are some difficulties in higherdimensional case. We employ various identities to overcome them.…”

“…Gardner and Peletier [11] obtained the exact number of solutions of problem (1.1) under the assumptions (f1), (f2), (f3) and λ sufficiently large, while Dancer [4] obtained the same result for domains with symmetry and λ large. On the other hand, when n = 1, exact multiplicity results are proved by J. Smoller and A. Wasserman [15] and S.-H. Wang [16] using phase-plane analysis, and by Korman, Yi Li, and Ouyang [8], [9] using bifurcation analysis.…”

“…As in [8] and [9], our main tool is the following bifurcation theorem due to Crandall-Rabinowitz [2].…”

On a Dirichlet problem involving p-Laplacian

“…In this paper, under novel assumptions, we are interested in ensuring the existence of at least three weak solutions for the problem (1). As usual, a weak solution of (1) Multiplicity results for problem (1) have been broadly investigated in recent years (see, for example, [1,2,4,5] ); for instance, in [1], using variational methods, the authors ensure the existence of a sequence of arbitrarily small positive solutions for problem (1) when the function f has a suitable oscillating behaviour at zero. Also, In [4], the author proves multiplicity results for the problem…”