Three solutions of a fourth order elliptic problem via variational theorems of mixed type

Abstract

Dip. Me.Mo.Mat.via B o~i a m o 25b, Abstract We prove the existence of two nontrivial ~olutions for the fourth orderXk(Xk -c) where 2 5 k 5 i or 6 > X,(X, -c) and 6 is close to X, (X, -c) where j 2 i + 1. (Here (A,),?, is the sequence of the eige~ivalues of -A in H;(R)). Moreover if c > XI, c is close to XI , b > X,(Xj -c) and b is close to Xj(X, -c ) where j 2 2 we get three non trivial solutions. AMS: 35J40KEY WORDS: Multiple solutions, variational theorems of mixed type, aeroelastic oscillations.

Four nontrivial solutions for subcritical exponential equations

“…when there are assumptions both on the values of the functional on some suitable sets and on the values of its gradient. Theorems of this kind were first introduced in [11] and then developed in [8][9][10]12] in a sequential case, [7] in a nonsmooth version, and they were fruitfully applied in many cases to establish multiplicity results ( [7,[13][14][15][16][17],…). For definition and proofs we refer to [8,9,12], where more general abstract results are proved.…”

“…Finally, exploiting an additional linking structure, we get the existence of a fourth nontrivial solution. This idea has been used in subcritical problems in R N , N ≥ 3 [15,17], also in presence of higher order differential operators [13,14,16]; in particular, the results showed in Theorem 2 are the counterpart in R 2 of the results found in [17] for subcritical power-like nonlinearities in R N , N ≥ 3, but in our case there are many difficulties due to the fact that we cannot control the growth of the nonlinearity with any power, and thus by the norm of the functions, as it is possible to do with power-like nonlinearities in R N , N ≥ 3.…”

Multiplicity of critical points in presence of a linking: application to a superlinear boundary value problem

“…Theorems of this kind were first introduced in [16] and then developed in [17] (see also [15] for a sequential version of a theorem of this kind) and they were fruitfully applied in many cases to establish multiplicity results (see also [18], [19], [20], [21]). For the definition and the proves we refer to [16].…”

Fourth order elliptic system with dirichlet boundary condition

“…Micheletti, Pistoia and Sacon [9] also proved that if c < l 1 and b ≥ l 2 (l 2 -c), then (1.2) has at least three solutions by variational methods. Choi and Jung [2] also considered the single fourth order elliptic problem…”

“…They [3] also proved that when c < l 1 , l 1 (l 1 -c) < b < l 2 (l 2 -c) and s <0, (1.3) has at least three solutions by using degree theory. In [7][8][9] the authors investigate the existence of solutions of jumping problems with Dirichlet boundary condition.…”