Volume 2009, Article ID 540360,8pages doi:10.1155/2009/540360
Research Article
Existence of Nontrivial Solution for
a Nonlocal Elliptic Equation with Nonlinear Boundary Condition
Fanglei Wang and Yukun An
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China
Correspondence should be addressed to Fanglei Wang,[email protected] Received 15 December 2008; Accepted 17 February 2009
Recommended by Zhitao Zhang
In this paper, we establish two different existence results of solutions for a nonlocal elliptic equations with nonlinear boundary condition. The first one is based on Galerkin method, and gives a priori estimate. The second one is based on Mountain Pass Lemma.
Copyrightq2009 F. Wang and Y. An. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
In this paper, we deal with the following elliptic equation with nonlinear boundary condition:
−Δuu fx, u M
Ω|∇u|2|u|2dx, inΩ,
∂u
∂γ gx, u, on∂Ω,
1.1
whereΩis a bounded domain inRN with smooth boundary∂Ω,N > 2,∂/∂γ is the outer unite normal derivative,M: R → Ris continuous,f :Ω×R → R,g :∂Ω×R → Rare Carath´eodory functions.
For1.1, if the nonlocal termM
Ω|∇u|2|u|2dxis replaced byM
Ω|∇u|2dx, then the equation
−M
Ω|∇u|2dx
Δufx, u, inΩ 1.2
is related to the stationary analog of the Kirchhoffequation:
utt−M
Ω|∇xu|2dx
Δxufx, t, 1.3
whereMs asb, a, b >0. It was proposed by Kirchhoff 1as an extension of the classical D’Alembert wave equations for free vibrations of elastic strings. The Kirchhoffmodel takes into account the length changes of the string produced by transverse vibrations. Equation 1.3received much attention and an abstract framework to the problem was proposed after the work 2. Some interesting and further results can be found in 3,4and the references therein. In addition,1.2has important physical and biological background. There are many authors who pay more attention to this equation. In particularly, authors concerned with the existence of solutions for1.2with zero Dirichlet boundary condition via Galerkin method, and built the variational frame in 5,6. More recently, Perera and Zhang obtained solutions of a class of nonlocal quasilinear elliptic boundary value problems using the variational methods, invariant sets of descent flow, Yang index, and critical groups 7,8.
If the nonlocal termM
Ω|∇u|2|u|2dxis replaced byM
Ω|u|2dx, then the equation
−M
Ω|u|2dx
Δufx, u, inΩ 1.4
arises in numerous physical models such as systems of particles in thermodynamical equilibrium via gravitationalCoulombpotential, 2-D fully turbulent behavior of real flow, thermal runaway in Ohmic Heating, shear bands in metal deformed under high strain rates, among others. Because of its importance, in 9,10, the authors similarly studied the existence of solution for1.4with zero Dirichlet boundary condition.
On the other hand, elliptic equations with nonlinear boundary conditions have become rather an active area of research; see 11–15and reference therein. Those references present necessary and sufficient conditions of solutions of elliptic equations with nonlinear boundary conditions. In 13, the authors study the elliptic equation
Δufx, u, inΩ, 1.5
with the nonlinear boundary condition
∂u
∂γ gx, u, on∂Ω. 1.6
They obtain various existence results applying coincidence degree theory and the method of upper and lower solutions.
Inspired by the above references, we deal with the existence of solutions for elliptic equation 1.1 with nonlinear boundary condition based on Galerkin method and the Mountain Pass Lemma.
The paper is organized as follows. InSection 2, we will give the existence of solution for 1.1 via Galerkin method. In Section 3, we will study the solution for1.1 using the Mountain Pass Lemma.
2. Existence
In this section, we state and prove the main theorem via Galerkin method whenΩis bounded.
For convenience, we give the following hypotheses.
H1A typical assumption forMis that there exists anm0>0 such thatMs≥m0, for alls≥0.
H2For alls∈R, assume that the functionsf,gsatisfying
|fx, s| ≤C1
1|s|p1−1
, a.e. inΩ,
|gx, s| ≤C2
1|s|p2−1
, a.e. on∂Ω, 2.1
where C1, C2 > 0 are constants, 2 < p1<2∗ 2N/N−2, 2 < p1 < 2∗ 2N− 1/N−2.
H3The functionx→fx,0 gx,0is not identically zero.
LetW1,2Ω {u∈ L2Ω : ∇u ∈L2Ω}be endowed with norm u 2
Ω|∇u|2
|u|2dx. ThenW1,2Ωis a Banach space.
A functionu∈W1,2Ωis a weak solution of1.1if
Ω∇u∇ϕ dx
Ωuϕ dx−
∂Ωgx, uϕ dx
Ω
f
M u 2ϕ dx, 2.2
for all ϕ∈W1,2Ω.
