In this work, we investigated the existence of nontrivial weak solutions for the equation
−div(w(x)∇u) = f(x,u),x∈R2,
where w(x) is a positive radial weight, the nonlinearity f(x,s) possesses growth at infinity of the type exp((α0+h(|x|))|s|2/(1−β)), with α0>0, 0<β<1 and h is a continuous radial function that may be unbounded at infinity. To show the existence of weak solutions, we used variational methods and a new type of the Trudinger-Moser inequality defined on the whole two-dimensional space.
Citation: Yony Raúl Santaria Leuyacc. Elliptic equations in R2 involving supercritical exponential growth[J]. Electronic Research Archive, 2024, 32(9): 5341-5356. doi: 10.3934/era.2024247
[1] | Ganglong Zhou . Group invariant solutions for the planar Schrödinger-Poisson equations. Electronic Research Archive, 2023, 31(11): 6763-6789. doi: 10.3934/era.2023341 |
[2] | Jiayi Fei, Qiongfen Zhang . On solutions for a class of Klein–Gordon equations coupled with Born–Infeld theory with Berestycki–Lions conditions on $ \mathbb{R}^3 $. Electronic Research Archive, 2024, 32(4): 2363-2379. doi: 10.3934/era.2024108 |
[3] | Quanqing Li, Zhipeng Yang . Existence of normalized solutions for a Sobolev supercritical Schrödinger equation. Electronic Research Archive, 2024, 32(12): 6761-6771. doi: 10.3934/era.2024316 |
[4] | Hui Jian, Min Gong, Meixia Cai . Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential. Electronic Research Archive, 2023, 31(12): 7427-7451. doi: 10.3934/era.2023375 |
[5] | Lingzheng Kong, Haibo Chen . Normalized solutions for nonlinear Kirchhoff type equations in high dimensions. Electronic Research Archive, 2022, 30(4): 1282-1295. doi: 10.3934/era.2022067 |
[6] | Xian Zhang, Chen Huang . Existence and multiplicity of sign-changing solutions for supercritical quasi-linear Schrödinger equations. Electronic Research Archive, 2023, 31(2): 656-674. doi: 10.3934/era.2023032 |
[7] | Ping Yang, Xingyong Zhang . Existence of nontrivial solutions for a poly-Laplacian system involving concave-convex nonlinearities on locally finite graphs. Electronic Research Archive, 2023, 31(12): 7473-7495. doi: 10.3934/era.2023377 |
[8] | Jingyue Cao, Yunkang Shao, Fangshu Wan, Jiaqi Wang, Yifei Zhu . Nonradial singular solutions for elliptic equations with exponential nonlinearity. Electronic Research Archive, 2024, 32(5): 3171-3201. doi: 10.3934/era.2024146 |
[9] | Changmu Chu, Shan Li, Hongmin Suo . Existence of a positive radial solution for semilinear elliptic problem involving variable exponent. Electronic Research Archive, 2023, 31(5): 2472-2482. doi: 10.3934/era.2023125 |
[10] | Jincheng Shi, Shuman Li, Cuntao Xiao, Yan Liu . Spatial behavior for the quasi-static heat conduction within the second gradient of type Ⅲ. Electronic Research Archive, 2024, 32(11): 6235-6257. doi: 10.3934/era.2024290 |
In this work, we investigated the existence of nontrivial weak solutions for the equation
−div(w(x)∇u) = f(x,u),x∈R2,
where w(x) is a positive radial weight, the nonlinearity f(x,s) possesses growth at infinity of the type exp((α0+h(|x|))|s|2/(1−β)), with α0>0, 0<β<1 and h is a continuous radial function that may be unbounded at infinity. To show the existence of weak solutions, we used variational methods and a new type of the Trudinger-Moser inequality defined on the whole two-dimensional space.
We begin recalling the following stationary Schrödinger equation:
{−Δu = f(x,u),in Ω⊂RN u=0on ∂Ω. | (1.1) |
To treat the Eq (1.1) variationally, the Sobolev embedding theorems restrict the nonlinearity f to be of the type |f(x,u)|≤c(1+|u|q−1), with 1<q≤2∗=2NN−2 and N≥3. Some pioneering results considering the above nonlinearity in a bounded domain Ω⊂RN were treated by Brézis [1], Brézis-Nirenberg [2], Bartsch-Willem [3], and Capozzi-Fortunato-Palmieri [4]. A natural extension of the equation defined on the whole space RN, considering the nonlinearity |f(x,u)|≤c(|u|+|u|q−1), with 1<q≤2∗=2NN−2 in N≥3, was studied by Kryszewski and Szulkin [5], and Ding and Ni [6], among others. For this case, the Eq (1.1) needs to be rewritten as −Δu+V(x)u=f(x,u) for x∈RN, where V(x) is used to address the compactness properties. Extensions of Eq (1.1) include the p-Laplacian operator, where Δu is replaced by Δpu:=div(|∇u|p−2∇u). For instance, equations with nonlinearities exhibiting critical Sobolev exponent growth are addressed in [7] for bounded domains in RN, with similar considerations in the whole space discussed in [8,9]. Critical exponential growth is considered in [10] for bounded domains and in [11] for the whole space. Additionally, equations involving the (p,q)-Laplacian operator, which address critical Sobolev exponents and related nonlinear growth, can be found in [12,13]. Another type of equation involves a weight operator div(w(x)∇u), as seen in [14,15], with Hamiltonian systems using this operator discussed in [16,17].
