Asymptotically linear elliptic problems at resonance

Journal of Differential Equations, 1978

We prove the existence of a nontrivial solution for the nonlinear elliptic problem −∆u = λh(x)u + a(x)g(u) in R N , where g is superlinear near zero and near infinity, a(x) changes sign, λ is positive, and h(x) 0 is a weight function. For g odd, we prove the existence of an infinite number of solutions. * Mathematics Subject Classifications: 35J25, 35J20, 58E05.

Acta Mathematica Et Informatica Universitatis Ostraviensis, 1994

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This paper studies the existence of bounded solutions of a forced non-linear differential equation of arbitrary order. Necessary and sufficient conditions for the existence of such solutions are obtained. These results are inspired by classical results on the periodic problem, both in the resonant and non-resonant cases.

Nonlinear Analysis: Theory, Methods & Applications, 1995

Proceedings of the American Mathematical Society, 1982

In solving equations of the form Lu-Nu = p in a Hubert space, where L is linear and N is nonlinear, the alternative method can sometimes be used to reduce the problem to one in a subspace. In this note previous reduction results are extended and at the same time the proofs are simplified. The approach is to use simple fixed point theorems in place of the traditional variational methods which are often quite delicate. 1. Introduction. This note is to extend previous results by Castro [6], Bates and Castro [3], Amann [1], Bates [2] and Mawhin [7, 8], while giving simpler proofs of the results in [1, 2, 3]. An application of the abstract results shows that under certain assumptions on g, the equation y" + g(t + v) = c = constant has no 2w-periodic solutions if c =£ 0 and if c = 0 has a continuum of such solutions. This example was chosen because it seemed interesting; it is not a general representative of the class of equations treated here. In fact the abstract results below may be applied to semilinear elliptic and hyperbolic PDE's. To proceed, let 77 be a (real or complex) Hubert space with inner product (•, •) and norm || • ||, let L be a linear operator in 77, TV be a nonlinear operator on 77 and let/? £ 77. Consider the question of solvability of (1.1) Lu-Nu=p. In [7] Mawhin has Theorem A. Suppose L is selfadjoint with spectrum a and TV has a selfadjoint Gâteaux derivative N'(u) which satisfies (1.2) There exist numbers a < b with [a, b] (~) a = 0 and ai < N'(u) < bl for each u E 77. Then (1.1) is uniquely solvable with the solution depending (Upschitz) continuously on p.

Nonlinear Analysis: Theory, Methods & Applications, 1999

Journal of Mathematical Analysis and Applications, 1990

Commentationes Mathematicae Universitatis Carolinae, 1999

Acta Mathematica Sinica, 1987

Abotrac*t. Existence of 2n-periodic solutions to the equation ~ + g(x)=p(t) is proved, under sharp nonresonance conditions on the interaction of g(x)/x with two consecutive eigenvalues m 2 and (m + 1)2: touch with the eigenvalues is allowed.

Topological Methods in Nonlinear Analysis, 1998