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Multiple solutions for resonant elliptic equations via local linking theory and Morse theory. (English) Zbl 0998.35021

The authors deal with the multiple solutions of the problem

$\left\{\begin{array}{cc}-{\Delta }u=g\left(x,u\right)\phantom{\rule{1.em}{0ex}}\hfill & \text{in}\phantom{\rule{4.pt}{0ex}}{\Omega }\hfill \\ u=0\phantom{\rule{1.em}{0ex}}\hfill & \text{on}\phantom{\rule{4.pt}{0ex}}\partial {\Omega }\hfill \end{array}\right\$

where $g\in {C}^{1}\left(\overline{{\Omega }}×ℝ,ℝ\right)$. They consider two classes of elliptic resonant problems, namely:

a) double-double resonant case, b) the authors introduce some new conditions and compute the critical groups both at zero and at infinity precisely. Combined local linking theory and Morse theory, the authors present three solutions for the completely resonant case.

##### MSC:
 35J65 Nonlinear boundary value problems for linear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE)