Existence of Nontrivial Solutions for Generalized Quasilinear Schrödinger Equations with Critical Growth

We study the following generalized quasilinear Schrödinger equations with critical growth −div(g2(u)∇u) + g(u)g󸀠(u)|∇u|2 + V(x)u = λf(x, u) + g(u)|G(u)|2−2G(u), x ∈ RN, where λ > 0, N ≥ 3, g(s) : R → R is a C1 even function, g(0) = 1, and g󸀠(s) ≥ 0 for all s ≥ 0, where G(u) fl ∫u 0 g(t)dt. Under some suitable conditions, we prove that the equation has a nontrivial solution by variational method.

The equations are related to the existence of solitary wave solutions for quasilinear Schrödinger equations where  : R × R  → C,  : R  → R is a given potential,  : R → R, and  : R  × R → R are suitable functions.The form of (2) has been derived as models of several physical phenomena corresponding to various types of ().For instance, the case () =  models the time evolution of the condensate wave function in superfluid film [1,2] and is called the superfluid film equation in fluid mechanics by Kurihara [1].In the case () = (1 + ) 1/2 , problem (2) models the self-channeling of a high-power ultra short laser in matter, the propagation of a high-irradiance laser in a plasma creates an optical index depending nonlinearly on the light intensity, and this leads to interesting new nonlinear wave equations; see [3][4][5][6].For more physical motivations and more references dealing with applications, we can refer to [7][8][9][10][11][12][13][14] and references therein.
In the past, the research on the existence of solitary wave solutions of Schrödinger equations ( 2) is for some given special function ().In this paper, we will use a unified new variable replacement to study (2), constructed by Shen and Wang in [16].Define the energy functional associated with (1) by where (, ) fl ∫  0 (, ).However,   is not well defined in  1 (R  ) because of the term ∫ R   2 ()|∇| 2 .To overcome this difficulty, we make a change of variable constructed by Shen and Wang in [16]: V fl () fl ∫  0 ().Then we obtain If  is a nontrivial solution of (1), then for all  ∈  ∞ 0 (R  ).Let  = (1/()).By [16] we know that (9) is equivalent to for all  ∈  ∞ 0 (R  ).Therefore, in order to find the nontrivial solution of (1), it suffices to study the existence of the nontrivial solutions of the following equations: Recently, the authors studied generalized quasilinear Schrödinger equations with subcritical growth [19,20], critical growth [21], and supercritical growth [22].
In order to reduce the statements for main results, we list the assumptions as follows: for all (, ) ∈ R  × R.
Set  =  1 (R  ) with the norm It is easy to prove that   is well defined on  and   ∈  1 (, R) under our assumptions and its Gateaux derivative is given by for all V,  ∈ .
Our main result of this paper is as follows.
From Remark 2 we obtain Corollary 3.
(ℎ 2 ) There exists 2 <  < 2 * such that for all  > 0. (ℎ 3 ) There exists  > 2 such that, for any  > 0, there holds As mentioned above, if we set  2 () = 1 + 2 2 , then we get the superfluid film equation in plasma physics whose nontrivial solutions were studied in [23].But our problem ( 1) is elliptic problem involving the critical exponent, so our result extends the results of the work [16,23] to a critical setting.Moreover, the assumptions about the nonlinearity in this paper are different from the assumptions about the nonlinearity in [16,23].
In this paper, we just assume that  is a continuous function.Moreover, there are functionals (, ) satisfying ( 3 ) but not satisfying the above Ambrosetti-Rabinowitz type condition (see Remark 1.2 in [25]).Hence, our result is different from the result there.
Then there exists a bounded Cerami sequence {V  } ⊂  for   with   (V  ) →   ≥  > 0, where is the constant appearing in Lemma 8.
Proof.By Lemma 8 and the mountain pass theorem without (PS) condition (see Theorem 4.1 in [26]), there exists a Cerami sequence {V  } ⊂  satisfying where is the constant appearing in Lemma 8.
In the following, we consider the case  = 3, 4. Indeed, if the conclusion is false, then there exists a sequence {  } with   → +∞ such that    ≥ (1/) /2 .Take V ∈ \{0}.Then by the proof of Lemma 8, there exists a unique    > 0 such that max >0    (V) =    (   V).Hence By Lemma 6 (6) and which implies that {   } is bounded.Hence, up to a subsequence, there exists  0 ≥ 0 such that    →  0 as  → ∞.If  0 > 0, then by ( 4 ) and Fatou lemma we have lim But, on the other hand, by Lemma 6(6) one has a contradiction.Hence  0 = 0 and by Lemma 7(4) we know that max as  → ∞.Consequently, a contradiction.This completes the proof.
Proof of Theorem 1.Since {V  } ⊂  is a bounded Cerami sequence for   at the level   > 0, there exists V ∈  such that Using a standard argument, we know that    (V) = 0, that is, V is a weak solution of (11).Indeed, for any  ∈  ∞ 0 (R  ), we have Advances in Mathematical Physics Consequently, for all  ∈  ∞ 0 (R  ).For any  ∈ , there exists a sequence Let  → ∞, we get that is, ⟨   (V), ⟩ = 0 for all  ∈ .Hence    (V) = 0; that is, V is a weak solution of (11).
In the following, we prove that V is nontrivial.With the aid of Lemma 10, the proof follows essentially the proof of Theorem 1.1 in [16].For completeness, we present the proof as follows.If the conclusion is false, we may assume V = 0. We divide the proof into four steps.
Step 1.We prove that {V  } ⊂  is also a Cerami sequence for the functional  ∞  :  → R, where By ( 2 ) and V  ⇀ 0 in , one has as  → ∞.Consequently, {V  } is also a Cerami sequence of  ∞  .