SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT

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1 Electronic Journal of Differential Equations, Vol (2012), No. 139, pp ISSN: URL: or ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT GUANGGANG LIU, SHAOYUN SHI, YUCHENG WEI Abstract. In this article we consider elliptic equations whose nonlinear term depends on the gradient of the unknown. We assume that the nonlinearity has a asymptotically linear growth at zero and at infinity with respect to the second variable. By applying Morse theory and an iterative method, we prove the existence of nontrivial solutions. 1. Introduction In this article we consider the following elliptic equation with dependence on the gradient, u = f(x, u, u), in, (1.1) u = 0, on, where R N (N 3) is a bounded domain with smooth boundary. Since the nonlinearity f depends on the gradient of the solution, solving (1.1) is not variational. In fact the well developed critical point theory cannot be applied directly. There have been several works on this equation, using sub and supersolution, topological degree, fixed point theorems and Galerkin method; see, for instance, [1, 2, 6, 11, 15, 16, 17]. In [5], de Figueiredo, Girardi and Matzeu developed a quite different method of variational type. Under the assumptions that f has a superlinear subcritical growth at zero and at infinity with respect to the second variable, they obtained the existence of a positive and a negative solutions of (1.1) by using the mountain pass theorem and iterative technique. Later, this method was applied to quasilinear elliptic equations [7, 8, 13], Hamiltonian systems [9] and impulsive differential equations [14]. In general the above papers which used mountain pass technique assume that the nonlinearity has a superlinear subcritical growth at zero and at infinity with respect to the second variable. Here we show that Morse theory and iterative method can be used to find solutions to (1.1) under the assumption that f has a asymptotically linear growth at zero and at infinity with respect to the second variable Mathematics Subject Classification. 35J20, 35J25, 35J60. Key words and phrases. Semilinear equation; Morse theory; critical group; iterative method. c 2012 Texas State University - San Marcos. Submitted April 17, Published August 19,

2 2 G. LIU, S. SHI, Y. WEI EJDE-2012/139 Let 0 < λ 1 < λ 2 λ 3 λ k... be the eigenvalues associated with the eigenvectors ϕ 1, ϕ 2, ϕ 3, ϕ 4,... of with Dirichlet boundary condition, and we make the following assumptions: (H0) f : R R N R is continuous; (H1) f(x, t, ξ) = λt+g 0 (x, t, ξ), where λ j < λ < λ j+1, g 0 (x, t, ξ) = o( t ) as t 0 uniformly for x, ξ R N ; (H2) f(x, t, ξ) = µt + g(x, t, ξ), where λ k < µ < λ k+1, k j, k 1, g(x, t, ξ) = o( t ) as t uniformly for x, ξ R N ; (H3) f(x, t, ξ) = µt + g(x, t, ξ), where µ = λ k and λ k l 1 < λ k l = λ k l+1 = = λ k 1 = λ k < λ k+1, k j, k l + 1, g(x, t, ξ) C and G(x, t, ξ) as t uniformly for x, ξ R N, where C > 0 is a constant, G(x, t, ξ) = t g(x, s, ξ)ds; 0 (H4) For any x, t 1, t 2 R, ξ 1, ξ 2 R N, f(x, t, ξ) satisfies the Lipschitz condition f(x, t 2, ξ 2 ) f(x, t 1, ξ 1 ) L( t 2 t 1 + ξ 2 ξ 1 ), where L > 0 is a constant. By (H1), zero is a solution of (1.1), called trivial solution. The purpose of this article is to find nontrivial solutions. Our main results as as follows: Theorem 1.1. Assume that (H0), (H1), (H2), (H4) hold. If 0 < L λ 1 λ 1 L (1.1) has at least a nontrivial weak solution. Theorem 1.2. Assume that (H0), (H1), (H3), (H4) hold. If 0 < L λ 1 λ 1 L (1.1) has at least a nontrivial weak solution. < 1, then < 1, then This article is organized as follows. In section 2 we give a simple revisit to Morse theory. In section 3 we prove Theorem 1.1 and Theorem 1.2 by using Morse theory and iterative method. An example will be given in section Preliminaries about Morse theory Let H be a real Hilbert space and J C 1 (H, R) be a functional satisfying the (PS) condition. Denote by H q (A, B) the q-th singular relative homology group of the topological pair with coefficients in a field G. Let u be an isolated critical point of J with J(u) = c. The group C q (J, u) := H q (J c, J c \ {u}), q Z, is called the q-th critical group of J at u, where J c = {u H J(u) c}. Denote K = {u H \ J (u) = 0}. Assume that K is a finite set. Take a < inf J(K). The critical groups of J at infinity are defined by C q (J, ) := H q (H, J a \ {u}), q Z. The following result is important in proving the existence of nontrivial critical points. Proposition 2.1 ([3, Proposition 3.6]). Suppose J satisfies the (PS) condition. If K =, then C q (J, ) = 0, q Z. If K = {u 0 }, then C q (J, ) = C q (J, u 0 ), q Z. Let A and A 0 be bounded self-adjoint operators defined on H. According to their spectral decomposition, H = H + H 0 H, where H +, H 0, H are invariant

