New multiple solutions for a Schrödinger–Poisson system involving concave-convex nonlinearities

In this paper, we study the following critical growth Schr ödinger–Poisson system with concave-convex nonlinearities term { −∆u+ u+ ηφu = λf(x)uq−1 + u, in R, −∆φ = u, in R, (0.1) where 1 < q < 2 , η ∈ R , λ > 0 is a real parameter and f ∈ L 6 6−q (R) is a nonzero nonnegative function. Using the variational method, we obtain that there exists a positive constant λ∗ > 0 such that for all λ ∈ (0, λ∗) , the system has at least two positive solutions.


Abstract:
In this paper, we study the following critical growth Schrö dinger-Poisson system with concave-convex where 1 < q < 2 , η ∈ R , λ > 0 is a real parameter and f ∈ L

Introduction and main results
In this paper, we are interested in the existence of multiple positive solutions to the following Schrödinger- where 1 < q < 2, η ∈ R, λ > 0 is a real parameter and f ∈ L 6 6−q (R 3 ) is a nonzero nonnegative function. The first Schrödinger equation coupled with a Poisson equation means that the potential is determined by the charge of the wave function. The general term f (x)u q−1 models the interaction between particles. The nonlocal term ηφu concerns the interaction with the electric field. For detailed mathematical and physical interpretation, we refer readers to [2,8,9] and the references therein.
In recent years, the following form of Schrödinger-Poisson system with critical growth has been investigated extensively. For previous related results, please refer to [1,4,12,15,[17][18][19]23]. Further, the Schrödinger-Poisson system with concave-convex nonlinearities has attracted much attention. on bounded domain, Guo and Liu in [12] were concerned about the following system where |f (u)| ≤ c 1 + c 2 |u| q−1 with 1 < q < 2 and infinitely many negative energy solutions were established for every µ > 0 small enough and λ > 0. In [15], Lei and Suo established two positive solutions to the system for λ > 0 enough small when 1 < q < 2 .
On unbounded domains, Sun et al. in [21] considered the following Schrödinger-Poisson system where λ > 0 is a parameter, 1 < q < 2 and f (x, u) is linearly bounded in u at infinity. They established the existence and multiplicity of solutions (when λ is enough small) under suitable assumptions on V, K, f .
Recently, Li and Tang [18] proved the existence of λ * > 0 such that the system , l ≥ 0, l ̸ ≡ 0 has two positive solutions for λ ∈ (0, λ * ) . However, the authors did not consider the cases of η = −1 or on the whole space H 1 (R 3 ) .
Indeed, it is very difficult to estimate the critical value level in the cases when η = −1 or on the space if we use the extremal function Observing these results, it is natural to ask if system (1.1) has multiple positive solutions since system (1.1) has a concave and convex nonlinearity. In this paper, we study the existence of multiple positive solutions of system (1.1) in the case of η ∈ R through variational method. We have the following results.
Then there exists a positive constant λ * > 0 such that for all λ ∈ (0, λ * ) , system (1.1) has at least two positive solutions. Remark 1.2 On one hand, as we shall see, Theorem 1.1 extends the result in [15] to unbounded domains.
Moreover, we get rid of the restriction of the coefficient of the nonlocal term. On the other hand, our result extends the result in [18] to the whole space H 1 (R 3 ) and two positive solutions are still obtained. Besides, we have the following result.
Then there exists a positive constant λ * > 0 such that for all λ ∈ (0, λ * ), system has at least two positive solutions.
Throughout this paper, we make use of the following notation: • We denote by B r (respectively, ∂B r ) the closed ball (respectively, the sphere) of center zero and radius • C, C 0 , C 1 , C 2 , ... denote various positive constants, which may vary from line to line; • For each p ∈ [2, 6), by the Sobolev embeddings, we denote

Proof of Theorem 1.1
With the help of the Lax-Milgram theorem, for every u ∈ H 1 (R 3 ) , the second equation of system (1.1) has a unique solution φ u ∈ H 1 (R 3 ) . We substitute φ u to the first equation of system (1.1), then system (1.1) transforms into the following equation The energy functional corresponding to equation (2.1) is given by Before proving our Theorem 1.1, we need the following lemma (see [8,10,11] ).

