Generalized Quasilinear Elliptic Equations in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^N$$\end{document}

In this paper, we aim at establishing a new existence result for the quasilinear elliptic equation -div(a(x,u,∇u))+At(x,u,∇u)+V(x)|u|p-2u=g(x,u)inRN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} - \textrm{div} (a(x,u,\nabla u)) + A_t(x,u,\nabla u) + V(x) {\vert u \vert }^{p-2} u= g(x,u) \quad \quad \hbox { in }{{\mathbb {R}}}^{N} \end{aligned}$$\end{document}with p>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p>1$$\end{document}, N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 2\ $$\end{document} and V:RN→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$\end{document} suitable measurable positive function. Here, we suppose A:RN×R×RN→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A: {\mathbb {R}}^N \times {\mathbb {R}}\times {\mathbb {R}}^N \rightarrow {\mathbb {R}}$$\end{document} is a given C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C}^{1}$$\end{document}-Carathéodory function which grows as |ξ|p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\xi |^p$$\end{document}, with At(x,t,ξ)=∂A∂t(x,t,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_t(x,t,\xi ) = \frac{\partial A}{\partial t}(x,t,\xi )$$\end{document}, a(x,t,ξ)=∇ξA(x,t,ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(x,t,\xi ) = \nabla _\xi A(x,t,\xi )$$\end{document}, V:RN→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V:{\mathbb {R}}^N\rightarrow {\mathbb {R}}$$\end{document} is a suitable measurable function and g:RN×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g:{\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$\end{document} is a given Carathéodory function which grows as |ξ|q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\xi |^q$$\end{document} with 1<q<p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<q<p$$\end{document}. Since the coefficient of the principal part depends on the solution itself, under suitable assumptions on A(x,t,ξ),V(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A(x,t,\xi ), V(x)$$\end{document} and g(x, t), we study the interaction of two different norms in a suitable Banach space with the aim of obtaining a good variational approach. Thus, a minimization argument on bounded sets can be used to state the existence of a nontrivial weak bounded solution on an arbitrary bounded domain. Then, one nontrivial bounded solution of the given equation can be found by passing to the limit on a sequence of solutions on bounded domains. Finally, under slightly stronger hypotheses, we can able to find a positive solution of the problem.

It is worth of knowing that equation (1.3) has a variational structure, but there is a lack of compactness as such a problem is settled in the whole Euclidean space R N and classical variational tools do not work easily; thus, suitable assumptions on potential V (x) and on the function g are required.
More recently, many authors investigated equation (1.1) when the function A(x, t, ξ) has the particular form 1 p A(x, t)|ξ| p with the coefficient A not constant. In this case, besides the difficulties arising for the problem (1.3) due to the lack of compactness, the presence of a coefficient which depends on the solution itself (and in the general case A(x, t, ξ) depends also on the derivatives of the solution) makes the variational approach more complicated. In fact, in this case, under suitable assumption on A, V and g, the "natural" functional associated to (1.3) is g(x, s)ds, but, even if A(x, t) is a smooth strictly positive bounded function, if A t (x, t) ≡ 0 functional I is well defined in W 1,p (R N ), while it is Gâteaux differentiable only along directions in X = W 1,p More in general, if A(x, t, ξ) grows as |ξ| p with respect to ξ, where p > 1, suitable additional growth assumptions on the functions involved guarantee that to the equation (1.1) can be associated the functional with η and q as above. The paper is organized as follows. In Sect. 2, we introduce the abstract framework and we recall a weaker version of the Cerami's variant of the Palais-Smale condition and the related Minimum Principle (see, Proposition 2.2). In Section 3 we introduce some preliminary assumptions on the functions A(x, t, ξ), V (x) and η(x) which allow to give a variational principle for equation (1.1). In Sect. 4, we consider some further hypotheses, we state our main result (see, Theorem 4.4) and we prove some properties for the action functional associated to the equation (1.1). Then, in Sect. 5, we prove Theorem 4.4 by means of a method of approximation on bounded domains, while in Sect. 6, a slightly strong assumption on the function A(x, t, ξ) allows us to state the existence of a nontrivial weak positive solution.

