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Session 38. Variational Methods in Nonlinear Analysis |
Three critical point theorems with applications to nonlinear BVPs |
| Marek Galewski, Technical Unviersity of Łódź, Poland |
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In this talk we are concerned with three critical theorems applicable for \(
C^{1}\) action functionals connected to anisotropic problems. Results are
based on recent investigations and on ideas developed by Ricerri which can
be summarized as follows: Let \((X,\left\Vert .\right\Vert )\) be a uniformly
convex Banach space with strictly convex dual, \(J\in C^{1}(X)\) be a
functional with compact derivative, \(x_{0},x_{1}\in X,\) \(p,r\in
\mathbb{R}
\), \(p>1\), \(r>0\).
Assume (A.1) \(\underset{\left\Vert x\right\Vert \rightarrow \infty }{\lim \inf }\frac{J(x)}{\left\Vert x\right\Vert ^{p}}\geq 0;\) (A.2) \(\underset{x\in X}{\inf }J(x)<\underset{\left\Vert x-x_{0}\right\Vert \leq r}{\inf }J(x);\) (A.3) \(\left\Vert x_{1}-x_{0}\right\Vert <r\) and \(J(x_{1})<\underset {\left\Vert x-x_{0}\right\Vert =r}{\inf }J(x)\). There exists a nonempty open set \(A\subseteq (0,+\infty )\) s. t. for all \( \lambda \in A\) the functional \(x\rightarrow \dfrac{\left\Vert x-x_{0}\right\Vert ^{p}}{p}+\lambda J(x)\) has at least three critical points in \(X\). Main idea used in this talk are concerned with the following
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