An existence result for a quasilinear degenerate problem in R N

In this paper we study an existence result of the quasilinear problem −div[φ′(|∇u|2)∇u] + a(x)|u|α−2u = |u|γ−2u + |u|β−2u in RN(N ≥ 3), where φ(t) behaves like tq/2 for small t and tp/2 for large t, a is a positive potential, 1 < p < q < N, 1 < α ≤ p∗q′/p′ and max {α, q} < γ < β < p∗ = pN/(N − p), with p′ and q′ the conjugate exponents of p, respectively q. Our main result is the proof of the existence of a weak solution, based on the mountain pass theorem.


Introduction and preliminary results
In this paper we are interested for a new type of operator, introduced by Azzollini in some recent papers [4,5] and by N. Chorfi and V. Rȃdulescu in [8].Their studies are based on the nonhomogeneous operators of the type div[φ (|∇u| 2 )∇u], where φ ∈ C 1 (R + , R + ) has a different growth near zero and at infinity.Such a type of behavior occurs when φ(t) = 2[(1 + t) 1/2 − 1], which corresponds to the operator div ∇u √ 1 + |∇u| 2 known as the prescribed mean curvature operator or the capillary surface operator.More precisely, φ(t) behaves like t q/2 for small t and t p/2 for large t, where 1 < p < q < N.Such behavior occurs, for example, when φ(t) = 2 p 1 + t q/2 p/q − 1 Email: iulia.stircu@hotmail.com
In [8] N. Chorfi and V. Rȃdulescu approached the quasilinear Schrödinger equation Their interest was studying the problem where N ≥ 3, a is a positive potential and f has a subcritical growth, problem studied by P. Rabinowitz in [15], in the new abstract setting introduced by Azzollini in [4,5] (see also [17]).
The importance of the Schrödinger type equation is obvious.This equation is fundamental for quantum mechanics, which together with general relativity represents the most useful current theories about the physical universe.
In 1927, elaborating research of many physicists, Erwin Schrödinger wrote a differential equation for any quantum waves function, namely where h is the Planck constant divided by 2Π, Ψ is the wave function, i is the square root of minus one and Ĥ is the Hamiltonian operator.
The classical wave equation defines waves in space and the solution is a numerical function depending on space and time.The same happens with the Schrödinger equation, but in this case the values of the wave function Ψ are also complex, not just real.
The applications of this equations are numerous, varying from Bose-Einstein condensates and nonlinear optics, propagation of the electric field in optical fibers, stability of Stokes waves in water to the behavior of deep water waves and freak waves in the ocean.For more applications to nonlinear equations with variable or constant exponents we refer [1,7,9,16,[18][19][20].
In this paper we are interested to study problem (1.1) in the particular case More precisely, we consider the quasilinear degenerate problem where a is a positive potential, 1 < p < q < N, 1 < α ≤ p * q /p and max {α, q} < γ < β < p * = pN/(N − p), with p and q the conjugate exponents, respectively, of p and q.
Our purpose is to prove, by means of the mountain pass theorem (see [12][13][14]), that problem (1.2) admits at last one weak solution.Now, we define the function space The property that L p (R N ) + L q (R N ) are Orlicz spaces, as well as others properties of these spaces, has been proved by M. Badiale, L. Pisani and S. Rolando in [6].
In order to use them throughout this paper, we state the following result that is also found in [5].
Finally, we define the function space We remark that X is continuously embedded in W defined by Azzollini in [5], where In the next section we introduce the main hypotheses and we state the basic results of this paper.The proof of the main result are developed in Section 3.