In dimension N=2, Sobolev embedding asserts that H10(Ω)⊂Lq(Ω) for q≥1. Therefore, there is no restriction on (1.1) for the values q>1 in |f(x,u)|≤c(1+|u|q−1). Additionally, some examples show that H10(Ω)⊄L∞(Ω). For this case, the maximal growth of the nonlinearity f is of the exponential type (see Pohozaev [18], Trudinger [19], and Yudovich [20]). More precisely, it has been proven that
eα|u|2∈L1(Ω),for allu∈H10(Ω) and α>0. | (1.2) |
Furthermore, Moser [21] showed that there exists a positive constant C=C(α,Ω) such that
supu∈H10(Ω)‖∇u‖2≤1∫Ωeα|u|2dx{≤C,α≤4π,+∞,α>4π. | (1.3) |
Equation (1.1) with nonlinearities involving exponential growth have been studied by Adimurthi [10], Adimurthi-Yadava [22], and de Figueiredo, Miyagaki, and Ruf [23], among others. Inequality (1.3) is called the Trudinger-Moser inequality. These types of results have been extensively investigated by various authors: in Sobolev spaces over the whole space R2 [24] and in Sobolev spaces for singular versions [25]; in Lorentz-Sobolev spaces within bounded domains [26,27], in Lorentz-Sobolev spaces over the whole space RN [28], and for singular versions in Lorentz-Sobolev spaces [29]; and in weighted Sobolev spaces [14,30]. Additionally, supercritical versions are discussed in [31].
Now, we introduce a supercritical version of the Trudinger-Moser inequality. Let Ω be a smooth domain in R2 and w be a weight defined on Ω. We shall denote by H10,rad(Ω,w) the radial Sobolev weighted space obtained as the closure of all the radially symmetric functions in C∞0(Ω) with respect to the norm
‖u‖Ω,w:=‖u‖H10,rad(Ω,w)=(∫Ωw(x)|∇u|2dx)12. |
In particular, if Ω is the whole space R2, we denote the above Sobolev space as H1rad(R2,w). Trudinger-Moser inequalities for radial Sobolev spaces with logarithmic weights defined on the unit ball B1 in R2 were treated by Calanchi and Ruf [14]. Considering w(x)=(log1/|x|)β and 0≤β<1, the mentioned authors proved that
∫B1eα|u|21−βdx<+∞, for all u∈H10,rad(B1,w) and for all α>0. | (1.4) |
Furthermore, setting α∗β=2[2π(1−β)]11−β, there exists C=C(α,β)>0 such that
supu∈H10,rad(B1,w)‖u‖B1,w≤1∫B1eα|u|21−βdx{≤C,α≤α∗β,+∞,α>α∗β. | (1.5) |
A supercritical version of the Trudinger-Moser inequality defined on H10,rad(B1):=H10,rad(B1,I), where the weight is the identity function on B1, was proved by Ngô and Nguyen [31]. The mentioned authors considered the following assumptions:
(h1)h:[0,1)→R is a radial function, h(0)=0 and h(r)>0 for r∈(0,1).
(h2) There exists some c>0 such that
h(r)≤c−lnr,near to 0. |
(h′3) There exists γ∈(0,1) such that
h(r)≤2γπln(1−r)lnr,near to 1. |
In [31], it was shown that
∫B1exp((α+h(|x|))|u|2)dx<+∞,for all u∈H10,rad(B1) and for all α>0. | (1.6) |
Furthermore, there exists C=C(α,h)>0 such that
supu∈H10,rad(B1)‖u‖B1,I≤1∫B1exp((α+h(|x|))|u|2)dx{≤C,α≤4π,=+∞,α>4π. | (1.7) |
Let us consider
(h″3)There exist γ∈(0,1) such that
h(r)≤γα∗βln(1−r)lnr,near to 1. |
The next proposition combines the above results.
