Multiplicity of Positive Solutions for a pq-Laplacian Type Equation with Critical Nonlinearities

and Applied Analysis 3 Proof. For u ∈ Mλ, (10) leads to Jλ (u) = ( 1 p − 1 r ) ‖u‖ p p + ( 1 q − 1 r ) ‖u‖ q q + ( 1 r − 1 p )∫ Ω f (x) |u| p ∗ dx > 0. (12) Define αλ := inf u∈M λ Jλ (u) . (13) Now we show that Jλ possesses the mountain-pass (MP, in short) geometry. Lemma 4. Suppose 1 < q < p < r < p and (H2) holds. Then for any λ > 0, one has that (i) there exist positive numbers R and d0 such that Jλ(u) ≥ d0 for ‖u‖p = R; (ii) there exists u ∈ W 0 (Ω) such that ‖u‖p > R and Jλ(u) < 0. Proof. (i) By (8), the Hölder inequality, and the Sobolev embedding theorem, we have that Jλ (u) ≥ 1 p ‖u‖ p p − 1 p ∫ Ω f (x) |u| p ∗ dx − 1 r ∫ Ω λg (x) |u| dx ≥ 1 p ‖u‖ p p − 1 p S ∗ /p ‖u‖ p ∗

Problem (  ) comes, for example, from a general reaction-diffusion system where () = |∇| −2 + |∇| −2 .This system has a wide range of applications in physics and related science such as biophysics, plasma physics, and chemical reaction design.In such applications, the function u describes a concentration, the first term on the right-hand side of (2) corresponds to the diffusion with a diffusion coefficient (), whereas the second one is the reaction and relates to sources and loss processes.Typically, in chemical and biological applications, the reaction term (, ) has a polynomial form with respect to the concentration .
The main purpose of this paper is to analyze the effect of the coefficient () of the critical nonlinearity to prove the multiplicity of positive solutions of problem (  ) for small  > 0. By the similar argument in [14], we can construct the  compact Palais-Smale sequences that are suitably localized in correspondence of  maximum points of .Under some assumptions (1)-(3), we could show that there are at least  positive solutions of problem (  ) for sufficiently small  > 0.
This paper is organized as follows.First of all, we study the argument of the Nehari manifold M  .Next, we prove the existence of a positive solution  0 ∈ M  .Finally, we show that the condition (3) affects the number of positive solution of (  ); that is, there are at least  critical points   ∈ M  of   such that   (  ) =    ((PS)-value) for 1 ≤  ≤ .The main results of this paper are given as follows.

Preliminaries
In what follows, we denote by ‖ ⋅ ‖  , | ⋅ |  the norms on  1, 0 (Ω) and   (Ω), respectively; that is, We denote the dual space of  1, 0 (Ω) by   (Ω).Set also equipped with the norm We will denote by  the best Sobolev constant as follows: It is well known that  is independent of Ω and is never achieved except when Ω = R  (see [15]).
It is well known that   is of  1 in  1, 0 (Ω) and the solutions of (  ) are the critical points of the energy functional   (see [16]).
We define the Nehari manifold where The Nehari manifold M  contains all nontrivial solutions of (  ).
Note that   is not bounded from below in  1, 0 (Ω).From the following lemma, we have that   is bounded from below on the Nehari manifold M  .Lemma 3. Suppose that 1 <  <  <  <  * and (H2) hold.Then for any  > 0, one has that   is bounded from below on M  .Moreover,   () > 0 for all  ∈ M  .
then  0 is a solution of (  ).
We will denote by α the MP level: Then we have the following result.
Remark 8.By Lemma 7 and the definition, it is apparent that Moreover, by Lemma 4(i), for any  0 > 0, there exists a  = ( 0 ), related to the MP geometry, such that Here  0 is the MP level associated to the functional

Existence of 𝑘 Positive Solutions
In this section, we first give some preliminary notations and useful lemmas.
Choose  0 > 0 small enough such that   0 (  ) ⊂ Ω and Then we have the following separation result.Proof.For any  ∈ which implies that Hence, from (39), we obtain which is a contradiction.
Next, we will investigate the effect of the coefficient () to find some Palais-Smale sequences which are used to prove Theorem 2. H3) hold, then for any  ∈ {1, 2, . . ., } and any  > 0, there exists a  0 > 0 such that for  ∈ (0,  0 ) one has