Lemma 2.1. Suppose thatF:Rm → Rmis a continuous function such thatFξ, ξ ≥0 on|ξ|r, where·,·is the usual inner product inRmand| · |its related norm. Then, there existsz0 ∈Br0 such thatFz0 0.
Lemma 2.2see 16. LetΩbe a domain inRnsatisfying the uniformCm-regularity condition, and suppose that there exists a simplem, p-extension operatorEforΩ. Also suppose thatmp < nand p≤q≤p∗ n−1p/n−mp. Then
Wm,pΩ→Lq∂Ω. 2.3
Ifmp n, then the embedding still holds forp ≤ q < ∞. Moreover, if 1 < p ≤ q < p∗, then the embedding is compact.
Theorem 2.3. Assume that (H1)–(H3) hold. In addition, we suppose that
H4there exist constants λ, η, μ, C3 such that fx, uu ≤ λ|u|2 η|u|, ∀x ∈ Ω, u ∈ R, gx, uu≤μC3m0−1|u|2, ∀x on ∂Ωwith
λμ < m0. 2.4
Then problem1.1has at least one weak solution. Besides, any solutionusatisfies the estimate
u ≤ m0η|Ω|1/2
m0−λ−μ. 2.5
Proof. Let{ψk}be different complete orthonormal systems forW1,2Ωand set
VnSpan{ψ1, . . . , ψn}. 2.6
ThenVnis isometric toRn. Then, eachu∈Vnis uniquely associated toξ ξ1, . . . , ξnby the relationu ξkϕk. Since{ψk}are, respectively, orthonormal inW1,2Ω, we get u 2 ξ 2Rn.
We search for solutionsun∈Vnof the approximate problem
Ω∇un∇ψkdx
Ωunψkdx−
Ω
fx, un
M un 2ψkdx−
∂Ωgx, unψkdy0,
∀ψk∈W1,2Ω, k1,2, . . . , n.
2.7
To solve this algebraic system we define the operatorPn:Rn → Rn
Pnuk
Ω∇un∇ψkdx
Ωunψkdx−
Ω
fx, un M un 2ψkdx
−
∂Ωgx, unψkdy0, ∀u∈Vn.
2.8
By conditionH2, the growth of functionfis subcritical, sou→f·, udefines a continuous Nemytskii mappingNf :Lp1Ω → Lp1Ω. Similarly, we also define a continuous mapping Ng:Lp2Ω → Lp2Ω.
From the continuity ofMandfx, u, gx, u, with respect tou, we denote thatPnis continuous. Therefore, fromH1,H2,H4and H¨older’s inequality, we note thatu∈Vn
Pnu, u ≥ u 2−λ u 2η|Ω|1/2 u
m0 −
∂Ω
μ m0C3|u|2
dy. 2.9
On the other hand, byLemma 2.2, we have
∂Ω
μC−13 m−10 |u|2
dyμC−13 m−10 u 2L2∂Ω≤μC3m0−1C3 u 2 μ
m0 u 2, 2.10
whereC3>0 is constant.
From2.9and2.10, we can prove that
Pnu, u ≥
1−λμ m0
u 2− η
m0|Ω|1/2 u . 2.11
This shows that there existsR >0, depending only onm0, λ, η, μ, C3,Ω, such thatPnu, u ≥0 if u R. Then system2.7has a solutionun∈Vnsatisfying
un ≤R. 2.12
From this bound estimate, going to a subsequence if necessary, there areνandusuch that un 2 −→ν, un uweakly inW1,2Ω. 2.13
Besides, sinceW1,2Ω → Lp1Ω,W1,2Ω → Lp2∂Ωcompactly and the mappingNf, Ng is, respectively, continuousLp1Ω → Lp1ΩandLp2∂Ω → Lp2∂Ω
un−→u inLp1Ω, fx, un−→fx, u inLp1Ω,
un−→u inLp2∂Ω, gx, un−→gx, u inLp2∂Ω. 2.14
Then fixingkin2.7and lettingn → ∞, we conclude that
Ω∇u∇ψkdx
Ωuψkdx−
Ω
fx, u Mν ψkdx−
∂Ωgx, uψkdy0. 2.15 From the completeness of ψk, identity holds with ψk replaced by any ψ ∈ W1,2Ω. In particularly, whenψu, we get
Ω∇u∇ψkdx
Ωuψkdx−
Ω
fx, u Mν ψkdx−
∂Ωgx, uψkdy0. 2.16 On the other hand, letψkunin2.7and passing to the limit, we get
ν−
Ω
fx, uu Mν dx−
∂Ωgx, uu dy0. 2.17 Then we conclude thatν u 2, which shows thatuis a solution of1.1. Finally, ifuis any solution of1.1anduis nontrivial, then
u 2−
Ω
fx, uu M u 2dx−
∂Ωgx, uu dy0, u ≤ η|Ω|1/2
m0−λ−μ.
