Existence of Classical Solutions for Nonlinear Elliptic Equations with Gradient Terms

This paper deals with the existence of solutions of the elliptic equation with nonlinear gradient term −Δu=f(x,u,∇u) on Ω restricted by the boundary condition u|∂Ω=0, where Ω is a bounded domain in RN with sufficiently smooth boundary ∂Ω, N≥2, and f:Ω¯×R×RN→R is continuous. The existence results of classical solutions and positive solutions are obtained under some inequality conditions on the nonlinearity f(x,ξ,η) when |(ξ,η)| is small or large enough.


Introduction and Main Results
Let Ω be a bounded domain in R N (N ≥ 2) whose boundary ∂Ω is C 2+µ -smooth for given µ ∈ (0, 1). In this paper, we discuss the existence of solutions of the elliptic boundary value problem (BVP) with gradient term −∆u = f (x, u, ∇u) , x ∈ Ω , where f : Ω × R × R N → R is the nonlinearity. This problem arises in many different areas of applied mathematics. Due to the appearance of the gradient term in the nonlinearity, BVP(1) has no variational structure, and the variational method and critical point theory cannot be applied to it directly. The authors of [1,2] proposed a method combining the mountain-pass theorem with an approximation technique to solve BVP (1). Firstly, for any given w ∈ H 1 0 (Ω), they considered the boundary value problem −∆u = f (x, u, ∇w) , x ∈ Ω , Note that BVP(2) has the variational structure. They established the existence of a solution u w of BVP(2) by using the mountain-pass theorem. Then, they constructed a sequence {u n } ⊂ H 1 0 (Ω) by the iterative equation −∆u n = f (x, u n , ∇u n−1 ) , x ∈ Ω , starting with an arbitrary u 0 ∈ H 1 0 (Ω) ∩ C 1 (Ω), and they proved that {u n } converges to a solution of BVP (1) in that f (x, ξ, η) satisfies Lipschitz conditions on (ξ, η) in the neighborhood of (0, 0) with appropriately small coefficients and certain growth conditions on ξ. Later, this iterative method based on the mountain-pass theorem was applied to many semilinear and quasilinear elliptic equations; see [3][4][5][6][7]. In [8], Ruiz obtained the existence of a positive solution for BVP(1) by combining Krasnoselskii's fixed-point theorem in cones with blow-up techniques when f (x, ξ, η) is a nonnegative function and satisfies a suitable growth condition on ξ and η. When Ω is a ball, annulus, or exterior domain of a ball, and f (x, ξ, η) is radially symmetric on x, the authors of [9][10][11][12][13] obtained the existence of positive radial solutions of BVP(1) by discussing the corresponding boundary value problem of second-order ordinary differential equations.
On the other hand, the lower-and upper-solutions method is an effective way to obtain the existence of solutions of BVP(1). In [14], Amann built a lower-and upper-solution theorem of BVP(1) in C 2+µ (Ω) in that f (x, ξ, η) has a continuous partial derivative with respect to ξ and η, and there is, at most, quadratic growth on η. He assumed BVP(1) has pair of ordered lower and upper solutions and proved the existence of a solution between the lower and upper solutions. In [15], Amann and Crandall slightly generalized the results of [14] by a more-direct argument. In [16], Pohozaev obtained the existence results for BVP(1) via the method of lower and upper solutions in the Sobolev space W 2,p (Ω) with p > N when f (x, ξ, η) is Lipschitzian with respect to η. In [17][18][19][20][21][22], the authors obtained the existence of solutions or positive solutions by using the lower-and upper-solutions method and fixed-point theorem under some growth condition of the nonlinearity.
Condition (F1) implies that f is continuous on Ω × R × R N and is a stronger regularity condition. Condition (F2) restricts f to at most quadratic growth with respect to η. If f grows at most like |η| 2−ε for some ε ∈ (0, 1), the regularity condition (F1) can be weakened as (F1) For every ρ > 0, there exists L := L(ρ) > 0 such that See [3,14]. Our existence results are related to the principle eigenvalue λ 1 of Laplace operator −∆ on the boundary condition u| ∂Ω = 0, which is given by Theorem 1. Let f : Ω × R × R N → R satisfy (F1) and (F2). If there exist constants a, b ≥ 0 satisfying a λ 1 and H > 0 such that and then, BVP(1) has at least one classical solution u ∈ C 2+µ (Ω).
In Theorem 1, if b = 0, the result is known (see [1, Theorem 1.2]), and if b = 0, the result is new. (7) and H > 0 such that f satisfies (8), and there exists a positive constant δ such that then, BVP(1) has at least one classical positive solution u ∈ C 2+µ (Ω).
If f satisfies the condition of Theorem 1, but assume that instead of (9), then v 0 ≡ 0 is a lower solution of BVP(1), and BVP(1) has at least one nonnegative solution, see [1, Theorem 1.3]. Theorem 2 is an addition of this result and uses (10) instead of (11) to obtain a positive solution of BVP(1).

