一類橢圓方程解的存在性

萬保成, 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

事實上, 文獻(xiàn)[4]在非線性項超線性增長時結(jié)合Ambrosetti-Rabinowitz(AR)條件或在漸近線性增長且在無窮遠(yuǎn)處關(guān)于λ1的共振條件滿足時分別得到了一個非平凡解. 文獻(xiàn)[5-6]在非線性項滿足超線性增長但(AR)條件不滿足的情形下, 結(jié)合非線性項在零點處的漸近性態(tài), 得到了問題(1)非平凡解的存在性.

2) 當(dāng)t→+∞時,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的正測度子集Ω0, 使得w0(x)≠0 a.e.x∈Ω0. 從而對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, ‖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.

[1] Andreu F, Mazón J M, Rossi J D, et al. A Nonlocalp-Laplacian Evolution Equation with Neumann Boundary Conditions [J]. J Math Pures Appl, 2008, 90(2): 201-227.

[2] Mashiyev R A, Alisoy G, Ogras S. Solutions to Semilinearp-Laplacian Dirichlet Problem in Population Dynamics [J]. Appl Math and Mech, 2010, 31(2): 247-254.

[4] JIU Quan-sen, SU Jia-bao. Existence and Multiplicity Results for Dirichlet Problems withp-Laplacian [J]. J of Math Anal and Appl, 2003, 281(2): 587-601.

[5] FANG Fei, LIU Shi-bo. Nontrivial Solutions of Superlinearp-Laplacian Equations [J]. J of Math Anal and Appl, 2009, 351(1): 138-146.

[6] LIU Shi-bo. On Superlinear Problems without the Ambrosetti and Rabinowitz Condition [J]. Nonlinear Analysis: Theory, Methods & Application, 2010, 73(3): 788-795.

[7] OU Zeng-qi, LI Chun. Existence of Solutions for Dirichlet Problems withp-Laplacian [J]. Nonlinear Analysis: Theory, Methods & Application, 2012, 75(13): 4914-4919.

[8] Schechter M. A Variation of the Mountain Pass Lemma and Applications [J]. J London Math Soc, 1991, 44(3): 491-502.