Abstract setting
In this section, we assume that • (X, · X ) is a Banach space with dual (X , · X ); • (W, · W ) is a Banach space such that X → W continuously, i.e., X ⊂ W and a constant σ 0 > 0 exists such that • J : D ⊂ W → R and J ∈ C 1 (X, R) with X ⊂ D. Nevertheless, to avoid any ambiguity, we will henceforth denote by X the space equipped with its norm · X , while, if the norm · W is involved, we will write it explicitly. Taking β ∈ R, we say that a sequence (u n ) n ⊂ X is a Cerami-Palais-Smale sequence at level β, briefly (CP S) β -sequence, if Moreover, β is a Cerami-Palais-Smale level, briefly (CP S)-level, if there exists a (CP S) β -sequence. The functional J satisfies the classical Cerami-Palais-Smale condition in X at the level β if every sequence (CP S) β -sequence converges in X up to subsequences. However, considering the setting of our problem, in general a (CP S) β -sequence may also exist which is unbounded in · X but which converges with respect to · W . Then, we are able to weaken the classical Cerami-Palais-Smale condition in an appropriate way according to ideas that have already been developed in previous papers (see, for example, [11][12][13]). Definition 2.1. The functional J satisfies the weak Cerami-Palais-Smale condition at level β (β ∈ R), briefly (wCP S) β condition, if for every (CP S) βsequence (u n ) n , a point u ∈ X exists such that (i) lim n→+∞ u n − u W = 0 (up to subsequences), If J satisfies the (wCP S) β condition at each level β ∈ I, I real interval, we say that J satisfies the (wCP S) condition in I.
Let us point out that, due to the convergence only in the norm of W , the (wCP S) β condition implies that the set of critical points of J at the β level is compact with respect to · W , so that we can state a Deformation Lemma and some abstract theorems about critical points (see [13]). In particular, the following Minimum Principle applies (for the proof, (see, [13, Theorem 1.6])).

Proposition 2.2. (Minimum Principle)
If J ∈ C 1 (X, R) is bounded from below in X and (wCP S) β holds at level β = inf X J ∈ R, then J attains its infimum, i.e., u 0 ∈ X exists such that J(u 0 ) = β.