2.18
The proof is complete.
3. Variational Method
In this section, we consider the following problem:
M
Ω|∇u|2|u|2dx
−Δuu aup1−1b, inΩ,
∂u
∂γ cup2−1d, on∂Ω,
3.1
wherea, b, c, dare constants, andp1, p2are defined inH2.
The nontrivial solution of3.1comes from the Mountain Pass Lemma in 17.
Lemma 3.1Mountain Pass Lemma. LetEbe a Banach space and letI ∈ C1E, Rsatisfy the Palais-Smale condition. Suppose also that
iI0 0,
iithere exist constantsr, a >0 such thatIu≥a, if u r, iiithere exists an elementv∈Hwith v > r, Iv≤0.
DefineΓ:{g∈C 0,1; H:g0 0, g1 v}. Then cinf
g∈Γmax
0≤t≤1I gt 3.2
is a critical value ofI.
Theorem 3.2. Assume the conditions (H1)–(H3) hold. In addition, the functionMsatisfies
H5there existm1≥m0withm0/2−m1/p>0 andt0>0, such thatMt m1,∀t≥t0, wherepmin{p1, p2}.
Then3.1has a nontrivial solution.
Proof. The weak solutions of 3.1 are critical points of the functionalJ : W1,2Ω → R defined by
Ju 1
2M u 2− 1 p1
Ωaup1dx− 1 p2
∂Ωcup2dy−
Ωbu dx−
∂Ωdu dy, 3.3 whereMt t
0Msds.
Let us check theP Scondition. Letψ ∈W1,2, we have
JuψM u 2
Ω∇u∇ψuψdx
−
Ωaup1−1ψ dx
−
∂Ωcup2−1ψ dy−
Ωbψ dx−
∂Ωdψ dy.
3.4
Let {un}be a Palais-Smale sequence inW1,2Ω, that is, Jun → cand Jun → 0 and assume the contradiction that un → ∞, then, fromH1,H5, we have
Jun−1
pJunun≥ m0
2 −m1
p
un 2− 1
p1 − 1 p
Ωaupn1dx
− 1
p2 −1 p
∂Ωcupn2dy−
1− 1 p
Ωbundx−
1−1 p
∂Ωdundy, 3.5
wherea, b, c, d >0. Then by the Sobolev embedding theorem andLemma 2.2, we can select C >0 such that
CC un ≥m0 1
2 −1 p
un 2, 3.6
which is a contradiction with un → ∞. Hence{un}is bounded inW1,2Ω. So{un}admits a weakly convergence subsequence. From H2, all the growth of f, gis subcritical, so the standard argument shows that{un}admits a strongly convergence subsequence.
Next we will verify the hypotheses of Lemma 3.1. By H¨older’s inequality, Sobolev embedding theorem, andLemma 2.2, we have
Ωa|u|p1dxa u pL1p1Ω≤N1 u p1,
∂Ωc|u|p2dyc u pL2p2∂Ω≤N2 u p2,
Ωb|u|dx≤bN3 u ,
∂Ωd|u|dy≤dN4 u .
3.7
So we obtain
Ju≥ 1
2m0 u 2−N1 u p1−N2 u p2−bN3 u −dN4 u . 3.8
Let u <1, we get
Ju≥ 1
2m0 u 2−N5 u p−bN3 u −dN4 u . 3.9
Lethr 1/2m0r2−N5rp−N6ε·r, then we taker r03N6ε/2m0such thathr0 a 3N62/m0ε2−N53PN6pεp>0, whenεis sufficient small.
So forbanddsmall enough, then we haveJu≥a >0 for all u r0.
On the other hand, takeω0∈W1,2Ωwith
Ωaω0p1dx1 fork >0, we have Jkω0
1
2M kω 0 2− kp1 p1
Ωaω0p1dx− 1 p2
∂Ωckω0p2dy−
Ωbkω0dx−
∂Ωkdω0dy
≤ 1
2m1 kω0 2−kp1 p1
Ωaωp01dx− 1 p2
∂Ωckω0p2dy−
Ωbkω0dx−
∂Ωkdω0dy.
3.10 Sincep1, p2>2, we obtainJkω0 → −∞whenk → ∞.
Letωkω0, withklarge enough, we have ω >max{t0, r0}andJω< a. So by the Mountain Pass Lemma andH3, we have a nontrivial solutionuxfor3.1. The proof is complete.
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