Existence results for nonlinear degenerate elliptic equations with lower order terms

Recently, much attention has been payed to partial di erential equations with lower order terms, not only for their physical relevance but also for their mathematical interest. From the mathematical point of view, it is well known that the lower order terms may a ect the existence, uniqueness, regularity and asymptotic behavior of solutions to partial di erential equations (see e.g. [1, 3–12, 19, 23]). In this paper, we are interested in the existence and regularity of solutions for a class of degenerate elliptic equations with two lower order terms, whose prototype is


Introduction
Recently, much attention has been payed to partial di erential equations with lower order terms, not only for their physical relevance but also for their mathematical interest. From the mathematical point of view, it is well known that the lower order terms may a ect the existence, uniqueness, regularity and asymptotic behavior of solutions to partial di erential equations (see e.g. [1, 3-12, 19, 23]). In this paper, we are interested in the existence and regularity of solutions for a class of degenerate elliptic equations with two lower order terms, whose prototype is a(x, u)|∇u| p− ∇u ( + |u|) θ(p− ) + ν|u| t− u = γ|∇u| pr + f in Ω, where Ω is a bounded open subset of R N ( < p < N), a(x, s) is a Carathéodory matrix and f is a measurable function.
As θ = , such kind of problems with di erent lower order terms were studied well in the literature. Without the aim to be complete, let us mention the following revelent works. The quasi-linear case p = was treated in [9], where it was shown that the term ν|u| t− u was in some sense to guarantee the existence of a solution when the growth of the gradient is superlinear. If there was an L interplay between the coe cient of the zero order term and the right-hand side f , the existence of bounded solutions was established in [4]. This result was improved in [5] and extended to the parabolic case in [16]. If p = , a(x, s) was an identity matrix, f = and γ = , the existence and non-existence results to problem (P) with two zero order terms were proved in [12]. The case ν = was considered in [1], where it was proved that smallness condition was asked on f to guarantee the existence of a solution. For other related results, see [3], where the existence, multiplicity and non-existence of solutions to a semilinear degenerate elliptic system of Hamiltonian type were proved; see also [19], where a Neumann problem driven by the p-Laplacian with singular and convection terms was investigated.
As θ ≠ , it is easy to see that the principal part of problem (P) degenerates, hence, a slow di usion e ect may appear as soon as the solution u becomes large. Such kind of equations could be seen as a reaction model which produces a saturation e ect. In case of ν = γ = , existence and regularity results of problem (P) were established in [2,13], while the parabolic case of such problem was just treated in [15](see also [21]). We also mention that the related obstacle problems with L data were investigated in [24,25].
When p = and γ = , the author in [11] proved that the zero order term ν|u| t− u may a ect the regularity of solutions of (P) under the assumption that f ∈ L m (Ω) with m ≥ . This result was extended to the general case p > in [10]. For this general case, the stability results were obtained in [14]. Also, it was shown in [7] that the existence of W , (Ω) solutions could be obtained by adding a zero order term. For other relevant papers, see [18,22] and the references therein.
Motivated by [2,9,11], this work studies the regularizing e ect of lower order terms on the solutions to problem (P) (that is ν ≠ and γ ≠ ). The main results obtained here generalize the previous result of [2,9,11] in some sense. The main di culties are the facts that the di erential operator is not coercive on W ,p (Ω) and the lower order terms have regularizing e ects on solutions. To overcome these di culties, we shall rst introduce a class of approximated problems and then establish some estimates for solutions by taking suitable test functions, and nally prove some convergence results to get the existence results.
This paper is organized as follows. In Section 2, we give the assumptions and the main results. In Section 3, we shall prove the main results.

Assumptions and the main results
Let Ω be a bounded open subset of R N with N ≥ , < p < N. Throughout this paper, c i orc i (i = , , . . . n) will denote a positive constant which only depends on the parameters of our problem. For any real number η > , we set η = η η− and η * = Nη N−η . For E ⊆ Ω, we also denote We shall use the truncation functions T k (s) and G k (s) de ned by Let us consider the following problem: There exist constants α, β, ν, γ ∈ R + , and a nonnegative function j ∈ L p (Ω) such that for almost every x ∈ Ω, for every s ∈ R and for every ξ , ζ ∈ R N with ξ ≠ ζ . Now we give the de nition of weak solutions to problem (P). De nition 2.1. A measurable function u ∈ W , (Ω) is called a weak solution of problem (P), if a(x, u, ∇u), g(x, u) and b(x, u, ∇u) are summable functions and The main results of this paper are stated as follows.
) N imposed here is to get the result (3.10) and then to obtain the existence result, see step 2 of the proof of Theorem 2.1. However to get the regularity result u ∈ W ,q (Ω) ∩ L t (Ω), it su ces to assume that F ∈ (L p p− (Ω)) N , see step 1 of the proof of Theorem 2.1. If θ = and p = , the assumption F ∈ (L pt (p− )(t−θ) (Ω)) N reduces to F ∈ (L (Ω)) N , which coincides the result of [9]. Remark 2.2. By (2.4) and (2.5) it easy to check that pt t+ +θ(p− ) > which implies that q > . We remark that the restriction of (2.4) listed here is to simplify the proof and can be replaced by p < r < . Indeed, using Young's inequality, it is easy to see that the above assertion remains true and our results in Theorem 2.1 and Theorem 2.2(see below) are still true in the case that p < r < . Thus our results Theorem 2.1 and Theorem 2.2 also cover the results of [9] where θ = and p = .

Proof of the main results
In order to prove the existence results, let us de ne We consider the following approximated problems The existence of weak solution un ∈ W ,p (Ω) of (3.2) follows by the classical result of [17].

. Proof of Theorem 2.1.
The proof relies on an approximation procedure which is divided into several steps.
Finally, the proof of (3.4) is similar at all to that of (3.3), except that the test function is changed to φ = [ − ( + |G k (un) −λ |)]sgn(G k (un)), so we omit the detail here.
Step 2: we now prove the following convergence result: To do this, rstly it is easy to derive from the estimates (3.3) that for a subsequence of {un} (still denote by {un}), and a function u ∈ W ,q (Ω) ∩ L t (Ω), such that as n → +∞ it results: un → u a.e. in Ω and weakly in W ,q (Ω) ∩ L t (Ω).
Step 3: End the proof.
To do this, for any h > let us take T h (un) as a test function in (3.2) , we easily obtain that (3.14) Then there exists a subsequence (still denoted {un}) such that Then use the argument of [20](see also [6,8,9]) one may get ∇un → ∇u a.e. in Ω. (3.16) By (3.14)- (3.16) and Fatou lemma, we conclude which gives where we have used the results (3.16)-(3.18) and Hölder inequality. Thus, we get . Proof of Theorem 2.2.
Step 1: we rst prove the result when F = .
Let us take φ = [( +|un|) δ − ]sgn(un) in (3.2), where δ > will be chosen later. We get after using Hölder's inequality and Young's inequality, (3.20) Observing that as lim  To end this proof in case F = , we shall choose a di erent δ in (3.22). Indeed, similar to (3.21) we infer that for S (independent of n) large enough, (Ω). Hence the assert follows as before. This completes the proof.