Variational framework and regularity result
Let N = {1, 2, ...} be the set of the strictly positive integers and, taking any Ω open subset of R N , N ≥ 2, we denote by: • B R (x) = {y ∈ R N : |x − y| < R} the open ball in R N with center in x ∈ R N and radius R > 0; • |D| the usual Lebesgue measure of a measurable set D in R N ; • L r (Ω) the Lebesgue space endowed with norm |u| Ω,r = Ω |u| r dx • W 1,p (Ω) and W 1,p 0 (Ω) the classical Sobolev spaces both equipped with the standard norm u Ω = (|∇u| p Ω,p + |u| both equipped with the norm For simplicity, we put B R = B R (0) and, if Ω = R N , we omit the subscript in the previous notations, i.e., we put • | · | r = | · | R N ,r for the norm in L r (R N ), for all 1 ≤ r < +∞; • | · | V,r = | · | R N ,V,r for the norm in L r V (R N ), for all 1 ≤ r < +∞; • · = · R N for the norm in W 1,p (R N ) and in W 1,p 0 (R N ); Remark 3.1. By assumption (V 1 ), the following continuous embeddings hold: and From now on, we suppose that V satisfies also the following additional hypothesis: Furthermore, if assumption (V 2 ) also occurs, the compact embedding holds, with From Proposition 3.2, it follows that for any r ≥ p as in (3.4), respectively (3.5) or (3.6), a constant τ r > 0 exists such that On the other hand, if we consider the case in which Ω is an open bounded domain of class C 1 in R N with ∂Ω bounded and p < N, from a classical embedding Theorem (see, e.g., [10, Corollary 9.14]) a constant σ * > 0, independent of Ω and depending only on p and N , exists such that |u| Ω,p * ≤ σ * u Ω for all u ∈ W 1,p 0 (Ω). From now on, we assume that V is a measurable function verifying (V 1 ) and we set In the following, we assume p ≤ N as, otherwise, from (3.6), it follows that X = W 1,p V (R N ) and all the arguments and the proofs can be simplified. The following lemmas hold.  and |u n | ∞ ≤ M for all n ∈ N, Proof. For the proof, (see, [18,Lemma 3.5]).
Here and in the following, let A : R N ×R×R N → R and g : R N ×R → R be such that, using the notation in (1.2), the following hypotheses hold: Remark 3.6. From (g 1 ), it results Moreover, from (g 0 )-(g 1 ) and Remark 3.6, we have that Clearly, it follows that u = 0 is a trivial solution of (1.1).
, by Hölder's inequality with p p−q and p q conjugate exponents and by (3.12) it follows that by applying again Hölder's inequality with p p−q , p q−1 and p conjugate exponents.
Remark 3.9. From (g 0 )-(g 1 ), we have that Indeed, Hölder's inequality with p−1 p−q and p−1 q−1 conjugate exponents implies that Therefore, from (3.9) and (3.13) it follows that the functional is well defined for all u ∈ X. Moreover, taking v ∈ X, from (3.14), the Gâteaux differential of functional J in u along the direction v is given by Now, we are ready to prove the following regularity result.

Proposition 3.10. Let p > 1 and assume that conditions
and (3.10), (3.11) hold for a constant M > 0, then Hence, J is a C 1 functional on X with Fréchet differential as in (3.16).

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Proof. For simplicity, we set Let (u n ) n ⊂ X, u ∈ X and M > 0 such that (3.10), (3.11) and (3.17) hold. Reasoning as in [14], we obtain that For completeness, we report the various computations. As a consequence of (3.11) and the growth assumption From (3.10) and (3.18), it follows that (J 1 (u n )) n is bounded. Arguing by contradiction, assume that the limit does not hold; then β ∈ R and a subsequence (J 1 (u kn )) n exist such that On the other hand, (3.10) implies that both u kn → u and ∇u kn → ∇u in L p (R N ) and also two functions (see, [10,Theorem 4.9]). Thus, from (h 0 ) and (3.17), we have where (3.18) and (3.22) imply in contradiction with (3.21). Now, from (3.10) and (3.19), (3.20) it follows that (dJ 1 (u n )) n is bounded in X . So, if by contradiction we assume that dJ 1 (u n ) − dJ 1 (u) X → 0 does not hold, thenε > 0, a subsequence (u kn ) n and a sequence (v n ) n ⊂ X exist such that v n X ≤ 1 and and from (3.10), (3.11), (3.19), (3.20), Hölder inequality, (3.22) and the same arguments pointed out previously, together with the Lebesgue's Dominated Convergence Theorem, it follows that, up to subsequences, Arguing as in [18,Proposition 3.11], we have that Finally, in order to state the regularity of J 3 (u), we have to prove that dJ 3 (u) is continuous or better if (u n ) n satisfies the assumptions (3.10), (3.11) and (3.17).

Set of hypotheses and the main result
From now on, besides (h 0 ) − (h 1 ), (g 0 ) − (g 1 ), (V 1 ) − (V 2 ) for the functions A(x, t, ξ), V (x) and g(x, t) we introduce the following set of hypotheses: (h 2 ) a constant α 0 > 0 exists such that (h 5 ) some constants μ > p and α 3 > 0 exist such that t p−1 = +∞ uniformly for a.e. x ∈ R N . We point out some direct consequences of the previous hypotheses.
We note that (4.2) implies hypothesis (H 5 ) in [12]. Now, we are ready to state our main result. Remark 4.5. The existence of solutions for quasilinear elliptic problems like (1.1), but in a bounded domain, has been established in [16]. On the other hand, equation (1.1) in the whole Euclidean space has been studied in the particular case A(x, t, ξ) = 1 p A(x, t)|ξ| p and g(x, t) = η(x)|t| p−1 (see [30]).
Hence, functional J is bounded from below, i.e., a constant α ∈ R exists such that Proof. From (h 2 ) and (3.13), we have Hence, the conclusion follows from From Proposition 4.6, the following convergence result can be stated. as n → +∞.

