Existence of Nontrivial Solution for a Nonlocal Elliptic Equation with Nonlinear Boundary Condition

Abstract

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.

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Recommended by Zhitao Zhang

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 15 December 2008; Accepted 17 February 2009

1. Introduction

In this paper, we deal with the following elliptic equation with nonlinear boundary condition: [figure omitted; refer to PDF] where Ω is a bounded domain in ^RN with smooth boundary ∂Ω , N>2 , ∂/∂γ is the outer unite normal derivative, M:^R+ [arrow right]R is continuous, f:Ω×R[arrow right]R , g:∂Ω×R[arrow right]R are Carathéodory functions.

For (1.1), if the nonlocal term M(_∫Ω |∇u^|2 +|u^|2 dx) is replaced by M(_∫Ω |∇u^|2 dx) , then the equation [figure omitted; refer to PDF] is related to the stationary analog of the Kirchhoff equation: [figure omitted; refer to PDF] where M(s)=as+b, a,b>0 . It was proposed by Kirchhoff [1] as an extension of the classical D'Alembert wave equations for free vibrations of elastic strings. The Kirchhoff model takes into account the length changes of the string produced by transverse vibrations. Equation (1.3) received 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, 4] and the references therein. In addition, (1.2) has 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 for (1.2) with 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 term M(_∫Ω |∇u^|2 +|u^|2 dx) is replaced by M(_∫Ω |u^|2 dx) , then the equation [figure omitted; refer to PDF] arises in numerous physical models such as systems of particles in thermodynamical equilibrium via gravitational (Coulomb) potential, 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 for (1.4) with zero Dirichlet boundary condition.

On the other hand, elliptic equations with nonlinear boundary conditions have become rather an active area of research; see [11-15] and 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 [figure omitted; refer to PDF] with the nonlinear boundary condition [figure omitted; refer to PDF] 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. In Section 2, we will give the existence of solution for (1.1) via Galerkin method. In Section 3, we will study the solution for (1.1) using the Mountain Pass Lemma.

2. Existence

In this section, we state and prove the main theorem via Galerkin method when Ω is bounded.

For convenience, we give the following hypotheses.

(H1) A typical assumption for M is that there exists an _m0 >0 such that M(s)≥_m0 , for all s≥0.

(H2) For all s∈R , assume that the functions f , g satisfying [figure omitted; refer to PDF] where _C1 ,_C2 >0 are constants, 2<_p1^{<2[low *]} =2N/(N-2) , 2<_p1 <^{2[low *]} =2(N-1)/(N-2) .

(H3) The function x...f(x,0)+g(x,0) is not identically zero.

Let ^W1,2 (Ω)={u∈^L2 (Ω):∇u∈^L2 (Ω)} be endowed with norm ||u^||2 =_∫Ω (|∇u^|2 +|u^|2 )dx . Then ^W1,2 (Ω) is a Banach space.

A function u∈^W1,2 (Ω) is a weak solution of (1.1) if [figure omitted; refer to PDF] for all [straight phi]∈^W1,2 (Ω) .

Lemma 2.1.

Suppose that F:^Rm [arrow right]^Rm is a continuous function such that ...F(ξ),ξ...≥0 on |ξ|=r , where ...[sm middot],[sm middot]... is the usual inner product in ^Rm and |[sm middot]| its related norm. Then, there exists _z0 ∈_Br ¯(0) such that F(_z0 )=0 .

Lemma 2.2 (see [16]).

Let Ω be a domain in ^Rn satisfying the uniform ^Cm -regularity condition, and suppose that there exists a simple (m,p) -extension operator E for Ω . Also suppose that mp<n and p≤q≤^p* =(n-1)p/(n-mp) . Then [figure omitted; refer to PDF] If mp=n , then the embedding still holds for p≤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

(H4) there exist constants λ,η,μ,_C3 such that f(x,u)u≤λ|u^|2 +η|u| , ∀x∈Ω, u∈R, g(x,u)u≤μ^{(_C3m0 )-1} |u^|2 , ∀x on ∂Ω with [figure omitted; refer to PDF]

Then problem (1.1) has at least one weak solution. Besides, any solution u satisfies the estimate [figure omitted; refer to PDF]

Proof.

Let {_ψk } be different complete orthonormal systems for ^W1,2 (Ω) and set [figure omitted; refer to PDF] Then _Vn is isometric to ^Rn . Then, each u∈_Vn is uniquely associated to ξ=(_ξ1 ,...,_ξn ) by the relation u=∑_{ξk[straight phi]k} . Since {_ψk } are, respectively, orthonormal in ^W1,2 (Ω) , we get ||u^||2 =||ξ^{||Rn 2} .

