Multiple positive solutions for a class of concave-convex Schrödinger-Poisson-Slater equations with critical exponent

Theorem 1.1

Assume that ( Q 1 ) and ( Q 2 ) hold, then there exists μ * > 0 such that equation (1.1) has at least two positive solutions for all μ ∈ ( 0 , μ * ) , and one of the solutions is a ground state solution.

Theorem 1.2

Assume that ( Q 1 ) , ( Q 3 ) , and ( Q 4 ) hold, then there exists μ * ∈ ( 0 , μ * ) small enough such that equation (1.1) has at least k + 1 distinct positive solutions for all μ ∈ ( 0 , μ * ) .

Remark 1.1

Since the lack of compactness of the Sobolev embedding E ↪ L s ( R 3 ) , ∀ s ∈ [ 3 , 6 ] , inspired by [20, 33], we adapt a measure representation concentration-compactness principle of Lions [18] to overcome the difficulties, and by exploring the parameter μ , we show that the associated energy functional satisfies, in general, the Palais-Smale condition at some level for μ > 0 small enough under our assumptions. Because the associated energy functional is not bounded below, we apply Ekeland’s variational principle to obtain that equation (1.1) has at least k + 1 positive solutions by the Nehari manifold.

Proposition 2.1

[10,20,25]   ( ‖ ⋅ ‖ E , E ) is a uniformly convex Banach space. Moreover, C 0 ∞ ( R 3 ) is dense in E . E ↪ L s ( R 3 ) continuously for s ∈ [ 3 , 6 ] and E ↪ L s ( Ω ) compactly for s ∈ [ 1 , 6 ) with bounded Ω ⊂ R 3 .

Proposition 2.2

Given a sequence { u n } in E ,

1. u n → u strongly in E if and only if u n → u and ϕ u n → ϕ u in D 1 , 2 ( R 3 ) .

2. u n ⇀ u weakly in E if and only if u n ⇀ u in D 1 , 2 ( R 3 ) and ∫ R 3 ϕ u n u n 2 d x is bounded. In such case, ϕ u n ⇀ ϕ u in D 1 , 2 ( R 3 ) .

3. Assume that we have three weakly convergent sequences in E, u n ⇀ u , v n ⇀ v , w n ⇀ w , and z ∈ E . Then, T ( u n , v n , w n , z ) → T ( u , v , w , z ) .

Proposition 2.3

[20,25,26] There exists C > 0 such that

for u ∈ E with s ∈ [ 3 , 6 ] .

Lemma 2.1

J μ is coercive and bounded from below on N μ .

Proof

For u ∈ N μ , by the Young and Hölder inequalities, we deduce from Proposition 2.3 that

which implies that J μ is coercive and bounded from below on N μ due to 1 < p < 2 . This completes the proof of Lemma 2.1.□

Lemma 2.2

For each u ∈ E with ∫ R 3 g ( x ) ∣ u ∣ 6 d x > 0 , then

1. Assume ∫ R 3 f ( x ) ∣ u ∣ p d x = 0 , then there exists a unique t − = t − ( u ) > t max > 0 such that t − u ∈ N μ − and

   J μ ( t − u ) = sup t ≥ t max J μ ( t u ) .

2. Assume ∫ R 3 f ( x ) ∣ u ∣ p d x > 0 , if μ ∈ ( 0 , μ 0 ) , then there exist unique t + and t − with 0 < t + = t + ( u ) < t max < t − such that t + u ∈ N μ + , t − u ∈ N μ − , and

   J μ ( t + u ) = inf 0 < t ≤ t max J μ ( t u ) , J μ ( t − u ) = sup t ≥ t + J μ ( t u ) .

Proof

For each u ∈ E with ∫ R 3 g ( x ) ∣ u ∣ 6 d x > 0 , let

obviously, m ( 0 ) = 0 and m ( t ) → − ∞ , as t → + ∞ . Moreover,

and therefore, m ′ ( t max ) = 0 and m ′ ( t ) > 0 for t ∈ ( 0 , t max ) and m ′ ( t ) < 0 for t ∈ ( t max , + ∞ ) , i.e., m ( t ) achieves its maximum at t max and t max is unique. Let

Then, by the Sobolev embedding theorem, it holds that

Case i. If ∫ R 3 f ( x ) ∣ u ∣ p d x = 0 , then there exists a unique t − = t − ( u ) > t max > 0 such that

and by simple computations, one has

which implies that t − u ∈ N μ − and

Case ii. If ∫ R 3 f ( x ) ∣ u ∣ p d x > 0 , then by the Sobolev embedding theorem and equation (2.10), for all μ ∈ ( 0 , μ 0 ) , it holds that

and thus, there exist unique t + and t − with 0 < t + = t + ( u ) < t max < t − such that

Similar to Case i, we can conclude that t + u ∈ N μ + , t − u ∈ N μ − , and

Then, the proof of Lemma 2.2 is complete.□

Lemma 2.3

If μ ∈ ( 0 , μ 0 ) , then N μ 0 = ∅ and N μ − is closed, where μ 0 as in equation (2.9).

Proof

(i) Assuming the contrary, there exist μ 0 ∈ ( 0 , μ 0 ) and 0 ≠ u 0 ∈ N μ 0 , and by the Sobolev embedding theorem and equation (2.8), one has

i.e.,

On the other hand, it follows from equation (2.7) that