Proposition 1.1 (See [30]). Assume that h satisfies (h1), (h2), and (h″3), and that w is the weight defined by w(x)=(log1/|x|)β for 0<|x|<1, where β∈[0,1). Then,
∫B1exp((α+h(|x|))|u|2/(1−β))dx<+∞,for all u∈H10,rad(B1,w)and for allα>0. |
Furthermore, there exists C=C(α,h)>0 such that
supu∈H10,rad(B1,w)‖u‖B1,w≤1∫B1exp((α+h(|x|))|u|2/(1−β))dx{≤C,α<α∗β,+∞,α>α∗β. |
We point out that conditions (h′3) or (h″3) allow the function h(r)→+∞ as r→1−, and this motivates us to say that a function f possesses supercritical exponential growth if there exists α0>0 such that
lims→+∞f(x,s)exp((α+h(|x|))|s|2/(1−β))={+∞,α<α0,0,α>α0, |
uniformly on x∈R2. The above limit implies that f(x,s)=g(x,s)exp((α0+h(|x|))|s|2/(1−β)), where
lims→+∞g(x,s)exp((α+h(|x|))|s|2/(1−β))=0,uniformly on x∈R2, for all α>0. |
Our first objective in this work is to extend Proposition 1.1, in the sense of obtaining a Trudinger-Moser inequality on the whole space R2. Following [32], we consider the weight
w(x)={[ln(1|x|)]β,0<|x|<1|x|a,|x|≥1, | (1.8) |
where 0≤β<1 and a>2. On h, we assume that
(h3) h(r)>0 for r∈[1,+∞). Moreover, there exist c>0 and ξ<a/(1−β)−2 such that
h(r)≤crξ,for r sufficiently large, |
where the constants a and β are given by (1.8).
In particular, (h3) allows us to consider the case where h(r)→+∞ as r→+∞. Next, we present our adaptation of the Trudinger-Moser inequality which will be utilized in our proof of the existence result.
Theorem 1.2. Suppose that h satisfies (h1)−(h3) and that w is the weight defined by (1.8). Then,
∫R2exp[((α+h(|x|))|u|2/(1−β))−1]dx<+∞,for all u∈H1rad(R2,w)andα>0. | (1.9) |
Moreover, if α<α∗β, there exists C>0 satisfying
sup‖u‖R2,w≤1∫R2exp[((α+h(|x|))|u|2/(1−β))−1]dx≤C. | (1.10) |
If α>α∗β, it holds that
sup‖u‖R2,w≤1∫R2exp[((α+h(|x|))|u|2/(1−β))−1]dx=+∞. | (1.11) |
In the subsequent section, we will outline the proof of Theorem 1.2. The aim of this study is to find a nontrivial weak solution to the following stationary Schrödinger equation:
−div(w(x)∇u) = f(x,u),x∈R2. | (1.12) |
Here, w represents the weight defined on (1.8) which allows that f possesses the maximal growth established in Theorem 1.2. More precisely, we assume the following hypotheses:
(H1) f:R2×R→R is continuous and possesses radial symmetry in its first variable, namely f(x,s)=f(y,s) whenever |x|=|y|. Additionally, f(x,s)=0 for all x∈R2 and s≤0. (H_2) The following limit holds:
lims→0f(x,s)s=0,uniformly on x∈R2. |
(H3) There exists a constant μ>2 such that
0<μF(x,s):=μ∫s0f(x,t)≤sf(x,s),for allx∈R2 and for all s>0. |
(H4) There exists a constant α0>0 such that
lims→+∞f(x,s)exp((α+h(|x|))|s|2/(1−β))={+∞,α<α0,0,α>α0, |
uniformly on x∈R2, where h satisfies (h1)−(h3).
(H5) There exist constants p>2 and Cp>0 such that
f(x,s)≥Cpsp−1,for alls≥0 and for all x∈R2, |
where
Cp>Spp(α0α∗β)(1−β)(p−2)/2(12−1p)(p−2)/2(12−1μ)(p−2)/2 |
and
Sp:=inf0≠u∈H1rad(R2,w) (∫R2w(x)|∇u|2dx)1/2(∫R2|u|pdx)1/p. |
In the forthcoming text, we shall denote the Hilbert space E:=H1rad(R2,w) equipped with the inner product defined as
⟨u,v⟩E=∫R2w(x)∇u∇vdx,for all u, v∈E, |
which induces the norm
‖u‖:=‖u‖E=(∫R2w(x)|∇u|2dx)1/2. |
Additionally, E∗ denotes the dual space of E equipped with its standard norm. We define u∈E to be a weak solution of (1.12) if
∫R2w(x)∇u∇ϕdx=∫R2f(x,u)ϕdx,for all ϕ∈E. | (1.13) |
To find weak solutions of our problem (1.12), we will employ variational methods. For this purpose, let us consider the functional J:E→R defined as:
J(u)=12∫R2w(x)|∇u|2dx−∫R2F(x,u)dx. |
Moreover, based on established arguments (see [33]), it follows that J belongs to C1(E,R) and
J′(u)ϕ=∫R2w(x)∇u∇ϕdx−∫R2f(x,u)ϕdx,for all u, ϕ∈E. |
The main result of this article is presented as follows:
Theorem 1.3. Suppose that f satisfies (H1)−(H5) and h satisfies (h1)−(h3). Then, problem (1.12) possesses a nontrivial weak solution.
