Elliptic problem driven by different types of nonlinearities

In this paper we establish the existence and multiplicity of nontrivial solutions to the following problem \begin{align*} \begin{split} (-\Delta)^{\frac{1}{2}}u+u+(\ln|\cdot|*|u|^2)&=f(u)+\mu|u|^{-\gamma-1}u,~\text{in}~\mathbb{R}, \end{split} \end{align*} where $\mu>0$, $(*)$ is the convolution operation between two functions, $0<\gamma<1$, $f$ is a function with a certain type of growth. We prove the existence of a nontrivial solution at a certain mountain pass level and another ground state solution when the nonlinearity $f$ is of exponential critical growth.


Introduction
The main objective of this paper is to establish the existence and multiplicity of nontrivial solutions for the following problem.
Similarly, f is said to be of critical exponential growth at ∞ if there exist θ ∈ (0, π] and β 0 ∈ (0, θ) such that lim t→∞ f (t) e βt 2 − 1 = 0, for every β > θ ∞, for every β < θ The following are some hypotheses which are commonly assumed for problems with the Moser-Trudinger inequality (cf. doÓ et al. [19] and Felmer et al. [21]): (A 1 ) f ∈ C(R, R), f (0) = 0, has critical exponential growth and F (t) ≥ 0, for every t ∈ R; there exists L > 4 such that f (t)t ≥ LF (t) > 0, for every t ∈ R (this condition is used to verify that a Cerami sequence is bounded in the Sobolev space H such that F (t) ≥ C q |t| q , for every t ∈ R, where S q , r 0 > 0 (S q will be defined in Lemma 3.12).
We now give a short review of the related results. The problem that inspired us to investigate the current problem is from the paper by Boër-Miyagaki [7]. The novelty addressed in this work is due to the presence of a singular term which is difficult to handle since the corresponding energy functional ceases to be C 1 . This poses an extra challenge in applying the Mountain pass theorem and other results in variational methods that demand the functional to be C 1 . To add to these difficulties, the logarithmic term poses a great challenge to establishing the existence of a convergent subsequence of a Cerami sequence. The idea of module translations from the paper by Cingolani-Weth [14] will be used. However, since the norm of the space X (the solution space which will be defined in the next section) is not invariant under translations, a new difficulty arises. The issues pertaining to the exponential term will be explained later. Problems involving nonlocal operators are important in many fields of science and engineering e.g. optimization, finance, phase transitions, stratified materials crystal dislocations, anomalous materials, semipermeable membranes, flame propagation, water waves, soft thin films, conservation laws, etc. We refer the reader to Caffarelli [8], Di Nezza et al. [15], and the references therein.
The literature pertaining to problems without the logarithmic Choquard and the singular term is quite vast and is impossible to list everything in this paper. A seminal work in the field of singularity driven problems is due to Lazer-Mc Kenna [27]. Thereafter the problems with singularity were studied by many researchers; see Ghanmi-Saoudi [22], Oliva-Petitta [31], Saoudi et al. [37], and the references therein. Some of the other works that the readers can consult involve the fractional Laplacian operator (−∆) s when 2s < N and s ∈ (0, 1) are Chang-Wang [10] and Felmer et al. [21]. We note that Felmer et al. [21] also studied some properties of the solutions besides regularity. Furthermore, doÓ et al. [18] studied the problem without the singular and the logarithmic term with potentials that vanish at infinity. Some more suggested papers are Autuori-Pucci [4], Cao [9], doÓ et al. [17], Iannizzotto-Squassina [25], Lam-Lu [26], Moser [30], and Pucci et al. [35]. We now turn our attention to the problems involving Choquard logarithmic term. Some related references are Alves-Figueiredo [3], Cingolani-Jeanjean [13], Cingolani-Weth [14], Du-Weth [20], and Wen et al. [41]. Furthermore, Cingolani-Weth [14] proved the existence of infinitely many distinct solutions and a ground-state solution, with V :  [28], and Panda et al. [33]. This paper is organized as follows. Section 2 is a quick look at the mathematical background, space description. In Section 3 we describe an application of the fractional Laplacian operator for dimension N = 1 along with a few auxiliary lemmas. In Section 4 we prove a few auxiliary lemmas and our main result. Finally, in Section 5, we give an appendix to the proofs of all results which have been used in the proof of the main theorem.

