Three problems about integral functionals on Sobolev spaces

BIAGIO RICCERI

Department of Mathematics, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Let Ω ⊂ R^{n} (n ≥ 3) be an open bounded set. Set X = W_{0}^{1,2}(Ω). For q > 0, denote by A_{q} the class of all Carathéodory functions f : Ω × R → R such that

sup_{(x,ξ)∈Ω×R} \frac{|f(x, ξ)|}{1 + |ξ|^{q}} < +∞ .

For 0 < q ≤ \frac{n+2}{n−2} and f ∈ A_{q}, put

Φ_{f}(u) = ∫_{Ω} (∫_{0}^{u(x)} f(x, ξ)dξ) dx

and

J_{f}(u) = \frac{1}{2} ∫_{Ω} |∇u(x)|^{2}dx − Φ_{f}(u)

for all u ∈ X.

So, the functional J_{f} is continuously Gâteaux differentiable on X and one has

J′_{f}(u)(v) = ∫_{Ω} ∇u(x)∇v(x)dx − ∫_{Ω} f(x, u(x))v(x)dx

for all u, v ∈ X. Further, if q < \frac{n+2}{n−2}, the functional Φ_{f} is sequentially weakly continuous, by Rellich-Kondrachov theorem.

The aim of this lecture is to motivate the interest for the study of the following problems.

PROBLEM 1. - Is there some f ∈ A_{q}, with q < \frac{n+2}{n−2}, which is not of the form f(x, ξ) = a(x), such that the functional Φ_{f} is continuous with respect to the weak topology of X ?

PROBLEM 2. - Denote by τ_{s} the topology on X whose members are the sets A such that X \ A is sequentially weakly closed. Is there some f ∈ A_{q}, with q < \frac{n+2}{n−2}, such that, for each λ > 0 and r ∈ R, the functional J_{λf} is unbounded below and the set J_{λf}^{−1}(r) has no isolated points with respect to the topology τ_{s} ?

PROBLEM 3. - Denote by τ the strongest vector topology of X. Is there some f ∈ A_{\frac{n+2}{n−2}} such that the set

{(u, v) ∈ X × X : J′_{f}(u)(v) = 1}

is disconnected in (X, τ) × (X, τ) ?

References

[1] A.J.B. LOPES-PINTO, On a new result on the existence of zeros due to Ricceri, J. Convex Anal., 5 (1998), 57-62.

[2] B. RICCERI, Existence of zeros via disconnectedness, J. Convex Anal., 2 (1995), 287-290.

[3] B. RICCERI, A further improvement of a minimax theorem of Borenshtein and Shul’man, J. Nonlinear Convex Anal., 2 (2001), 279-283.