We point out that equations or systems with nonlinearities involving the classical Trudinger-Moser inequalities imply that the growth of f is of type exp(|s|2) as s tends to infinity (see [23,24,25,34,35,36], among others). Equations considering Trudinger-Moser inequalities on Lorentz-Sobolev spaces allow us to consider f of the type exp(|s|p) with p>1 as s tends to infinity (see [1,37,38,39]). Equations with logarithmic weight defined on the unit ball in R2 may have nonlinearities of the form exp(|s|2/(1−β)) for 0≤β<1 (see [14,16]), exp((α+h(|x|))|s|2) (see [31,40]), or exp(α+h(|x|)|s|2/(1−β)) (see [16,30,41]). Furthermore, our existence theorem complements the work in [30] since we consider the whole space R2. Our main contribution is given by the assumption (H4), which allows us to consider the behavior of f(x,s) as exp(α+h(|x|)|s|2/(1−β)) for some α>0, as s tends to infinity, where the radial function h may be unbounded at infinity. Finally, note that the class of functions which satisfy conditions (H1)−(H5) is not empty, for instance, consider the following function f:R2×R→R defined by
f(x,s)={Asp−1+p(1+|x|ξ)sp−1exp((1+|x|ξ)sp),s≥00,s<0, |
for some positive constants a>2, 0<β<1, 0<ξ<a/(1−β)−2, p=2/(1−β), and A sufficiently large.
We begin this section by presenting a version of the Strauss result [42], which follows from [14,32] and plays an important role to prove our version of the supercritical Trudinger-Moser inequality.
Lemma 2.1 (See [14,32]). Let u be a function in E. Then,
|u(x)|≤{(−ln|x|)1−β2√2π(1−β)‖u‖,if0<|x|<1,1√2πa|x|a/2‖u‖,if|x|≥1. |
The next lemma is related to the embeddings of the space E into Lebesgue spaces.
Lemma 2.2 (See [32]). The space E is continuously and compactly embedded in Lp(R2) for p>4/a.
Proof. Let us consider u∈E with ‖u‖≤1 and α<α∗β. By Lemma 2.1, we have
∫R2∖B1[exp((α+h(|x|))|u|2/(1−β))−1]dx=+∞∑k=11k!∫R2∖B1[α+h(|x|)]k|u|2k/(1−β)dx≤+∞∑k=11k!∫R2∖B1[α+h(|x|)]k|x|ak1−βdx≤+∞∑k=12kαkk!∫R2∖B11|x|αk1−βdx++∞∑k=12kk!∫R2∖B1hk(|x|)|x|ak1−βdx. | (2.1) |
Since a>2(1−β), there exists C1>0 such that
∫R2∖B11|x|ak1−βdx≤∫R2∖B11|x|a1−βdx=C1,for allk≥1. | (2.2) |
From (h3), there exist c1>0 and R0>1 such that
h(|x|)≤c1|x|ξ,for all|x|≥R0. |
Since a>(2+ξ)(1−β), we can get C2>0 such that
∫R2∖BR0hk(|x|)|x|ak1−βdx≤∫R2∖BR0ck1|x|(a1−β−ξ)kdx≤ck1∫R2∖BR01|x|a1−β−ξdx=C2,for all k≥1. | (2.3) |
Using the continuity of h, we can find c2>0 such that h(|x|)≤c2 for 1≤|x|≤R0. Then, we can get C3>0 such that
∫BR0∖B1hk(|x|)|x|αk1−βdx≤∫BR0∖B1ck2|x|αk1−βdx≤ck2∫BR0∖B11|x|a1−βdx=C3,for all k≥1. | (2.4) |
Replacing (2.2)–(2.4) in (2.1), one has
∫R2∖B1[exp((α+h(|x|))|u|2/(1−β))−1]dx≤C1e2α+(C2+C3)e2. | (2.5) |
On the other hand, consider v(x)=u(x)−u(e) for |x|<1 and v(x)=0 for |x|≥1, where e is fixed in R2 such that |e|=1. Then, v∈H10,rad(B1,w) for each u∈E. Moreover, using the fact that ‖u‖≤1, we have that ‖v‖H10,rad(B1,w)≤1. Taking ϵ>0 sufficiently small satisfying α(1+ϵ)<α∗β, we can find Cϵ>0 such that
|u(x)|2/(1−β)≤(1+ϵ)|v(x)|2/(1−β)+Cϵ|u(e)|2/(1−β). |
Then,