Preliminaries
This section is devoted to presentation of the most important notations, results, remarks that will be used in our study of problem (1.1) (for the remaining background material we refer the reader to the comprehensive monograph by Papageorgiou-Rȃdulescu-Repovš [34]), and the statement of the main result. We begin by defining the Hilbert space equipped with the norm We denote the Schwartz class of functions by S(R). Thus for any u ∈ S(R), the Fourier transform of (−∆) 1 2 u is given by |ξ|û, whereû denotes the Fourier transform of u. Also, by Proposition 3.6 given in Di Nezza et al. [15], we have The factor 1 2π will be ignored in the paper throughout. Next, we define a slightly smaller space that will make the associated energy functional well-defined (cf. Stubbe [40, Lemma 2.1]): endowed with the norm Then X is a Hilbert space as well. We define three auxiliary bilinear forms as follows We further define the functionals Clearly, a combination of the Hardy-Littlewood-Sobolev inequality (HLS) Lieb [29], 0 ≤ ln(1+r) ≤ r, for any r > 0, leads to the inequality.
The associated energy functional is well-defined due to the lemma above and the space definition. However, the functional is not C 1 which disallows the use of the basic results of variational analysis. To tackle this, we define the cutoff functionalẼ as followsẼ in Ω.
By Lemmas 2.1, 3.1 we can now conclude that (2.14) Remark 2.2. Under the hypothesis (A 1 ) − (A 2 ), there exists, for q > 2, ǫ > 0 and β > θ, a constant c 2 > 0 such that Furthermore, there exists a constant c 3 > 0 satisfying the following inequality An important consequence of (2.15) is the following: Remark 2.3. Henceforth, 1. the notation of a subsequence will be the same as its sequence; 2. the notation for cutoff energy functionalẼ will be continued to be denoted by E.
We are now in a position to state our main result.
Theorem 2.4. Assume that hypotheses (A 1 )−(A 4 ) are satisfied and let q > 4 and C q > 0 be chosen sufficiently large. Then is the class of paths on X joining γ(0) and γ (1); An important application of the fractional Laplacian operator for dimension N = 1 can be found in Dipierro et al. [16]. We give a gist of the version of a model for the dynamics of the dislocation of atoms in crystals. The model is related to the Peierls-Nabarro energy functional. The system is a hybrid combination in which a discrete dislocation occurring along dislocation dynamics in crystals a slide line is incorporated in a continuum medium. The problem is as follows where s ∈ 1 2 , 1 , P is a 1-periodic potential and σ ǫ plays the role of external stress acting on the material. Setting v ǫ (t, x) = v t ǫ 1+2s , x ǫ , equation (2.18) can be recast as follows For a suitable choice of v 0 ǫ the basic layer solution u is introduced, which happens to be a solution to the following problem One can see that the problem considered in this article has a proper physical application and is a testimony to the importance of the problem considered in this paper.

Auxiliary Lemmas
In this section we will discuss some auxiliary lemmas.
Then there exist n 0 ∈ N and C > 0 such that u n * < C, for any n ≥ n 0 . Moreover, if A(u 2 n , v 2 n ) → 0 and v n 2 → 0, as n → ∞, then v n * → 0 as n → ∞.
Hence, from Lemma 3.3 and equation (2.15), for any u ∈ X we havê The following are some useful lemmas which will be used in the paper.
, and (w n ) be bounded sequences in X such that u n ⇀ u in X. Then for every z ∈ X, we have A(v n w n , z · (u n − u)) → 0 as n → ∞. (i) The functionals U, V, W are of class C 1 on X. In fact, (ii) V is continuously differentiable on L 4 (R).
(iii) U is a weakly lower semicontinuous functional on H 1 2 (R).
(iv) E is lower semicontinuous on H 1 2 (R). We now check that the energy functional E depicts the Mountain pass geometry. This will be required to obtain a Cerami sequence for a certain mountain pass energy level d (given in Theorem 2.4). Following is the definition of a Cerami sequence pertaining to a C 1 -functional.
Y is a normed linear space with the norm · Y . Then Φ is said to satisfy the Cerami condition at a level c ∈ R if any sequence (u n ) ⊂ Y such that Φ(u n ) → c and (1 + u n Y )Φ ′ (u n ) → 0 as n → ∞ has a convergent subsequence in Y .