We search for solutions _un ∈_Vn of the approximate problem [figure omitted; refer to PDF] To solve this algebraic system we define the operator _Pn :^Rn [arrow right]^Rn [figure omitted; refer to PDF] By condition (H2), the growth of function f is subcritical, so u...f([sm middot],u) defines a continuous Nemytskii mapping _Nf :^L_p1 (Ω)[arrow right]^{Lp1[variant prime]} (Ω) . Similarly, we also define a continuous mapping _Ng :^L_p2 (Ω)[arrow right]^{Lp2[variant prime]} (Ω) .

From the continuity of M and f(x,u),g(x,u) , with respect to u , we denote that _Pn is continuous. Therefore, from (H1), (H2), (H4) and Hölder's inequality, we note that u∈_Vn [figure omitted; refer to PDF] On the other hand, by Lemma 2.2, we have [figure omitted; refer to PDF] where _C3 >0 is constant.

From (2.9) and (2.10), we can prove that [figure omitted; refer to PDF] This shows that there exists R>0 , depending only on _m0 ,λ,η,μ,_C3 ,Ω , such that ..._Pn u,u...≥0 if ||u||=R . Then system (2.7) has a solution _un ∈_Vn satisfying [figure omitted; refer to PDF] From this bound estimate, going to a subsequence if necessary, there are ν and u such that [figure omitted; refer to PDF] Besides, since ^W1,2 (Ω)...^L_p1 (Ω) , ^W1,2 (Ω)...^L_p2 (∂Ω) compactly and the mapping _Nf ,_Ng is, respectively, continuous ^L_p1 (Ω)[arrow right]^{Lp1[variant prime]} (Ω) and ^L_p2 (∂Ω)[arrow right]^{Lp2[variant prime]} (∂Ω) [figure omitted; refer to PDF] Then fixing k in (2.7) and letting n[arrow right]∞ , we conclude that [figure omitted; refer to PDF] From the completeness of _ψk , identity holds with _ψk replaced by any ψ∈^W1,2 (Ω) . In particularly, when ψ=u , we get [figure omitted; refer to PDF] On the other hand, let _ψk =_un in (2.7) and passing to the limit, we get [figure omitted; refer to PDF] Then we conclude that ν=||u^||2 , which shows that u is a solution of (1.1). Finally, if u is any solution of (1.1) and u is nontrivial, then [figure omitted; refer to PDF] The proof is complete.

3. Variational Method

In this section, we consider the following problem: [figure omitted; refer to PDF] where a,b,c,d are constants, and _p1 ,_p2 are defined in (H2).

The nontrivial solution of (3.1) comes from the Mountain Pass Lemma in [17].

Lemma 3.1 (Mountain Pass Lemma).

Let E be a Banach space and let I∈^C1 (E,R) satisfy the Palais-Smale condition. Suppose also that

(i) I(0)=0,

(ii) there exist constants r,a>0 such that I(u)≥a , if ||u||=r ,

(iii): there exists an element v∈H with ||v||>r, I(v)≤0 .

Define Γ:={g∈C[0,1]; H:g(0)=0, g(1)=v} . Then [figure omitted; refer to PDF] is a critical value of I .

Theorem 3.2.

Assume the conditions (H1)-(H3) hold. In addition, the function M satisfies

(H5) there exist _m1 ≥_m0 with (_m0 /2)-(_m1 /p)>0 and _t0 >0 , such that M(t)=_m1 , ∀t≥_t0 , where p=min{_p1 ,_p2 } .

Then (3.1) has a nontrivial solution.

Proof.

The weak solutions of (3.1) are critical points of the functional J:^W1,2 (Ω)[arrow right]R defined by [figure omitted; refer to PDF] where M^(t)=^∫0t M(s)ds .

Let us check the (PS) condition. Let ψ∈^W1,2 , we have [figure omitted; refer to PDF] Let {_un } be a Palais-Smale sequence in ^W1,2 (Ω) , that is, J(_un )[arrow right]c¯ and ^{J[variant prime]} (_un )[arrow right]0 and assume the contradiction that ||_un ||[arrow right]+∞ , then, from (H1), (H5), we have [figure omitted; refer to PDF] where a,b,c,d>0 . Then by the Sobolev embedding theorem and Lemma 2.2, we can select C>0 such that [figure omitted; refer to PDF] which is a contradiction with ||_un ||[arrow right]+∞ . Hence {_un } is bounded in ^W1,2 (Ω) . So {_un } admits a weakly convergence subsequence. From (H2), all the growth of f,g is subcritical, so the standard argument shows that {_un } admits a strongly convergence subsequence.

