Under certain assumptions, by virtue of the variational methods, the multiple weak solutions of the systems are obtained.
Also, exploiting variational methods and the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the p-biharmonic operator are investigated in [16].
Molica Bisci and Repovs̆ in [37], exploiting variational methods, investigated the existence of multiple weak solutions for a class of elliptic Navier boundary problems involving the p-biharmonic operator and presented a concrete example of an application.
In [9], Yin and Yang established the existence of multiple weak solutions in W 0 1, p for (6) where the nonlinearity f ( x, t ) is of concave-convex type, λ, θ > 0 are parameters, g ∈ L ∞ + and 1 < r < q < p < N.
By using a sub-supersolution approach and a mountain pass theorem, Giacomoni et al. [13] proved, among other things, that when (h(x equivlambda), (k(x equiv1), (0<alpha<1) and (p-1<beta <p^-1) ((p^) is the critical Sobolev exponent of p), problem (1) has <span class="lh">multiple weak solutions (depending on certain value of the parameter λ).
If (I u)) satisfies the ((PS _{c}) condition, by using Ljusternik-Schnirelman's theory for (Z_{2}) invariant functional (see [18]), we get a sequence of critical points ({u_{n}}) of (I u)), which implies that (I u)) has multiple weak solutions.