Proposition 4.7. Suppose that hypotheses
Proof. The conditions (4.3) -(4.5) follow from Proposition 4.6, the reflexivity of the space W 1,p V (R N ) and (3.7). Remark 4.8. We note that the thesis of Proposition 4.7 holds even if we replace · V with the usual norm · in W 1,p (R N ). Remark 4.9. We are not able to prove that J satisfies the condition (wCP S) in the whole space X. In particular, we are not able to prove that the weak limit u ∈ L ∞ (R N ) because the result of Ladyzhenskaya-Ural'tseva (see, [25, Theorem II 5.1]) holds for bounded domains. In order to overcome this problem, we use an approximation method on bounded domains. and we denote by X Ω its dual.
which, together with (3.3), implies that the norms · Ω,V and · Ω are equivalent, too. We have, in particular, that the norm in (4.6) can be replaced with the equivalent one, still denoted by · XΩ , given by and it results Actually, since any function u ∈ X Ω can be trivially extended to a functionũ ∈ X just assumingũ(x) = 0 for all x ∈ R N \Ω, then Thus, we can consider the restriction J Ω of the functional J to X Ω , i.e., Hence, by simplifying the arguments in the proof in [12,Proposition 3.1], it follows that the functional J Ω : X Ω → R is C 1 and, for any u, v ∈ X Ω , its Fréchet differential in u evaluated in v is given by (4.9) We need to state the following result.
Reasoning as in Proposition 4.7 with J Ω instead of J and using the Sobolev Embedding Theorem for bounded domains, we have that (u n ) n is bounded Page 15 of 27 205 in W 1,p 0 (Ω). Hence, there exists u ∈ W 1,p 0 (Ω) such that, up to subsequences, if n → +∞, then u n u weakly in W 1,p 0 (Ω), u n → u strongly in L r (Ω) for each r ∈ [1, p * [, u n → u a.e. in Ω.
and (g 0 ) − (g 1 ). Then functional J Ω has at least a critical point u Ω ∈ X Ω . Moreover, if (g 2 ) holds, then u Ω is not trivial.

Proof of the Theorem 4.4
For the proof of our main result, we follow an approach similar to those in [14] and [18]. Anyway, the different assumptions for our problem requires to rehash the proofs and provide them with all the details.
Throughout this section, we suppose that all the hypotheses in Theorem 4.4 are satisfied. Thus, for any k ∈ N, we can consider the spaces X B k as in (4.7) and the related functionals as in (4.8). For the sake of convenience, since any u ∈ X B k can be trivially extended setting u = 0 a.e. in R N \B k , we still denote by u such an extension.
Thus, from Propositions 4.6 and 4.12, a sequence (u k ) k ⊂ X exists such that for every k ∈ N it results: where, from Proposition 4.6, in (ii) we can choose α independent of k. Now, our aim is proving that sequence (u k ) k is bounded in X. First of all, we recall the following result which is a refinement of [25, Theorem II.51].