∫B1[exp((α+h(|x|))|u|2/(1−β))−1]dx≤∫B1exp((α+h(|x|))|u|2/(1−β))dx≤∫B1exp((α+h(|x|))((1+ϵ)|v(x)|2/(1−β)+Cϵ|u(e)|2/(1−β)))dx≤sup|x|≤1exp((α+h(|x|))Cϵ|u(e)|2/(1−β))∫B1exp(((1+ϵ)α+(1+ϵ)h(|x|))|v(x)|2/(1−β))dx. |
Using the continuity of h and Lemma 2.1, there exists C4>0 such that
sup|x|≤1exp((α+h(|x|))Cϵ|u(e)|2/(1−β))≤C4. |
Therefore,
∫B1[exp((α+h(|x|))|u|2/(1−β))−1]dx≤C4∫B1exp(α∗β+(1+ϵ)h(|x|))|v(x)|2/(1−β))dx. | (2.6) |
Note that the function hϵ(r)=(1+ϵ)h(r) defined on r∈[0,1) satisfies the conditions of Proposition 1.1 and using the fact that v∈H10,rad(B1,w), we can find C5>0 such that
∫B1[exp((α+h(|x|))|u|2/(1−β))−1]dx≤C4supv∈H10,rad(B1,w)‖v‖B1,w≤1∫B1exp((α∗β+hϵ(|x|))|v(x)|2/(1−β))dx≤C5. | (2.7) |
Using the above inequality and (2.5), we obtain C>0, independent of the election of u∈E, satisfying
∫R2[exp((α+h(|x|))|u|2/(1−β))−1]dx≤C. |
Therefore, the inequalities (1.9) and (1.10) follow. Moreover, we consider the sequence (ψk)⊂E defined as
ψk(x)=(1α∗β)(1−β)/2{k21−βln(1|x|2)1−β,0≤|x|≤e−k/2,k1−β2,e−k/2≤|x|≤1,0,|x|>1. |
Note that ‖ψk‖=1 for each k≥1, and for α>α∗β, it follows that
∫R2exp((α+h(|x|))|ψk|2/(1−β))dx≥∫B1exp(α|ψk|2/(1−β))dx≥2π∫1e−k/2exp(αα∗βk)rdr. |
Consequently,
∫R2exp((α+h(|x|))|ψk|2/(1−β))dx≥πek(αα∗β−1)(ek−1)→+∞,as k→∞, |
and the proof of the last assertion follows.
Remark 2.3.
(a) An example of a function h that satisfies conditions (h1)−(h3) is given by h(r)=rξ for some 0<ξ<a/(1−β)−2 where a and β are given in (1.8).
(b) As it was observed in [31], the assertions of Theorem 1.2 are no longer valid when considering the space of nonradial functions H1(R2,w).
We now outline several results necessary for utilizing variational methods.
Lemma 3.1. Assume that (H1),(H2), and (H4) hold. Then, there exist σ,ρ>0, such that
J(u)≥σ,for allu∈E with ‖u‖=ρ. |
Proof. Given q>4/a and ϵ>0, from (H1), (H2), and (H4), there exists c>0 such that
|F(x,s)|≤ϵ|s|2+c|s|qexp[((2α0+h(|x|))|s|2/(1−β))−1],for all(x,s)∈R2×R. |
By the Cauchy-Schwarz inequality and the inequality (ew−1)2≤e2w−1 for all w≥0, we obtain
∫R2F(x,u)dx≤ϵ‖u‖22+c‖u‖q2q(∫R2[exp((4α0+2h(|x|))|u|2/(1−β))−1]dx)1/2. | (3.1) |
Using Lemma 2.1, for u in E with ‖u‖≤1, one has
|u(x)|≤1√2πa|x|a/2,for all |x|≥1. |
By (h3), there exist R0>1 and c1>0 such that
h(|x|)≤c1|x|ξ,for all |x|≥R0. |
Therefore, we can get C1>0 such that
(4α0+2h(|x|))|u|2/(1−β)≤4α0(2πa)1/(1−β)|x|a1−β+2c1(2πa)1/(1−β)|x|a1−β−ξ≤C1|x|η,for all|x|≥R0, |
where η=min{a/(1−β)−ξ,a/(1−β)}>2, which implies the existence of C2>0 such that
∫R2∖BR0[exp((4α0+2h(|x|))|u|2/(1−β))−1]dx≤2π∫+∞R0r(exp(C1r−η)−1)dr=C2. | (3.2) |
Let h0=max0≤r≤R0h(r). Using Theorem 1.2, we can get C3>0 such that
∫BR0[exp((4α0+2h(|x|))|u|2/(1−β))−1]dr≤∫BR0[exp((4α0+2h0)|u|2/(1−β))−1]dx≤∫BR0[exp((4α0+2h0)‖u‖2/(1−β)(|u|‖u‖)2/(1−β))−1]dx≤C3, | (3.3) |
provided that ‖u‖≤ρ1 for some ρ1>0 such that (4α0+2h0)ρ2/(1−β)1<α∗β. From (3.1)–(3.3), and Lemma 2.2, there exists C>0 such that
∫R2F(x,u)dx≤ϵC‖u‖2+C‖u‖q, |
provided that ‖u‖≤ρ0 for some 0<ρ0≤min{1,ρ1}. Then,
J(u)≥12‖u‖2−∫R2F(x,u)dx≥(12−ϵC)‖u‖2−C‖u‖q. |
Note that we can assume that ϵ>0 satisfies (1/2−ϵC)≥1/4. Consequently, it is possible to choose ρ>0 and σ>0 with 0<ρ≤ρ0 such that J(u)≥σ>0, for all u∈E with ‖u‖=ρ.
