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A three critical points theorem revisited. (English) Zbl 1214.47079

Let $X$ be a reflexive Banach space, $I\subset ℝ$ an interval; ${\Phi }:X\to ℝ$ a sequentially weakly lower semicontinuous ${C}^{1}$ functional, bounded on each bounded subset of $X$, whose derivative admits a continuous inverse on ${X}^{*}$; $J:X\to ℝ$ a ${C}^{1}$ functional with compact derivative. Assume that ${lim}_{\parallel x\parallel \to \infty }\left({\Phi }\left(x\right)+\lambda J\left(x\right)\right)=+\infty$ for all $\lambda \in I$, and there exists $\rho \in ℝ$ such that ${sup}_{\lambda \in I}{inf}_{x\in X}\left({\Phi }\left(x\right)+\lambda \left(J\left(x\right)+\rho \right)\right)<{inf}_{x\in X}{sup}_{\lambda \in I}\left({\Phi }\left(x\right)+\lambda \left(J\left(x\right)+\rho \right)\right)$.

Then there exists a subset $A\subset I$, $A\ne \varnothing$, and $r>0$ with the following property: for every $\lambda \in A$ and every ${C}^{1}$ functional ${\Psi }:X\to ℝ$ with compact derivative, there exists $\delta >0$ such that, for each $\mu \in \left[0,\delta \right]$, the equation ${{\Phi }}^{\text{'}}\left(x\right)+\lambda {J}^{\text{'}}\left(x\right)+\mu {{\Psi }}^{\text{'}}\left(x\right)=0$ has at least three solutions in $X$ whose norms are less than $r$.

MSC:
 47J30 Variational methods (nonlinear operator equations) 58E05 Abstract critical point theory 49J35 Minimax problems (existence) 35J60 Nonlinear elliptic equations