Proof. Let u ∈ X \ {0} and such that u > u λ a.e. in Ω. Choose ρ 1 , ρ 2 > 1 such that ρ 1 ∼ 1, ρ 2 > 2 and ρ 1 β u 2 in order to apply the exponential estimates in (2.17). Furthermore, on using the Sobolev embeddings, we get Thus, for a pair of sufficiently small positive numbers (µ 0 , R), we get E(u) > 0, whenever u = r < R and µ ∈ (0, µ 0 ). Similarly,  Proof. Suppose that u ∈ X \ {0}. By (A 4 ) and q > 4 we get Proof. Using (2.14), the condition in (3.5), (A 3 ), and the embedding of H 1 2 (R) ֒→ L q (R) for q ∈ [1, ∞) from Lemma 3.1, we obtain (3.6) The inequality in (3.6) clearly shows that the sequence (u n ) is bounded in H 1 2 (R). For if not, then on dividing (3.6) by u n q and then passing the limit n → ∞, yields a contradiction to 0 ≥ C 6 L 4 − 1 . Thus, for a small range of µ, say (0, µ 0 ), we have The following lemma shows that any sequence (u n ) ⊂ X such that E(u n ) ≤ d for all n ∈ N can be taken to be of sufficiently small norms.
Lemma 3.12. Let (u n ) ⊂ X satisfy the Cerami condition in (3.5) with q > 4. Then for some sufficiently small r 0 > 0, we have lim sup n u n 2 < r 2 0 .
Proof. By Lemma 3.11, we know that (u n ) is bounded in H 1 2 (R). Certainly, lim sup n u n 2 ≤ 4c+o (1) is bounded above (and of course, below). We will find an estimate of the upper bound for this quantity. Consider the set S = {u ∈ X : u = 0, W (u) ≤ 0} and define u t (x) = t 2 u(tx) for all t > 0, u = 0 ∈ X, x ∈ R. We have W (u t ) = t 6 C(U) − t 6 ln t u 4 2 → −∞ as t → ∞. This shows that S is nonempty. By the Sobolev embedding theorem we have u ≥ C u q , for all u ∈ H We will now estimate the energy level d. Let v ∈ S, and T > 0 be sufficiently small. Then E(T v) < 0. Take a path α : [0, 1] → X defined as Consequently, for w ∈ S, we have where we have used (A 4 ). On taking infimum over w ∈ S, we obtain lim sup Before we state and prove the next lemma we need to recall the following two theorems.
Then T is pre-compact in L q (Ω). |u n (x)| 2 dx > 0. (3.12) Then there exist u ∈ H 1 2 (R) \ {0} and (y n ) ⊂ Z such that up to a subsequence,ũ n = u n (· − y n ) ⇀ u ∈ H 1 2 (R). Here, B 3 2 (y) = {x ∈ R : |x − y| < 3 2 }. Proof. The property of lim inf and sup together produces the sequence (y n ) such that |y n | → ∞ and the boundedness of (u n ) in H 1 2 (R) produces u such that u n ⇀ u ∈ H 1 2 (R). Also, u = 0 given condition (3.12). By Lemma 3.1 we have that u n → u in L 2 (R) and hence from Theorem 3.13 we have u n → u in L 2 (B 3 2 (y n )). Therefore, there exists a subsequence such that u n → u a.e. in B 3 2 (y n ). Theseũ n = u n (· − y n ) are nothing but the restrictions of the functions in (u n ) over B 3 2 (y n ). Thus by Theorem 3.14, we can consider the functions O 00ũn that will still be denoted byũ n . Therefore, u n ⇀ u ∈ H 1 2 (R).

Proof of the Main Theorem
We now give a proof of Theorem 2.4. First, we will prove another lemma. (i) u n → 0 and E(u n ) → 0 as n → ∞.
(ii) There exists (y n ) ⊂ Z such that |y n | → ∞ such thatũ n = u n (·−y n ) → u in X, for a nontrivial critical point u ∈ X of E.
Proof. By Lemma 3.12, we see that the Cerami sequence (u n ) is bounded in H 1 2 (R). Therefore there exists a subsequence from Lemma 3.15 such that R (e ρ 1 βu 2 n − 1)dx ≤ C β , for every n ∈ N.  This further implies that u n , U(u n ) → 0 as n → ∞. By the embedding H 1 2 (R) ֒→ L 2 (R), we have that u n 2 → 0 as n → ∞. Also by Remark 2.2, we have´R F (u n )dx → 0 as n → ∞. Thus E(u n ) → 0 as n → ∞ which is a contradiction. Thus This fact combined with Lemma 3.15, helps us to produce a sequence (y n ) ⊂ Z and u ∈ H 1 2 (R) \ {0} such that u n (· − y n ) =ũ n ⇀ u in H 1 2 (R). Therefore (ũ n ) is bounded in L p (R) for any p ≥ 2 and henceũ n (x) → u(x) a.e. in R. Now, observe that for q > 2, Therefore, U(ũ n ) < ∞. Hence, since (ũ n ) is bounded in L 2 (R) it follows by Lemma 3.2 that, ( ũ n * ) is bounded in X. By the reflexivity of X, we getũ n ⇀ u in X. From Lemma 3.1 we obtainũ n → u in L p (R) for all p ≥ 2.

