Existence Results for a Class of the Quasilinear Elliptic Equations with the Logarithmic Nonlinearity

School of Economics and Management, Jiangsu Maritime Institute, Nanjing 211170, China State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China Department of Basic Courses, The Army Engineering University of PLA, Nanjing 211101, China School of Business Administration, Nanjing University of Finance and Economics, Nanjing 210046, China School of Teacher Education, Nanjing Normal University, Nanjing 210046, China


Introduction
In this paper, we consider the existence of solution to the following problem where Ω ⊂ R n , φ p ðzÞ = jzj p−2 z, ψ p ðzÞ = jzj p−1 z, p > 2, and n ≥ 1. We always suppose that aðxÞ is a sign-changing function; hðxÞ ≥ 0 is a ∈C 1 function. Equations of the above form are mathematical models occurring in studies of the p-Laplace equation, generalized reaction-diffusion theory [1], non-Newtonian fluid theory [2,3], non-Newtonian filtration theory [4,5], and the turbulent flow of a gas in porous medium [6]. In the non-Newtonian fluid theory, the pair p is a characteristic quantity of the medium. Media with p > 2 are called dilatant fluids, and those with p < 2 are called pseudoplastics. If p = 2, they are Newtonian fluids. When p ≠ 2, the problem becomes more complicated since certain nice properties inherent to the case p = 2 seem to be lose or at least difficult to verify. The main differences between p = 2 and p ≠ 2 can be founded in [7,8].
In recent years, logarithmic nonlinearity is widely used in pseudo-parabolic equations which describe the mathematical and physical phenomena. Equations of the type ð 1Þ have been studied by many authors when p = 2 (see, for example, [9][10][11][12] and the reference therein). To do so, the authors always use the nice properties of Δ, such that, maximum principle and comparison principle and so on. Meanwhile, existence and structure of solutions for such equations with p > 1 in bounded domains have also attracted much interest (see [13,14]).
In the following discussion, we consider two different situations. Firstly, we consider the existence of positive solution for problem ð1Þ with Neumann boundary conditions. In this case, suppose that Ω = B R = B R ð0Þ ⊂ R n , aðxÞ > 0, hðxÞ ≥ 0 are also radial functions, aðxÞ = aðjxjÞ, hðxÞ = hðjxjÞ in B R . Our strategy in the study of problem ð1Þ is to adopt a double perturbation argument. First, following [15,16] (see also [17]), for each 0 < ε < 1, we consider a family of approximate problems Then, it is natural to look for a family of solutions of ð2Þ and then to pass the limit as ε → 0 to obtain a solution to ð1Þ.
Here, θ > 0 is an appropriate constant. When r → 0 + , we get a solution to ð2Þ. The role of problem ð3Þ is that we cannot use Poincare inequality to solve ð2Þ directly by variational methods.
Secondly, we consider the multiple solutions for problem ð 1Þ with Dirichlet boundary conditions. In this case, we consider aðxÞ is a sign-changing function, hðxÞ = 0. The method is based on Nehari manifold and logarithmic Sobolev inequality.
Remark 2. Theorem 1 is valid even if we change the logarithm by a more general singular function. In fact, suppose g : ð0, 1Þ → R is a smooth function such that for some m ∈ ð0, 1Þ. Then, we can perturb g by a family g ε of smooth functions decreasing in ε, such that g ε ð0Þ = 0 and g ε ðsÞ → gðsÞ pointwise in s ∈ ð0,∞Þ as ε → 0. This perturbation can be done in such a way that g ε0 ≥ 0 for some ε n > 0, and then, all the results in Section 2 hold with little modification.
The paper is organized as follows. In Section 2, we construct a sub-and a supersolution for 3 and finish the proof of Theorem 1. In Section 3, we prove Theorem 3 by the method of Nehari manifold and logarithmic Sobolev inequality. Proof. We just need to see that, since aðxÞ > 0, hðxÞ ≥ 0 in B R , the following inequality holds independently of 0 < ε ≤ 1 and θ > 1: We proceed to find a supersolution for 3. Denote by X r , the following subspace of H 1 ðA rR Þ: For υ ∈ X r , we define the expression: Remark. The expression j⋅j r defines a norm on X r , and ðX r , j⋅j r Þ is a reflexive Banach space. Furthermore, by ([23], (7.44)), the Poincare inequality holds on X r , that is, there exists η > 0 such that Next, we work with the radial formulation for E ε,r in the specific case that ε = 1, Journal of Function Spaces where ϕ p ðsÞ = jsj p−2 s. Notice that For simplicity, denote Then, if υ solves we will have that υ + θ is a solution of Eq. (10). In order to prove existence of such υ, we find a minimum of the functional in the sequel. Let S ⊂ X r denote the set of symmetric functions with respect to the origin. We define Φ : where Fðs, υðsÞÞ = Ð t 0 f ðs, ðz + θÞ + Þdz and z + ≔ max fz, 0g.

Lemma 5.
The functional Φ is C 1 , weakly lower semicontinuous and coercive so that there exist υ ∈ X r such that The proof is standard by (9). Also, since υ is a weak solution of (13), we have in which Then, we define Lemma 6. Suppose that θ > 1. Then, the function u ≡ u r is a supersolution for ð3Þ which does not depend on 0 < ε ≤ 1.

Lemma 7.
There exists a constant M > 0 such that ju r j ∞ ≤ M and the constant M does not depend on r ∈ ð0, RÞ. Moreover, for each ρ ∈ ð0, RÞ, there exist a constant C ρ and r ρ ∈ ð0, RÞ such that we have the following estimates: Proof or Lemmas 6 and 7 can be found in [18], we omit them here.

