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Landesman-Lazer type problems at an eigenvalue of odd multiplicity. (English) Zbl 0780.35043

One typical result of this paper is that the equation \(Lu-\lambda u+ F(u)=h\) has at least one solution if \(\lambda\in [0,\lambda_ +]\), and at least two solutions if \(\lambda\in [\lambda_ -,0)\) provided that \(\langle h,y\rangle > \liminf_{n\to\infty} \langle F(t_ n y_ n+t^ \alpha_ n z_ n),y\rangle\) for \(y_ n\to y\in N(L)\), \(\| y_ n\|=1\), \(t_ n\to\infty\), \(z_ n\) some bounded sequence, \(\alpha\in [0,1)\). Here \(L\) is a linear Fredholm operator (with index zero) on a Banach space \(X\) having zero as an eigenvalue of odd multiplicity. The perturbation \(F\) is sublinear, i.e. \(\| F(u)\|\leq c\| u\|^ \alpha+d\), and \(\langle\cdot,\cdot\rangle\) denotes some bilinear form on \(X\times X\). The proof is based on the theory of topological degree and bifurcation theory. The result applies to semilinear ODE and PDE.

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
47J05 Equations involving nonlinear operators (general)
58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces
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References:

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