On the number of solutions of Dirichlet problems for some quasilinear elliptic equations

In this dissertation, the number of weak solutions to the problem Dpu+fu =t^p-2th&d4; x+hx inW u=0on6W is investigated where p > 1; Ω ⊂ $RN$, N ≥ 2, is a bounded domain with a sufficiently smooth boundary; ĥ, h ∈ C(); ĥ ≥ 0, and ĥ $\not\equiv$ 0 in .; Under the assumptions, lim s→-∞fs s^p-2s =a, lim s→∞fs s^p-1=b, , and $b>l1$, where $l1$ is the principal eigenvalue of Dpu+lu ^p-2u=0inW u=0on 6W, we show there exists t_h $\gg$ 1 such that if t ≥ t_h, the problem has at least two solutions. In addition, if ĥ > 0 in , we show that there exist −$\infty$ < t₁ ≤ t₂ < $\infty$ such that if t < t₁ the problem has no solution, if t₁ ≤ t ≤ t₂, the problem has at least one solution, and if t > t₂, the problem has at least two solutions.