3 EJDE-2012/139 SEMILINEAR ELLIPTIC EQUATIONS 3 subspaces corresponding to the positive, zero and negative spectrum of A, respectively, similarly, H = H 0 + H0 0 H0, where H+ 0, H0 0, H0 are invariant subspaces corresponding to the positive, zero and negative spectrum of A 0, respectively. Let P 0 : H H 0 be the orthogonal projector. Consider the functionals Φ(u) = 1 2 A u, u + ϕ(u), Φ 0 (u) = 1 2 A 0u, u + ϕ 0 (u). We make the following assumptions: (A1) (A ) ± := A H ± has a bounded inverse on H ±. (A2) γ := dim(h H 0 ) <. (A3) ϕ C 1 (H, R) has a compact gradient mapping ϕ(u), and ϕ(u) = o( u ) as u. In addition, if dim H 0 0, we assume ϕ(u) C, u H, ϕ(p 0 u) as P 0 u. (A4) (A 0 ) ± := (A 0 ) H ± has a bounded inverse on H 0 ±. 0 (A5) β := dim(h0 ) <, and dim H0 0 = 0. (A6) ϕ 0 C 1 (H, R) has a compact gradient mapping ϕ 0 (u), and Also we use the following results. ϕ 0 (u) = o( u ) as u 0. Theorem 2.2 ([4, Lemma 5.1]). Assume that (A1) (A3) hold, then Φ satisfies the (PS) condition, and { G, q = γ, C q (Φ, ) = 0, q γ. Theorem 2.3 ([4, Theorem 4.1]). Assume that (A4) (A6) hold, then { G, q = β, C q (Φ 0, 0) = 0, q β. 3. Proof of Theorems 1.1 and 1.2 Let H0 1 () be the usual Sobolev space with the inner product u, v = u(x) v(x) dx, u, v H0 1 (). For w H0 1 (), consider the problem u = f(x, u, w), in, u = 0, on (3.1) and the associated functional I w : H0 1 () R, I w (v) = 1 v(x) 2 dx F (x, v(x), w(x)) dx. 2 By (H0) (H2) or (H0) (H3), I w C 1 (H 1 0 (), R), and the weak solutions of the problem (3.1) corresponds to the critical points of the functional I w, see [12].

