一類橢圓方程解的存在性

萬(wàn)保成, 李 健, 李士軍

(吉林農(nóng)業(yè)大學(xué) 信息技術(shù)學(xué)院, 長(zhǎng)春 130118)

考慮如下擬線性橢圓Dirichlet邊值問題:

(1)

(H1) 存在C>0和q∈(p,p*)(如果1

事實(shí)上, 文獻(xiàn)[4]在非線性項(xiàng)超線性增長(zhǎng)時(shí)結(jié)合Ambrosetti-Rabinowitz(AR)條件或在漸近線性增長(zhǎng)且在無(wú)窮遠(yuǎn)處關(guān)于λ1的共振條件滿足時(shí)分別得到了一個(gè)非平凡解. 文獻(xiàn)[5-6]在非線性項(xiàng)滿足超線性增長(zhǎng)但(AR)條件不滿足的情形下, 結(jié)合非線性項(xiàng)在零點(diǎn)處的漸近性態(tài), 得到了問題(1)非平凡解的存在性.

2) 當(dāng)t→+∞時(shí),J(tφ1)→-∞.

引理2在定理1的假設(shè)下, 函數(shù)J滿足(C)c條件.

J(un)→c∈R, (1+‖un‖)‖J′(un)‖→0,n→+∞.

(2)

(3)

由(H4)知, 對(duì)任意的>0, 存在M3>0, 使得

(4)

令wn=un/‖un‖p, 則存在{wn}的子列(不妨仍記為{wn})及w0∈W01,p(Ω), 使得wn?w0, 并且wn(x)→w0(x) a.e.x∈Ω. 由式(4)可得1≤(η+). 因此, 存在Ω的正測(cè)度子集Ω0, 使得w0(x)≠0 a.e.x∈Ω0. 從而對(duì)a.e.x∈Ω0, 有un(x)→∞(n→∞). 由(H3)和Fatou引理可得這與式(3)矛盾. 因此{(lán)un}在中有界. 從而存在使得當(dāng)n→∞時(shí), ‖un‖→‖u0‖.

由引理1與引理2, 并應(yīng)用推廣形式的山路定理[8], 即可完成定理1的證明.

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