The next lemma follows the same lines as [30, Lemma 3.3].
Lemma 3.2. Suppose that (H1)−(H2) hold. If e0≠0 in E, then there exists t>0 large enough such that e=te0 satisfies
J(e)<0and‖e‖>ρ, |
where ρ>0 is given by Lemma 3.1.
In this section, we show some results related to the Palais-Smale sequences. Let us recall that we say that (un)⊂E is a (PS)c sequence for the functional J if
J(un)→cand‖J′(un)‖E∗→0. | (4.1) |
Moreover, if (un) satisfying (4.1) possesses a convergent subsequence, we say that (un) satisfies the Palais-Smale condition at the level c.
The following lemma asserts that each Palais-Smale sequence associated with J is bounded.
Lemma 4.1. Assume (H1)−(H4). Then any Palais-Smale sequence for the functional J is bounded in E.
Proof. Using (H3), we obtain
J(un)−1μJ′(un)un=(12−1μ)‖un‖2−1μ∫R2(μF(x,un)−f(x,un)un)dx≥(12−1μ)‖un‖2. |
Using (4.1), we have
J(un)=c+on(1)and‖J′(un)‖E∗=on(1). |
Therefore, for n sufficiently large, we obtain
(12−1μ)‖un‖2≤c+on(1)+on(1)‖un‖. |
Consequently, the sequence (un) is bounded in E.
Lemma 4.2. Assume that (H1)−(H4) are satisfied. Then, J satisfies the Palais-Smale condition at the level c, where
c<(12−1μ)(α∗βα0)1−β. |
Proof. Take a Palais-Smale sequence (un)⊂E for J at the level c of J. Using Lemma 4.1, we can find u∈E, up to a subsequence, such that un⇀u weakly in E. Setting vn:=un−u, we have that vn⇀0 weakly in E. Then,
∫R2w(x)∇un∇vndx−∫R2f(x,un)vndx=J′(un)vn=on(1) |
and
∫R2w(x)∇un∇vndx=‖un‖2−‖u‖2+on(1). |
Therefore,
‖un‖2−‖u‖2=∫R2f(x,un)vndx+on(1). | (4.2) |
It remains to show that, up to a subsequence, the integral in (4.2) tends to zero as n→+∞. From Lemma 4.1 and the assumption on c, we obtain
(12−1μ)‖un‖2=c+on(1)<(12−1μ)(α∗βα0)1−β+on(1). |
Thus, without loss of generality, we can find δ>0 such that
‖un‖2/(1−β)≤α∗βα0−δ,for all n∈N. | (4.3) |
Now, take m>1 and ϵ>0 such that
m(α0+2ϵ)(α∗βα0−δ)<α∗β. | (4.4) |
From assumptions on f, there exists Cϵ>0 such that
|f(x,s)|≤ϵ|s|+Cϵ[exp((α0+ϵ+h(|x|))|s|2/(1−β))−1],for all(x,s)∈R2×R. |
By the Hölder inequality with 1/m+1/m′=1 and the identity (er−1)m≤erm−1 for all r≥0, we obtain
∫R2|f(x,un)vn|dx≤ϵ‖un‖2‖vn‖2+Cϵ‖vn‖m′(∫R2[exp(m(α0+ϵ+h(|x|))|un|2/(1−β))−1]dx)1/m. | (4.5) |
Using the continuity of h and h(0)=0, there exists 0<r1<1 such that
h(|x|)<ϵ,for all|x|≤r1. |
Thus,
∫Br1[exp(m(α0+ϵ+h(|x|))|un|2/(1−β))−1]dx≤∫Br1[exp(m(α0+2ϵ)‖un‖2/(1−β)(|un|‖un‖)2/(1−β))−1]dx. |
Using (4.3), (4.4), and Theorem 1.2, we can get C1>0 such that
∫Br1[exp(m(α0+ϵ+h(|x|))|un|2/(1−β))−1]dx≤∫Br1[exp(α∗β(|un|‖un‖)2/(1−β))−1]dx≤C1. | (4.6) |
By (h3), there exist c>0 and r2>1 such that
h(r)≤c|x|ξ,for all|x|≥r2. |
Using the above inequality, the boundedness of the sequence (‖un‖), and Lemma 2.1, there exists C2>0 such that
m(α0+ϵ+h(|x|))|un(x)|2/(1−β)≤C2|x|η,for alln≥1and|x|≥r2, |
where η=min{a/(1−β)−ξ,a/(1−β)}>2, which implies the existence of C3>0 such that
∫R2∖Br2[exp(m(α0+ϵ+h(|x|))|un|2/(1−β))−1]dx≤2π∫+∞r2[exp(C2|x|−η)−1]dr=C3. | (4.7) |
Since the sequence (un) is bounded in E, by Lemma 2.1, one has
|un(x)|≤M0,for allr1≤|x|≤r2and for alln≥1. |
Additionally, since h is continuous, there exists C3>0 such that
∫Br2∖Br1[exp(m(α0+ϵ+h(|x|))|un|2/(1−β))−1]dx≤C3. | (4.8) |
Using (4.6)–(4.8), the integral on the right-hand side of (4.5) is bounded. Moreover, by the compact embeddings E↪L2(R2) and E↪Lm′(R2), and the weakly convergence vn⇀0 in E, up to a subsequence, we obtain
∫R2|f(x,un)vn|dx≤ϵ‖u‖2‖vn‖2+C‖vn‖m′→0,asn→+∞, |
and the lemma follows.