4 4 G. LIU, S. SHI, Y. WEI EJDE-2012/139 Define the operators L, L 0 : H0 1 () H0 1 () by L u = u µ( ) 1 u and L 0 u = u λ( ) 1 u. Obviously, L and L 0 are bounded self-adjoint operators. Let φ w (u) = g(x, u(x), w(x)) dx, φ w0 (u) = g 0 (x, u(x), w(x)) dx. It is well known that φ w, φ w0 are compact mappings. By (H1), (H2) or (H1), (H3), φ w (u) = o( u ) as u, and φ w0 = o( u ) as u 0. We can rewrite the functional I w by I w (u) = 1 2 L u, u φ w (u) = 1 2 L 0u, u φ w0 (u). According to the spectral decomposition of the operator L, H0 1 () = H + H 0 H, where H +, H 0, H are invariant subspaces corresponding to the positive, zero and negative spectrum of L respectively. If (H2) holds, then H = span{ϕ 1, ϕ 2,... ϕ k }, H + = (H ). If (H3) holds, then H = span{ϕ 1, ϕ 2,... ϕ k l 1 }, H 0 = span{ϕ k l,... ϕ k }, H + = (H 0 H ). Similarly, according to the spectral decomposition of the operator L 0, H0 1 () = H 0 + H0 0 H0, where H+ 0, H0 0, H0 are invariant subspaces corresponding to the positive zero and negative spectrum of L 0 respectively. If (H1) holds, then L 0 is invertible and H0 = span{ϕ 1, ϕ 2,... ϕ j }, H 0 + = (H 0 ). Lemma 3.1. Assume that (H0) (H2) hold. Then for any w H 1 0 (), (3.1) has at least a nontrivial weak solution. Proof. By (H2), dim H 0 = 0, L H ± has a bounded inverse on H ± and dim H = k, thus (A1), (A2) and (A3) hold. So by Theorem 2.2, I w satisfies (PS) condition and { G, q = k, C q (I w, ) = 0, q k. By (H1), dim H 0 = 0, L 0 H ± has a bounded inverse on H 0 ± 0 (A4), (A5) and (A6) hold. By Theorem 2.3 { G, q = j, C q (I w, 0) = 0, q j. and dim H = j, thus Since k j, C q (I w, ) C q (I w, 0) for some q Z, hence by Proposition 2.1, I w has at least a nontrivial critical point and (3.1) has at least a nontrivial weak solution. Lemma 3.2. Assume that (H0), (H1), (H3) hold. Then for any w H 1 0 (), (3.1) has at least a nontrivial weak solution. Proof. By (H3), L H ± has a bounded inverse on H ±, and dim(h H 0 ) = k, so (A1), (A2) hold. On the other hand, dim H 0 > 0, but it can be checked that φ w (u) C for any u H0 1 () and a constant C > 0, and φ w (u) with u H 0 as u. Indeed, by (H3), Hölder inequality and Sobolev inequality, for any u, v H0 1 (), we have φ w (u), v g(x, u, w) v dx C( v 2 ) 1/2 C v,

5 EJDE-2012/139 SEMILINEAR ELLIPTIC EQUATIONS 5 where C, C > 0 are constants, this implies φ w (u) C for all u H0 1 (). We claim that φ w (u) with u H 0 as u. If this is not true, then there exists a sequence {u n } and constant M > 0 such that u n H 0, u n as n, and φ w (u n ) M. Let ũ n = un u, then ũ n n H 0 and ũ n = 1. By dim H 0 <, there exists a subsequence of {ũ n } still denoted by {ũ n }, and ũ such that ũ n converges strongly to ũ H 0 as n, then ũ satisfies the equation ũ = λ k ũ, in, ũ = 0, on. (3.2) Since ũ 0, by the unique continuation property as in [10], ũ 0 a.e. in, which implies u n a.e. in. Hence by (H3), G(x, u n (x), w(x)) a.e. in, then φ w (u n ) = G(x, u n (x), w(x)) dx as n, we obtain a contradiction. Therefore, (A3) holds. Next by using the argument used in the proof of Lemma 3.1 we complete the proof. Lemma 3.3. There exists a constant c 1 > 0 independent of w such that u w c 1 for all solutions u w obtained in Lemma 3.1 or Lemma 3.2. Proof. First we decompose u w as u w = u + w + u w H 0 + H 0. Since u w is a weak solution of the problem (3.1), one has u w φ dx = (λu w + g 0 (x, u w, w))φ dx, φ H0 1 (). (3.3) Particularly, take φ = u + w u w into (3.3), we have u w (u + w u w) λu w (u + w u w) dx = g 0 (x, u w, w)(u + w u w) dx. (3.4) By (H1), λ j 1 < λ < λ j, then we have u w (x) (u + w u w) λu w (x)(u + w u w) dx = ( u + w 2 λ u + w 2 ) ( u w 2 λ u w 2 ) dx (1 λ ) u + λ w 2 λ dx + ( 1) u j λ w 2 dx j 1 m u w 2 dx, (3.5) where m = min{(1 λ λ j ), ( λ λ j 1 1)} > 0. Fix (N + 2)/(N 2) > p > 1, by (H1) (H2) or (H1) (H3), for any ɛ > 0, there exists constant k ɛ > 0 such that

