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Existence theorems for nonselfadjoint semilinear elliptic boundary value problems. (English) Zbl 0535.35026

In this paper, following certain new arguments already implicitly applied for certain hyperbolic problems by L. Cesari and R. Kannan [Periodic solutions of nonlinear wave equations, Arch. Ration Mech. Anal. 82, 295-312 (1983; Zbl 0521.35037)], and by the authors [Global periodic solutions of the nonlinear wave equation, ibid. (to appear)] and more recently by Cesari and Bowman for nonselfadjoint nonlinear ordinary differential equations, we prove existence theorems for weak and strong solutions of problems of the form \(Ex=Nx\). Here E is a real nonselfadjoint elliptic operator of order 2m, \(m\geq 1\), in a bounded domain G in \({\mathbb{R}}^ n\) with a given system of linear homogeneous conditions \(Bx=0\) on the boundary \(\partial G\) of G. Then N is a Nemitsky operator of the form \((Nx)(t)=f(t)+g(t,D^{\alpha}x),t\in G,\) where f:\(G\to {\mathbb{R}}\) and \(g:G\times {\mathbb{R}}^{\mu}\to {\mathbb{R}}\) are supposed bounded and g continuous and depending on the \(\mu\) derivatives of x in G of order \(\alpha\), with \(0\leq | \alpha | \leq k_ 0<m\). Furthermore the boundary conditions given by B are expressed in terms of partial derivatives of order \(\beta\), \(0\leq | \beta | \leq k_ 0.\)
The remarkable Theorem of Landesman and Lazer for selfadjoint real elliptic partial differential equations of order 2 was first extended by S. A. Williams [J. Differ. Equations 8, 580-586 (1970; Zbl 0209.130)] and D. G. De Figueiredo [Partial differ. Equat. relat. Top., Tulane Univ. 1974, Lect. Notes Math. 446, 144-165 (1975; Zbl 0312.35032)] to adjoint elliptic problems of order 2m and nonlinearity g depending on derivatives of orders \(\alpha\), \(0\leq | \alpha |<2m-1\), and then by H. C. Shaw [J. Differ. Equations 26, 335-346 (1977; Zbl 0341.35039)] to the nonselfadjoint case with \(p=q<\infty\), where \(p=\dim Ker E\) and \(q=\dim Ker E^*\) under the assumption that the corresponding eigenfunctions have the same regions of positivity and negativity.
In the present paper we obtain some sufficient conditions of the Landesman-Lazer type for existence of solutions for elliptic problems which are not necessarily selfadjoint and not necessarily satisfying the requirement adpoted by Shaw.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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