- Existence Result for Differential Inclusion with p(x)-Laplacian
- A system of evolutionary problems driven by a system of hemivariational inequalities
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Existence Result for Differential Inclusion with p(x)-Laplacian
Theory Appl. Xiao, N. Huang and M. Wong, Well-posedness of hemivariational inequalities and inclusion problems, Taiwanese J. Zeng, S. Li, W. Zhang and X. Xue, Hadamard well-posedness for a set-valued optimization problem, Optim. Zolezzi, Extended well-posedness of optimization problems, J. The first is a topological minimax theorem: Theorem 2. The next lemma summarizes the properties of J : Lemma 4. We begin by giving an estimate of the integral which defines J : from the Lebourg mean value theorem [13, Theorem 1. Hence, J is well defined.
The main result Following , we first prove our main result in the form of an alternative: Theorem 4. The condition A1 is trivial. Thus, condition A2 is fulfilled. This concludes the proof. Moreover, assume that condition C is satisfied. Now we prove that the functional J admits a non-convex superlevel set. By reducing ourselves to considering symmetric functions, as said before, we easily overcome the problem of verifying condition E : Lemma 6. Now we define the class of functionals we will be dealing with: Definition 5. An important property of invariant functionals is expressed by the principle of symmetric crit- icality.
Then, every critical point of h XG is also a critical point of h. From our general result we deduce an analogous of Corollary 2, which assures the existence of at least three symmetric solutions: Corollary 3. Let F be as in Theorem 4 and satisfying condition C. Now we prove that I is G-invariant on X.
Then, applying Theorem 5, we deduce that the critical points of I XG are actually critical points of I. References  T. Bartsch, Z. Wang, Existence and multiplicity results for some superlinear elliptic problems on RN , Comm. Partial Differential Equations 20 — Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Varga, An existence result for hemivariational inequalities, Electron. Differential Equations 37 1— Efimov, S. Stechkin, Approximate compactness and Chebyshev sets, Soviet Math.
A system of evolutionary problems driven by a system of hemivariational inequalities
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