6 6 G. LIU, S. SHI, Y. WEI EJDE-2012/139 g 0 (x, t, ξ) ɛ t + k ɛ t p. By Hölder inequality and Sobolev inequality g 0 (x, u w (x), w(x))(u + w u w) dx (ɛ u w (x) + k ɛ u w (x) p )( u + w + u w ) dx ɛ u w L 2 () u + w L 2 () + ɛ u w L 2 () u w L 2 () + k ɛ u w p L p+1 () u+ w L p+1 () + k ɛ u w p L p+1 () u w L p+1 () ɛ u w u + λ w + ɛ u w u 1 λ w + Ck ɛ u w p u + w + Ck ɛ u w p u w 1 2ɛ u w 2 + 2Ck ɛ u w p+1. λ 1 (3.6) Combining (3.4), (3.5) and (3.6) we obtain (m 2ɛ λ 1 ) u w (x) 2 2Ck ɛ u w (x) p+1. Since m > 0, we can take ɛ > 0 sufficiently small such that m 2ɛ λ 1 > 0, note that p + 1 > 2, thus there exists a constant c 1 > 0 independent of w such that u w c 1. Proof of Theorem 1.1. First take u 0 H 1 0 (), by Lemma 3.1 we can construct a sequence {u n } such that for n 1 u n is a nontrivial solution of the equation u n = f(x, u n, u n 1 ), in, u n = 0, on. (3.7) From (3.7), u n+1 and u n satisfy u n+1 ( u n+1 u n ) = f(x, u n+1, u n )(u n+1 u n ), (3.8) u n ( u n+1 u n ) = f(x, u n, u n 1 )(u n+1 u n ), (3.9) By (3.8), (3.9), (H4), Sobolev inequality, and Hölder inequality, we obtain u n+1 u n 2 = (f(x, u n+1, u n ) f(x, u n, u n 1 ))(u n+1 u n ) dx L( u n+1 u n + u n u n 1 ) u n+1 u n dx L (u n+1 u n ) 2 dx + L( (u n u n 1 ) 2 dx) 1/2 ( λ 1 L λ 1 u n+1 u n 2 + thus L λ1 u n u n 1 u n+1 u n ; u n+1 u n L λ 1 λ 1 L u n u n 1. u n+1 u n 2 dx) 1/2 Since 0 < L λ 1 λ < 1, {u 1 L n} is a Cauchy sequence in H0 1 (), so {u n } converges strongly to some u H0 1 ().

7 EJDE-2012/139 SEMILINEAR ELLIPTIC EQUATIONS 7 We claim that u is a weak solution of (1.1). Indeed for any φ C0 (), by (H4), (f(x, u n, u n 1 ) f(x, u, u ))φ dx L φ L () ( u n u + u n 1 u ) dx C φ ( u n u + u n 1 u ) 0 as n ; thus by Lemma 3.1, 0 = lim u n φ f(x, u n, u n 1 )φ dx n = u φ f(x, u, u )φ dx. Hence the claim is proved. By Lemma 3.3, u n c 1 ; thus u c 1. Therefore, u is a nontrivial weak solution of the problem (1.1). Proof of Theorem 1.2. By Lemma 3.2 and using the argument as used in the proof of Theorem 1.1 we complete the proof. Consider the equation 4. Examples u = mu au 1 + u 2 u3 1 + u 4 sin2 u, in, u = 0, on, (4.1) where R N (N 3) is a bounded domain with smooth boundary. Suppose that a > 0, m a < λ k m for some k 1, m a λ j for any j N, and 0 < (m+a+3) λ 1 λ 1 (m+a+3) < 1. We will show that (4.1) has at least one nontrivial weak solution. Let f(t, ξ) = mt at 1 + t 2 t3 1 + t 4 sin2 ξ. Then f C(R R N, R), so (H0) holds. Let g 0 (t, ξ) = at( t 2 ) t3 1 + t 4 sin2 ξ. Then f(t, ξ) = (m a)t + g 0 (t, ξ). It is not difficult to see that g 0 (t, ξ) = o( t ) as t 0. Since m a λ j for any j N, then (H1) holds. Let g(t, ξ) = at 1 + t 2 t3 1 + t 4 sin2 ξ. Then f(t, ξ) = mt + g(t, ξ). It is not difficult to see that g(t, ξ) = o( t ) as t. Note that m a < λ k m for some k 1. If m λ k+l for any l 0, then (H2) holds. If m = λ k+l for some l 0, then (H3) holds, in fact, since a > 0, we have and at g(t, ξ) 1 + t 2 + t t 4 sin2 ξ a t G(t, ξ) = 0 as t 1 + s 2 ds 0 s s 4 sin2 ξ ds