The main results
We assume that a in problem (1.2) is a singular potential satisfying the following hypotheses: In the following, we assume that the function φ, which generates the differential operator in problem (1.2), has the next properties: Our first hypothesis which asserts that φ approaches 0 ensures us that problem (1.2) is degenerate and no ellipticity condition is assumed.
We also remark that, because of the presence of the general potential a, the solutions of problem (1.2) cannot be reduced to radially symmetric solutions, like in [5].A frequently used property in [5] by Azzollini was the continuously embedding of the space W in L p * (R N ), provided that 1 < p < min {q, N}, 1 < p * q /p and α ∈ (1, p * q /p ).By interpolation, for every We define the energy functional I : X → R by Proposition 2.2.The functional I is well-defined on X and I ∈ C 1 (X, R), with the Gâteaux derivative given by This result can be easily ensured by standard arguments and [3, Lemma 2.2].We notice that our hypotheses imply that Now, we give a version of the mountain pass lemma of A. Ambrosetti and P. Rabinowitz [2] (see also [8]).
Lemma 2.3.Let X be a Banach space and assume that I ∈ C 1 (X, R) satisfies the following geometric hypotheses: (a) I(0) = 0 (b) there exist two positive numbers a and r such that I(u) ≥ a for any u ∈ X with u = r; (c) there exists e ∈ X with e > r such that I(e) < 0. I(p(t)).
Then there exists a sequence (u n ) ⊂ X such that Moreover, if I satisfies the Palais-Smale condition at the level c, then c is a critical value of I.
Finally, the main result of this paper is given by the following theorem.

Proof of Theorem 2.4
It is obvious that I(0) = 0. Now we check (b), the first geometrical condition of the mountain pass lemma, more exactly the existence of a "mountain" around the origin.Let be u ∈ X, r ∈ (0, 1) a fixed point and u = r.
Using (φ 3 ), (iv) of Proposition 1.1 and the continuously embeddings of the spaces where c 3 and c 4 are two positive constants.
So, there exists t 0 > 0 such that I(tu) < 0. Thus, we have checked the second geometrical hypothesis of the mountain pass lemma, or the existence of a "valley" over the chain of mountains.Now, we prove that the corresponding setting is non-degenerate, namely, the associated min-max value given by Lemma (2.By contradiction, we suppose that c = 0, that is, for all > 0 there exists q ∈ P such that If we fix < a, where a is given by (3.2), then q(0) = 0 and q(1) = t 0 u.Therefore, q(0) = 0 and q(1) > r.
Since q is continuous, there exists t 1 ∈ (0, 1) such that q(t 1 ) = r, so The above inequality is a contradiction, which shows that our claim (3.4) is true.
By Lemma (2.3), we obtain a Palais-Smale sequence (u n ) ∈ X for the level c > 0 such that Finally, we prove that this sequence (u n ) is bounded in X.Using relation (3.5), we obtain that By relation (φ 5 ) and hypothesis max {α, q} < γ < β < p * we have for all n ∈ N, with c 0 > 0 arbitrary.By the above inequality we deduce that (u n ) is bounded in X.
We know that X is a closed subset of W.Then, by Proposition 2.5 in [5] we deduce that Now, we are concerned to prove that u 0 is a solution of problem (1.2).Fix ϕ ∈ C ∞ 0 (R N ) and set Ω := supp(ϕ).We can write and for this purpose, we define the energy functionals By (3.9) and (3.10) we obtain It follows from (3.12) and (3.13) that Since A is convex (by (φ 6 )), The functional A is convex and continuous, thus it is lower semicontinuous, so Combining (3.16) and (3.17) we obtain that Making use of the same arguments as in [5, p. 210], we deduce that We can conclude now that for all ϕ ∈ X, then u 0 is a solution of problem (1.2).
Proof of Theorem 2.4 completed.We have previously shown, by means of mountain pass lemma, that problem (1.2) has a weak solution.It remains to argue that the solution u 0 found above is nontrivial.So, in order to complete the proof of theorem (2.4), we use some methods developed in [10] and [11].
Since the secquence (u n ) satisfies the Palais-Smale condition, relation (3.5) leads to for n a positive integer large enough.By (φ 6 ) we deduce that φ(t 2 ) − φ(0) ≤ φ (t 2 )t 2 and applying now (φ 2 ), φ(t 2 ) ≤ φ (t 2 )t 2 , which means we can write that From now, we split the proof in two cases.First, we suppose that α ≥ 2. Combining relations (3.18) and (3.19) we obtain where c 3 and c 4 are positive constants.Our aim is to show that u 0 = 0.For this purpose we suppose by contradiction that u 0 = 0.This both relation (3.9) implies that hence,  We argue again by contradiction and assume that u 0 = 0.In particular, this implies that u n → 0 in L α loc (R N ).
where M = sup n R N a(x)|u n | α dx.
If we choose k large enough and take into account hypothesis (a 2 ), we obtain by (3.26) that c = 0, a contradiction.
Resuming, we have obtained that u 0 is a nontrivial solution of problem (1.2) and this concludes our proof.