Existence of Solution for 3.
In this section, we use the suband supersolution from Section 2.1 (u and u r , respectively) to obtain a solution for the problem 3. Define the function where we choose b in such a way that the function u → g ε ðs , uÞ is increasing in u for all s ∈ ½r, R. Now, starting with u 0 = u, we define a sequence u n such that each u n satisfies For all nonnegative υ ∈ W 1,m 0 ðΩÞ. Then, the inequality implies that Lemma 9. The sequence fu n g is nondecreasing and satisfies u 0 ðsÞ ≤ u n ðsÞ ≤ u n+1 ðsÞ ≤ u r ðsÞ for all s ∈ ½r, R and all n ∈ N.

Journal of Function Spaces
Proof. We just need to see that u 0 ≤ u 1 ≤ u r and the general case follows by induction in an analogous way. We have So, we can apply Lemma 8 and obtain that u 0 ≤ u 1 in ½r, R. On the other hand, Again, Lemma 8 implies u 1 ≤ u r in ½r, R. By Lemma 9, we define the pointwise limit and we see that The function u ε r is in fact a solution of 3.

Lemma 10.
The function u ε r is a solution of 3, and it belongs to C 1 ½r, R.
Proof. For each n ∈ N, we have Since we have we obtain, as in Lemma 7, that jϕ p ðu′ p ÞjC 1 ½ρ, R is bounded. Then, for a subsequence that we still denote by u n , we have the convergence 2.3. Obtaining a Solution for E ε . In this section, we pass the limit as r → 0 + and then obtain a solution for 2.
Lemma 11. For a fixed 0 < ε ≤ 1, the problem ð2Þ has a solution u ε which is obtained as the limit of u ε r as r → 0 + .
Proof. For simplicity, we omit the dependence on ε > 0 for u ε r . We know that Also, we have As in Lemma 7, we can prove, for each ρ ∈ ð0, RÞ, there exist a constant C ρ > 0 and r ρ ∈ ð0, RÞ such that we have the following estimates: Then, from the compact imbedding C 1 ½ρ, R → C 0 ½ρ, R, we see that there exist a sequence r n and u ε defined on ð0, R such that, if we define w n ≔ u r n , then 2.4. Concluding the Proof of Theorem 1. Now, we would like to pass the limit in the family u ε obtained in Section 2.3 and get a solution to ð1Þ. In order to do that, we need some estimates like the ones in Lemma 7 independently of ε. First, we observe that the following estimate holds in ð0 , R Notice that the family ðu ε Þ 0<ε≤1 satisfies ε > 0 for u ε r . We know that if s ∈ ½R/2, R,and if s ∈ ð0, R/2.

Journal of Function Spaces
From Eqs. (36)-(38) we see, as in Lemma 7 that, for each for each ρ ∈ ð0, RÞ, there exist a constant C ρ > 0 and ε ρ ∈ ð0, RÞ such that we have the following estimates: Now, arguing as in Section 2.4, we can find a function u which satisfies that is, u is a radial solution for the problem ð1Þ. We see that u ∈ C 1 ð0, RÞ ∩ Cð0, R. Now, extend continuously u to the whole interval ð0, R. Indeed, let r i be a sequence in ð0, R/2Þ with r i → 0 as i → ∞. From Eq. (13) (after we have passed the limit in ε) Then, if r j > r i , we get From Eq. (36), we obtain that there exists a constant C > 0 such that so uðr i Þ is a Cauchy sequence in R. Let L be the limit of such sequence. By a similar argument, we conclude that if s i is another sequence in ð0, R/2Þ converging to 0, then we necessarily have uðs i Þ → L. So, we have proved that finishing the proof of Theorem 1.

Proof of Theorem 3
3.1. Preliminaries. In this section, we consider the multiple solutions for problem ð1Þ with Dirichlet boundary conditions. In this case, we consider aðxÞ is a sign-changing function, hðxÞ = 0. Moreover, it is necessary to note that the presence of the logarithmic nonlinearity leads to some difficulties in deploying the potential well method. In order to handle this situation, we need the following logarithmic Sobolev inequality which was introduced by [25].
We start by considering the energy functional J by in which kuk p = kuk L p ðΩÞ.
It is clear that all nontrivial critical points of J must lie on N, and as we will see below, local minimizers on N are usually critical points of J. Also, we can see that Let u ∈ W 1,p ðR n Þ \ f0g and consider the real function j : λ → JðλuÞ for λ > 0 defined by Ð Ω aðxÞjuj p dx:Such maps are known as fibering maps which were introduced by Drabek and Pohozaev [26].
Then, by direct calculations, we have Since λ > 0, then λu ∈ N if and only if j′ðλÞ = 0.
Proof. If u 0 is a local minimizer for J on N, by Lagrange multipliers, there exists κ ∈ R such that where χðuÞ = k∇uk p p − Ð Ω aðxÞjuj p log jujdx. Proof. From (62), j′ðλÞ has a unique turning point at Since aðxÞ is sign-changing, then we can take u 1 such that ð Ω a x ð Þ u 1 j j p dx < 0, and then λ u 1 ð Þu 1 ∈ N + : ð70Þ Also, we can take u 2 such that ð Ω a x ð Þ u 2 j j p dx < 0, and then λ u 2 ð Þu 2 ∈ N − : Then, both N + and N − are nonempty.
Just like [19], by Lemmas 13-16, we can get the following results.