8 8 G. LIU, S. SHI, Y. WEI EJDE-2012/139 = a 2 ln(1 + t2 ) 1 4 ln(1 + t4 ) sin 2 ξ as t, uniformly in ξ R n. Finally, we show that (H4) holds. By Lagrange mean value theorem, for any t 1, t 2 R, ξ 1, ξ 2 R N we have f(t 2, ξ 2 ) f(t 1, ξ 1 ) = (mt 2 at t 2 t t 4 sin 2 ξ 2 ) (mt 1 at t 2 t t 4 sin 2 ξ 1 ) 1 at 2 m t 2 t t 2 2 t 3 1 at t 2 1 t ( 1 + t t 4 (sin 2 ξ 2 sin 2 ξ 1 ) 1 m t 2 t 1 + a t 2 t t 2 t ξ 2 ξ 1 (m + a + 3)( t 2 t 1 + ξ 2 ξ 1 ), t t 4 ) sin 2 ξ 2 1 so (H4) holds. By Theorem 1.1 and Theorem 1.2, Equation (4.1) has at least one nontrivial weak solution. Acknowledgments. This research is supported by the following grants: from NSFC, 2012CB from National 973 project of China, from SRFDP. Also supported by the Program for New Century Excellent Talents in University, 985 project of Jilin University, the outstanding young s project of Jilin University, and Graduate Innovation Fund of Jilin University ( ). References [1] C. O. Alves, P. C. Carrião, L. F. O. Faria; Existence of solutions to singular elliptic equations with convection terms via the Galerkin method, Electronic Journal of Differential Equations Vol. 2010, (2010), No.12, [2] H. Amann, M. G. Crandall; On some existence theorems for semilinear elliptic equations, Indiana Univ. Math. J 27 (1978), [3] T. Bartsch, S. J. Li; Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Analysis TMA 28 (1997), [4] K.C. Chang; Infinite Dimensional Morse Theory and Multiple Solution Problem, Birkhäuser, Boston, [5] D. G. de Figueiredo, M. Girardi, M. Matzeu; Semilinear ellptic equations with dependence on the gradient via mountain-pass techniques, Differential and Integral Equations 17 (2004), [6] D. G. de Figueiredo, J. Sánchez, P. Ubilla; Quasilinear equations with dependence on the gradient, Nonlinear Analysis 71 (2009), [7] G. M. de Figueiredo; Quasilinear equations with dependence on the gradient via Mountain Pass techniques in R N, Applied Mathematics and Computation 203 (2008), [8] M. Girardi, M. Matzeu; A compactness result for quasilinear elliptic eqations by mountain pass techniques, Rend. Math. Appl. 29 (2009), [9] M. Girardi, M. Matzeu; Existence of periodic solutions for some second order quasilinear Hamiltonian systems, Rend. Lincei. Mat. Appl. 18 (2007), 1 9. [10] H. P. Heinz; On the number of solutions of nonlinear Schrödinger equations and on unique continuation, J. Differ. Equ. 116 (1995), [11] Pohozaev S; On equations of the type u = f(x, u, Du), Mat. Sb. 113 (1980), [12] P. Rabinowitz; Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conf. Ser. Math., no. 65, Amer. Math. Soc., Providence, R. I., 1986.

9 EJDE-2012/139 SEMILINEAR ELLIPTIC EQUATIONS 9 [13] R. Servadei; A semilinear elliptic PDE not in divergence form via variational methods, J. Math. Anal. Appl. 383 (2011), [14] K. Teng, C. Zhang; Existence of solution to boundary value problem for impulsive differential equations, Nonlinear Anal. R.W.A. 11 (2010), [15] X. Wang, Y. Deng; Existence of multiple solutions to nonlinear elliptic equations in nondivergence form, J. Math. Anal. and Appl. 189 (1995), [16] J. B. M. Xavier; Some existence theorems for equations of the form u = f(x, u, Du), Nonlinear Analysis T.M.A. 15 (1990), [17] Z. Yan; A note on the solvability in W 2,p () for the equation u = f(x, u, Du), Nonlinear Analysis TMA 24 (1995), Guanggang Liu College of Mathematics, Jilin University, Changchun , China address: lgg112@163.com Shaoyun Shi College of Mathematics, Jilin University, Changchun , China address: shisy@jlu.edu.cn Yucheng Wei College of Mathematics, Jilin University, Changchun , China Department of Mathematics, Hechi University, Yizhou , China address: ychengwei@126.com