Lemma 2.1 For every
and the following results hold: Proof By the Sobolev and Hölder inequalities, we obtain Thus, there exists u small enough such that I λ (u) ≜ d < 0. The proof is complete.
Indeed, by (2.2), we can deduce that In view of the definition of d , there exists a bounded minimizing sequence From [18], it holds that ∫ Set w n = u n − u 1 . By Brézis-Lieb's Lemma (see [6]), one has If u 1 = 0, then w n = u n , which follows that w n ∈ B R (0). If u 1 ̸ = 0, we also get w n ∈ B R (0) for n large sufficiently. Hence, from (2.3) one has Therefore, by Lemma 2.1, it follows from (2.4) that As n → ∞ , it holds that d ≥ I λ (u 1 ) . Since B R (0) is closed and convex; thus, u 1 ∈ B R (0) . Hence, we obtain I λ (u 1 ) = d < 0 and u 1 ̸ ≡ 0 . It follows that u 1 is a local minimizer of I λ . Then for any ψ ∈ H 1 (R 3 ), ψ ≥ 0, setting t > 0 small enough such that u 1 + tψ ∈ B R (0) , one obtains (2.5) Dividing by t > 0 and passing to the limit as t → 0 + in (2.5), we get By the arbitrariness of ψ , the above inequality also holds for −ψ . Hence, (u 1 , φ u1 ) is a nonzero solution of system (1.1). Moreover, similar to the discussion of Theorem 3.3 in [14], we can also obtain that (u 1 , φ u1 ) is a positive solution of system (1.1) with I λ (u 1 ) < 0. This completes the proof of Theorem 2.3. Consequently, Otherwise, there exists a subsequence (still denoted by itself) such that From (2.6), as n → ∞ , for every φ ∈ H 1 (R 3 ), it follows Take the test function φ = v * in (2.7), then it holds that Putting φ = v n in (2.6) and using the Brézis-Lieb's lemma, we obtain (2.9) It follows from (2.8) and (2.9) such that On the one hand, from (2.8), by the Young inequality, it holds On the other hand, by (2.6) and (2.10), it follows This is a contradiction. Therefore, l = 0 implies that We know that the extremal function and |∇U | 2 in Ω . ζ(x) = 1 near x = 0 and it is radially symmetric. We define Besides, since (u 1 , φ u1 ) is a positive solution of system (1.1), we can see that there exist m, M > 0 such that

Lemma 2.5 Under the assumptions of Theorem 1.1, it holds
Proof It is known (see [7]) that According to the definition of u ε , it follows where the last equality holds provided p < 3. For 1 < q < 2, one has It is obvious that the following inequality (a + b) 6 ≥ a 6 + b 6 + 6a 5 b + 6ab 5 , holds for each a, b ≥ 0.
Using the inequalities above, for all t ≥ 0 , we obtain where It follows from Hölder's inequality and (2.12) such that According to [20], we have ∫ Therefore, one obtains Now, we prove that there exist t ε > 0 and positive constants t 1 , t 2 independent of ε, λ , such that sup t≥0 h ε (t) = h ε (t ε ) and Indeed, we see that lim t→+∞ h ε (t) = −∞ and h ε (0) = 0 . Then there exists t ε > 0 such that It is similar to the paper [16] that (2.13) holds. Note that . Then it holds that which implies that sup t≥0 h ε (t) < 1 3 S 3 2 − Dλ 2 2−q . From the above information, we can deduce that (2.11) holds true when λ < Λ 1 . The proof is complete. } . By Lemma 2.5, we can choose a sufficiently large T 0 > 0 such that I λ (u 1 + T 0 u ε ) < 0 , with the fact that I λ (u 1 ) < 0 . Then we apply the mountain-pass lemma [5] to obtain that there is a sequence {u n } ⊂ H 1 (R 3 ) such that  such that u n → u 2 in H 1 (R 3 ) . Moreover, we can obtain (u 2 , φ u2 ) is a nonnegative weak solution of system (1.1) and I λ (u 2 ) = lim n→∞ I λ (u n ) = c > 0.
Therefore, we infer that u 2 ̸ ≡ 0. It is similar to Theorem 2.3 that u 2 > 0 in R 3 . The proof is complete.