Theorem 3.
Let the conditions of Theorem 1 be satisfied, and there exists a positive constant δ such that (10) and hold. Then, BVP(1) has at least one positive solution u 1 ∈ C 2+µ (Ω) and one negative solution u 2 ∈ C 2+µ (Ω).
The proofs of Theorems 1-3 are based on the method of lower and upper solutions built by Amann [14]. A lower solution v of BVP(1) means that v ∈ C 2+µ (Ω) and satisfies and an upper solution w of BVP(1) means that w ∈ C 2+µ (Ω) and satisfies By [1, Theorem 1.1], we have the following existence result: . If BVP(1) has a lower solution v 0 and an upper solution w 0 such that v 0 ≤ w 0 , then BVP(1) has at least one solution u 2 ∈ C 2+µ (Ω) between v 0 and w 0 .
Theorem 4 is a special case of [1, Theorem 1.1]. In Section 3, we use Theorem 4 to prove Theorems 1-3. Some preliminaries to discuss BVP(1) are presented in Section 2.
Next, consider the nonlinear elliptic equation BVP(1). We have the following existence result of the classical solution: Theorem 5. Let f : Ω × R × R N → R satisfy (F1) and in the following growth condition (F3) let there exist constants a, b ≥ 0 satisfying (7) and c > 0 such that Then, BVP(1) has a unique classical solution u ∈ C 2+µ (Ω).
Proof. We first show that BVP(1) has an L 2 solution u 0 ∈ H 2 (Ω) ∩ H 1 0 (Ω). Since the solution operator of LBVP(13) S : L 2 (Ω) → H 2 (Ω) ∩ H 1 0 (Ω) is a linear bounded operator, by the compactness of the Sobolev embedding H 2 (Ω) → H 1 (Ω), S : By Condition (F3), F : H 1 0 (Ω) → L 2 (Ω) is continuous, and it maps every bounded set of H 1 0 (Ω) into a bounded set of L 2 (Ω). Hence, the composite mapping is completely continuous. By the definition of S, the strong L 2 solution of LBVP(13) is equivalent to the fixed point of A. We use the Leray-Schauder fixed point theorem [25] to show that A has a fixed point. For this, we consider the equation family and show that the set of their solutions is bounded in H 1 0 (Ω). Let u ∈ H 1 0 (Ω) be a solution of (17) for λ ∈ (0, 1). Set h = λ F(u). Since h ∈ L 2 (Ω), by the definition of S, u = Sh ∈ H 2 (Ω) ∩ H 1 0 (Ω) is the unique solution of LBVP (13). Hence, u satisfies the differential equation By this equation and Condition (F3), we have By this inequality and (14), we obtain that From this, it follows that Hence, the set of the solutions of Equation Family (17) is bounded in H 1 0 (Ω). By the Leray-Schauder fixed-point theorem, A = S • F has a fixed point u 0 ∈ H 1 0 (Ω), which belongs to H 2 (Ω) and is an L 2 solution of BVP(1).
Strengthen Condition (F3) of Theorem 5; we have following existence and uniqueness result.
The proof of Theorem 6 is completed.

Proof. Consider the elliptic boundary value
Corresponding to BVP (1), the nonlinearity f of BVP (21) is given by It is easy to verify that the function f defined by (22) satisfies Conditions (F1) and (F4). Hence, by Theorem 6, BVP(21) has a unique solution w 0 ∈ C 2+µ (Ω). Set then, w 0 is the classical solution of LBVP (13). Since −∆w 0 = h > 0, by the maximum principle of the elliptic operators, w 0 (x) > 0 for every x ∈ Ω. Hence, w 0 is a positive classical solution of BVP (20). On the other hand, the positive solution of BVP (20) is also a solution of BVP (21). By the uniqueness of the solution of BVP (21), w 0 is the unique positive classical solution of BVP (20).

Proofs of the Main Results
Proof of Theorem 1. Let a, b, H be the constants in the condition of Theorem 1. Choose a positive constant by then, from Conditions (8) and (9), it follows that and respectively. By Theorem 7, BVP(20) has a unique positive solution w 0 ∈ C 2+µ (Ω). By Equation (20) and Inequality (25), we easily see that w 0 is an upper solution of BVP(1), and by (20) and (26), −w 0 is a lower solution of BVP(1). Since −w 0 ≤ w 0 , by Theorem 4, BVP(1) has at least one solution u ∈ C 2+µ (Ω) between −w 0 and w 0 .

Proof of Theorem 2.
Let c be the positive constant defined by (24) and w 0 be the unique positive solution of BVP (20). Then by the proof of Theorem 1, w 0 ∈ C 2+µ (Ω) is a upper solution of BVP(1). It is well-known that the minimum positive real eigenvalue λ 1 of the elliptic eigenvalue problem −∆u = λ u, x ∈ Ω, has a positive unit eigenfunction; that is, there exists a function ϕ 1 ∈ C 2 (Ω) ∩ C + (Ω) with ϕ 1 C(Ω) = 1 that satisfies the equation Let δ be the constant in (10), and choose Set v 0 = δ 0 ϕ 1 . By the regularity of the solution of the linear equation LBVP (13), v 0 ∈ C 2+µ (Ω). For every x ∈ Ω, since v 0 (x) ≥ 0 and by the inequality (10) and Equation (28), we have Hence, v 0 is a lower solution of BVP(1). We show that v 0 ≤ w 0 . Consider the function u = w 0 − v 0 . Since w 0 satisfies Equation (20) and v 0 satisfies Equation (28), it follows that Since u| ∂Ω = 0, by the maximum principle of the elliptic operators, u(x) > 0 for every x ∈ Ω. Hence, v 0 ≤ w 0 .
Therefore by Theorem 4, BVP(1) has at least one solution u 0 ∈ C 2+µ (Ω) between v 0 and w 0 that is a positive solution of BVP(1).

Proof of Theorem 3.
Let c be the positive constant defined by (24) and w 0 be the unique positive solution of BVP (20). Then by the proof of Theorem 1, w 0 is an upper solution and −w 0 is a lower solution of BVP(1).