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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