Next we will verify the hypotheses of Lemma 3.1. By Hölder 's inequality, Sobolev embedding theorem, and Lemma 2.2, we have [figure omitted; refer to PDF] So we obtain [figure omitted; refer to PDF] Let ||u||<1 , we get [figure omitted; refer to PDF] Let h(r)=(1/2)_m0^r2 -_N5^rp -_N6 [straight epsilon][sm middot]r , then we take r=_r0 =3_N6 [straight epsilon]/2_m0 such that h(_r0 )=a¯=(3^N62 /_m0 )^{[straight epsilon]2} -_N5^{3PN6p[straight epsilon]p} >0 , when [straight epsilon] is sufficient small.

So for b and d small enough, then we have J(u)≥a¯>0 for all ||u||=_r0 .

On the other hand, take _ω0 ∈^W1,2 (Ω) with _∫Ω a^ω0_p1 dx=1 for k>0 , we have [figure omitted; refer to PDF] Since _p1 ,_p2 >2 , we obtain J(k_ω0 )[arrow right]-∞ when k[arrow right]+∞ .

Let ω=k_ω0 , with k large enough, we have ||ω||>max{_t0 ,_r0 } and J(ω)<a¯ . So by the Mountain Pass Lemma and (H3), we have a nontrivial solution u(x) for (3.1). The proof is complete.

References

[1] G. Kirchhoff Mechanik , Teubner, Leipzig, Germany, 1883.

[2] J.-L. Lions, G. de la Penha, L. A. Medeiros, "On some questions in boundary value problems of mathematical physics," Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977) , vol. 30, of North-Holland Mathematics Studies, pp. 284-346, North-Holland, Amsterdam, The Netherlands, 1978.

[3] A. Arosio, S. Panizzi, "On the well-posedness of the Kirchhoff string," Transactions of the American Mathematical Society , vol. 348, no. 1, pp. 305-330, 1996.

[4] K. Ono, "On global solutions and blow-up solutions of nonlinear Kirchhoff strings with nonlinear dissipation," Journal of Mathematical Analysis and Applications , vol. 216, no. 1, pp. 321-342, 1997.

[5] C. O. Aives, F. J. S. A. Corrêa, T. F. Ma, "Positive solutions for a quasilinear elliptic equation of Kirchhoff type," Computers & Mathematics with Applications , vol. 49, no. 1, pp. 85-93, 2005.

[6] T. F. Ma, "Remarks on an elliptic equation of Kirchhoff type," Nonlinear Analysis: Theory, Methods & Applications , vol. 63, no. 5-7, pp. e1967-e1977, 2005.

[7] K. Perera, Z. Zhang, "Nontrivial solutions of Kirchhoff-type problems via the Yang index," Journal of Differential Equations , vol. 221, no. 1, pp. 246-255, 2006.

[8] Z. Zhang, K. Perera, "Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow," Journal of Mathematical Analysis and Applications , vol. 317, no. 2, pp. 456-463, 2006.

[9] R. Stanczy, "Nonlocal elliptic equations," Nonlinear Analysis: Theory, Methods & Applications , vol. 47, no. 5, pp. 3579-3584, 2001.

[10] F. J. S. A. Corrêa, D. C. de Morais Filho, "On a class of nonlocal elliptic problems via Galerkin method," Journal of Mathematical Analysis and Applications , vol. 310, no. 1, pp. 177-187, 2005.

[11] J. F. Bonder, J. D. Rossi, "Existence results for the p -Laplacian with nonlinear boundary conditions," Journal of Mathematical Analysis and Applications , vol. 263, no. 1, pp. 195-223, 2001.

[12] K. Chaïb, "Necessary and sufficient conditions of existence for a system involving the p -Laplacian (1<p<N) ," Journal of Differential Equations , vol. 189, no. 2, pp. 513-525, 2003.

[13] P. Amster, M. C. Mariani, O. Méndez, "Nonlinear boundary conditions for elliptic equations," Electronic Journal of Differential Equations , vol. 2005, no. 144, pp. 1-8, 2005.

[14] S.-Z. Song, C.-L. Tang, "Resonance problems for the p -Laplacian with a nonlinear boundary condition," Nonlinear Analysis: Theory, Methods & Applications , vol. 64, no. 9, pp. 2007-2021, 2006.

[15] J.-H. Zhao, P.-H. Zhao, "Existence of infinitely many weak solutions for the p -Laplacian with nonlinear boundary conditions," Nonlinear Analysis: Theory, Methods & Applications , vol. 69, no. 4, pp. 1343-1355, 2008.

[16] R. A. Adams, J. J. F. Fournier Sobolev Spaces , Academic Press, Amsterdam, The Netherlands, 2003.

[17] L. C. Evans Partial Differential Equations , American Mathematical Society, Providence, RI, USA, 1998.