First, we will show that Sp is attained in a function in E. Consider a sequence (uk)⊂E such that
∫R2|uk|pdx=1and(∫R2w(x)|∇uk|2dx)1/2→Sp. |
Therefore, (uk) is bounded in E. Thus, we can assume that there exists some up∈E such that uk⇀up weakly in E, uk→up strongly in Lp(R2), and uk(x)→up(x) almost everywhere in R2. Hence, ‖up‖p=1 and ‖up‖≤lim infk→+∞‖uk‖=Sp. Noticing that Sp≤‖up‖, and taking the absolute value of the functions, we can guarantee that up≥0. Thus there exists up∈E such that u(x)≥0 in R2 with ‖up‖p=1 satisfying
Sp=inf0≠u∈H1rad(R2,w)(∫R2w(x)|∇u|2dx)1/2(∫R2|u|pdx)1/p=‖up‖. |
This will be the element e0 considered in Lemma 3.2. From Lemmas 3.1 and 3.2, based on the well-known pass mountain theorem by Ambrosetti-Rabinowitz [43,44]), we obtain a Palais-Smale (un)⊂E at the level d≥σ, where σ is given by Lemma 3.1, and d>0 is given by
d=infγ∈Γmaxt∈[0,1]J(γ(t)), |
and
Γ={γ∈C([0,1],E):γ(0)=0,γ(1)=e}. |
From (H5), we get
J(tup)=t22‖up‖2−∫R2F(x,tup)dx≤t22‖up‖2−Cptpp∫R2|up|pdx. |
By the assumption on Cp, we obtain
supt≥0J(tup)≤maxt≥0{t2S2p2−Cptpp}=(p−2)S2p/(p−2)p2pC2/(p−2)p<(12−1μ)(α∗βα0)1−β. | (5.1) |
Note that e=t0up with t0>0 is given by Lemma 3.2. Consider γ0∈Γ defined by γ0(t)=tt0up. By (5.1), we get
d=infγ∈Γmaxt∈[0,1]J(γ(t))≤maxt∈[0,1]J(γ0(t))≤maxt∈[0,1]J(tt0up)≤maxt≥0J(tup)<(12−1μ)(α∗βα0)1−β. |
Using Lemma 4.2, the sequence (un), up to a sequence, is convergent, that is, we can get u∈E such that un→u in E. By the continuity of J and J′, we have that J(u)=d and J′(u)=0. Therefore, u is a solution of the problem (1.12). Moreover, using the fact that J(u)=d≥σ, we conclude that u is nontrivial.
In this paper, we presented a new type of Trudinger-Moser inequality defined on a radial weighted Sobolev space. Additionally, as an application of the above result, by applying the mountain pass theorem, we found nontrivial weak solutions for a nonlinear equation. Our main contribution is to extend previous results by establishing equations defined on R2, involving a nonlinear equation with supercritical exponential growth.
The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.
This research was supported by the Universidad Nacional Mayor de San Marcos RR N° 05557-R-22 and project number B22140231. The author would like to thank the anonymous reviewers for all remarks that corrected and improved the previous version of the paper.
The authors declare there are no conflicts of interest.
