Positive solutions with changing sign energy to a nonhomogeneous elliptic problem of fourth order

M Talbi,N Tsouli

doi:10.5269/bspm.v29i1.11491

Abstract

In this paper, we study the existence for two positive solutions toa nonhomogeneous elliptic equation of fourth order with a parameter lambda such that 0 < lambda < lambda^. The first solution has a negative energy while the energy of the secondone is positive for 0 < lambda < lambda_0 and negative for lambda0 < lambda < lambda^. The values lambda_0 and lambda^ are given under variational form and we show that every corresponding critical point is solution of the nonlinear elliptic problem (with a suitable multiplicative term).

Highlights

May satisfy a variational problem similar to (V )
We consider the transformation of Poisson problem used by P.Drábek and M.Ôtani: We recall some properties of the Dirichlet problem for the Poisson equation:
Since the functional Eλ is even in t and that we are interested by the positive solutions, we limit our study for t > 0

Summary

Sobolev injection theorem we deduce that

Combining the two last inequalities and by Sobolev injection theorem there exist a constant c′ such that for every n we have (p − q)||Wn||pp < (r − q)||ΛWn||rr ≤ c′||Wn||rp. Lemma 2.4 The functionals v → Eλ(t1(v, λ), v) and v → Eλ(t2(v, λ), v) are bonded bellow in Lp(Ω). Proof : Let (vn) be a minimizing sequence of the functional v → Eλ(t1(v, λ), v). We know that ∂tEλ(t1(vn, λ), vn) = 0, [t1(vn, λ)]p||vn||pp = λ[t1(vn, λ)]q||Λvn||qq + [t1(vn, λ)]r||Λvn||rr. Sobolev injection of X in Lq(Ω) and the fact that lim sup ||Vn||p < +∞, implies n→+∞. Lemma 2.6 Let (vn) ⊂ S be a minimizing sequence of (2.12) (Wn) := (t2(vn, λ)vn)) are Palais-Smale sequences for the functional Eλ.

Now we show that

Let us put

We conclude that