Appendix
Lemma 5.2 will establish the existence of a positive solution to (2.13) and Lemma 5.4 will guarantee that a solution to (1.1) is greater than or equal to the solution to (2.13).
We choose the test function φ = (u − v) + . We express, 3) The equation in (5.3) implies This leads to the conclusion about the Lebesgue measure of R + , i.e., |R + | = 0. In other words v ≥ u a.e. in R.
Lemma 5.2. Let µ > 0. Then the following problem has a unique weak solution in X 0 . This solution is denoted by u µ , satisfies u µ ≥ ǫ µ v 0 a.e. in Ω, where ǫ µ > 0 is a constant.
Proof. We follow the proof in Choudhuri [11] and Choudhuri-Saoudi [12]. First, we note that an energy functional on X formally corresponding to (5.5) can be defined as follows.
for all u ∈ X where u + (x) = max{u(x), 0}. By using the Poincaré inequality, this functional is coercive and continuous on X. It follows that E possesses a global minimizer u 0 ∈ X. Obviously, u 0 = 0 since E(0) = 0 > E(ǫv 0 ), for sufficiently small ǫ and some v 0 > 0 in R.
Next if u 0 is a global minimizer for E, then |u 0 | is also a global minimizer. This is because E(|u 0 |) ≤ E(u 0 ). Clearly, the equality holds if and only if u − 0 = 0 a.e. in R. Here u − (x) = min{−u(x), 0}. In other words we must have u 0 ≥ 0, i.e. u 0 ∈ X + where X + = {u ∈ X : u ≥ 0 a.e. in R} is the positive cone in X. Furthermore, we will show that u 0 ≥ ǫv 0 > 0 holds a.e. in R for small enough ǫ. We observe that, whenever ǫ ∈ (0, ǫ µ ], for some sufficiently small ǫ µ . We now show that u 0 ≥ ǫ µ v 0 . Suppose we assume the contrary that w = (ǫ µ v 0 − u 0 ) + does not vanish identically in R. We denote R + = {x ∈ R : w(x) > 0}. We will analyye the function ζ(t) = E(u 0 + tw) of t ≥ 0. This function is convex when defined over X + being convex. Furthermore ζ ′ (t) = E ′ (u 0 + tw), w is nonnegative and nondecreasing for t > 0. Consequently, for 0 < t < 1 we have by inequality (5.7) and ζ ′ (t) ≥ 0 with ζ ′ (t) being nondecreasing for every t > 0. which leads to a contradiction. Therefore w = 0 in R and hence u 0 ≥ ǫ µ v 0 a.e. in R. Moreover, since the functional E is strictly convex on X + , we conclude that u 0 is the only critical point of E in X + with the property ess inf V u 0 > 0, for any compact subset V ⊂ R. Thus we choose u µ = u 0 in the cutoff functional. Remark 5.3. We now perform an analysis on a solution (if it exists). Suppose that u is a solution to (1.1). Then we observe the following 1. If u is a global minimizer, then clearly E(u) ≤ E(|u|). Further, E(u) ≥ E(|u|) is always true due to the first term of the energy functional. Thus u − = 0 a.e. in R.
2. In fact, a solution to (1.1) can be considered to be positive, i.e. u > 0 a.e. in R, due to the presence of the singular term.
Therefore without loss of generality, we may assume that the solution is positive.
We finally have the following result.
Proof. Fix µ ∈ (0, µ 0 ) and let u ∈ X be a positive solution to (1.1) and u µ > 0 be a solution to (5.5). We will show that u ≥ u µ a.e. in R. Thus, we let R * = {x ∈ R : u(x) < u µ (x)} and from the equation satisfied by u, u µ , we have Hence, by (5.9) and (5.10), we obtain u ≥ u µ a.e. in R. Now suppose S = {x ∈ Ω : u(x) = u λ (x)}. Clearly S is a measurable set and hence for any δ > 0 there exists a closed subset F of S such that |S \ F | < δ. Furthermore, let |S| > 0. Define a test function ϕ ∈ C 1 c (R) such that This is a contradiction. Therefore, |S| = 0. Hence, u > u λ a.e. in R.