Lemma 5.2. Let
Ω be an open bounded domain in R N and consider p, s so that 1 < p ≤ N and p ≤ s < p * (if N = p we just require that p * is any number larger than s) and take u ∈ W 1,p 0 (Ω). If a * > 0 and m 0 ∈ N exist such that with Ω + m = {x ∈ Ω : u(x) > m}, then ess sup Ω u is bounded from above by a positive constant which can be chosen so that it depends only on |Ω + m0 |, N , p, s, a * , m 0 , u Ω , or better by a positive constant which can be chosen so that it depends only on N , p, s, a * , m 0 and a * 0 for any a * 0 such that Proof. From Proposition 4.6 and (ii), since X B1 ⊂ X B k for any k ∈ N, it is , so the condition q < p implies that ( u k V ) k is bounded. Furthermore, by (3.3), the sequence ( u k ) k is bounded, too. Now, we need to prove that (|u k | ∞ ) k is bounded.
To reach our aim, we suppose that |u k | ∞ > 1, i.e., ess sup R N u k > 1 or ess sup R N (−u k ) > 1.
Assume that ess sup R N u k > 1 and consider the set Now, for any m ∈ N, we define the function R + m : R → R such that It follows that R + m u k ∈ X B k . Hence, from condition (iii), (4.9), (h 4 ), Remark 4.2, (h 3 ), (V 1 ) and from direct computations we obtain where B + m,k = x ∈ R N : u k (x) > m and obviously it is B + m,k ⊂ B + 1,k . Since (g 1 ) implies that g(x, t) ≥ 0 if t > 0 for a.e. x ∈ R N , then g(x, u k (x))m ≥ 0 for a.e. x ∈ B + m,k . So, applying again (g 1 ) it results Thus, from (5.2), it follows that Since q < p and u k (x) > 1 for any x ∈ B + m,k , it follows that where c * > 0 is a constant independent of m and k. From Lemma 5.2 with Ω = B k and from the boundedness of ( u k ) k and |B + m,k | k , it follows that a constant M > 0, independent of k ∈ N, exists such that ess sup B k u k ≤ M. Similar arguments apply if ess sup R N (−u k ) > 1. Thus, it results that (|u k | ∞ ) k is bounded and the proof of (5.1) is complete.
Remark 5.4. From (5.1) and (3.9), it follows that for all τ ≥ p We observe that the boundedness of ( u k V ) k stated in the proof of Proposition 5.3 and the reflexivity of W 1,p V (R N ) ensure the existence of u * ∈ W 1,p V (R N ) such that, up to subsequences, From (3.7), it results u k → u * strongly in L r (R N ) for any r ∈ [p, p * ), (5.5) and u k → u * a.e. in R N . (5.6) Proof. For the proof, see, [18,Proposition 6.4].

Proposition 5.7. It results
Fixing any R ≥ 1, from (5.7), it is enough to prove that To achieve this aim, following an idea introduced in [8], let us consider the real map ψ(t) = teη t 2 , whereη > β 2 2β 1 2 will be fixed once β 1 , β 2 > 0 are chosen in a suitable way later on. By definition, Defining v k = u k − u * , we have that (5.4) implies v k 0 weakly in W 1,p V (R N ), while from (5.5), respectively (5.6), it follows that v k → 0 strongly in L p (R N ), (5.8) respectively v k → 0 a.e. in R N . (5.9) Moreover, from (5.1) and Proposition 5.5, it is v k ∈ X and |v k | ∞ ≤ M 0 for all k ∈ N (5.10)
Then, Theorem 4.4 can be improved as follows.
From now on, we assume that all the hypotheses in Theorem 6.1 are verified. Clearly, (g 1 ) implies that g(x, 0) = 0 for a.e. x ∈ R N .
Then, we introduce the new function g + : R N × R → R defined as g + (x, t) = g(x, u) a.e. in R N , for all t ≥ 0 0 a.e. in R N , for all t < 0 and the related primitive G + (x, t) = G(x, u) a.e. in R N , for all t ≥ 0 0 a.e. in R N , for all t < 0 Remark 6.2. We note that g + (x, t) satisfies conditions (g 0 )-(g 2 ).
From the previous remark and Proposition 3.10, it follows that the functional is of class C 1 on X and for any u, v ∈ X it is a(x, u, ∇u)∇vdx +