[1] | H. Brézis, Elliptic equations with limiting Sobolev exponents, Comm. Pure Appl. Math., 39 (1986), 517–539. |
[2] |
H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983), 437–477. https://doi.org/10.1002/cpa.3160360405 doi: 10.1002/cpa.3160360405
![]() |
[3] | T. Bartsh, M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc., 123 (1995), 3555–3561. |
[4] | A. Capozzi, D. Fortunato, G. Palmieri, An existence result for nonlinear elliptic problems involving critical Sobolev exponent, Ann. Henri Poincaré, 2 (1985), 463–470. |
[5] |
W. Kryszewski, A. Szulkin, Generalized linking theorem with an application to a semilinear Schrödinger equation, Adv. Differ. Equations, 3 (1998), 441–472. https://doi.org/10.57262/ade/1366399849 doi: 10.57262/ade/1366399849
![]() |
[6] |
W. Y. Ding, W. M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Rational Mech. Anal., 91 (1986), 283–308. https://doi.org/10.1007/BF00282336. doi: 10.1007/BF00282336
![]() |
[7] |
J. P. G. Azorero, I. P. Alonso, Existence and non-uniqueness for the p-Laplacian, Comm. Partial Differ. Equations, 12 (1987), 1389–1430. https://doi.org/10.1080/03605308708820534 doi: 10.1080/03605308708820534
![]() |
[8] |
J. V. Gonçalves, C. O. Alves, Existence of positive solutions for m-Laplacian equations in RN involving critical Sobolev exponents, Nonlinear Anal., 32 (1998), 53–70. https://doi.org/10.1016/S0362-546X(97)00452-5 doi: 10.1016/S0362-546X(97)00452-5
![]() |
[9] |
E. A. B de Silva, S. H. M. Soares, Quasilinear Dirichlet problems in RN with critical growth, Nonlinear Anal., 43 (2001), 1–20. https://doi.org/10.1016/S0362-546X(99)00128-5 doi: 10.1016/S0362-546X(99)00128-5
![]() |
[10] | A. Adimurthi, Existence of positive solutions of the semilinear Dirichlet Problem with critical growth for the N-Laplacian, Ann. Sc. Norm. Sup. Pisa, 17 (1990), 393–413. |
[11] |
J. M. do Ó, N-Laplacian equations in RN with critical growth, Abstr. Appl. Anal., 2 (1997), 301–315. https://doi.org/10.1155/S1085337597000419 doi: 10.1155/S1085337597000419
![]() |
[12] |
L. Baldelli, R. Filippucci, Existence of solutions for critical (p,q)-Laplacian equations in RN, Commun. Contemp. Math., 9 (2022), 2150109. https://doi.org/10.1142/S0219199721501091 doi: 10.1142/S0219199721501091
![]() |
[13] |
L. Gongbao, Z. Guo, Multiple solutions for the p & q-Laplacian problem with critical exponent, Acta Math. Sci. Ser. B Engl. Ed., 29 (2009), 903–918. https://doi.org/10.1016/S0252-9602(09)60077-1 doi: 10.1016/S0252-9602(09)60077-1
![]() |
[14] |
M. Calanchi, B. Ruf, On a Trudinger-Moser type inequality with logarithmic weights, J. Differ. Equations, 258 (2015), 1967–1989. https://doi.org/10.1016/j.jde.2014.11.019 doi: 10.1016/j.jde.2014.11.019
![]() |
[15] |
Y. R. S. Leuyacc, Standing waves for quasilinear Schrodinger equations involving double exponential growth, AIMS Math., 8 (2023), 1682?1695. https://doi.org/10.3934/math.2023086 doi: 10.3934/math.2023086
![]() |
[16] |
Y. R. S. Leuyacc, Hamiltonian elliptic system involving nonlinearities with supercritical exponential growth, AIMS Math., 8 (2023), 19121–19141. https://doi.org/10.3934/math.2023976 doi: 10.3934/math.2023976
![]() |
[17] |
Y. R. S. Leuyacc, Singular Hamiltonian elliptic systems involving double exponential growth in dimension two, Partial Differ. Equations Appl. Math., 10 (2024), 100681. https://doi.org/10.1016/j.padiff.2024.100681 doi: 10.1016/j.padiff.2024.100681
![]() |
[18] | S. Pohožaev, The Sobolev embedding in the special case pl=n, in Proceedings of the Technical Science Conference on Advance Science Research Mathematics Sections, (1965), 158–170. |
[19] | N. Trudinger, On embedding into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473–483. |
[20] | V. Yudovich, Some estimates connected with integral operators and with solutions of elliptic equations, Dokl. Akad. Nauk SSSR, 138 (1961), 805–808. |
[21] | J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970), 1077–1092. |
[22] | S. L. Yadava, Multiplicity results for semilinear elliptic equations in a bounded domain of R2 involving critical exponent, Ann. Sc. Norm. Super. Pisa-Cl. Sci., 27 (1990), 481–504. |
[23] |
D. G. de Figueiredo, O. H. Miyagaki, B. Ruf, Elliptic equations in R2 with nonlinearities in the critical growth range, Calc. Var. Partial Differ. Equations, 3 (1995), 139–153. https://doi.org/10.1007/BF01205003 doi: 10.1007/BF01205003
![]() |
[24] |
D. M. Cao, Nontrivial solution of semilinear elliptic equation with critical exponent in R2, Comm. Partial Differ. Equations, 17 (1992), 407–435. https://doi.org/10.1080/03605309208820848 doi: 10.1080/03605309208820848
![]() |
[25] |
M. de Souza, J. M. do Ó, On a class of singular Trudinger-Moser type inequalities and its applications, Math. Nachr., 284 (2011), 1754–1776. https://doi.org/10.1016/j.aml.2012.05.007 doi: 10.1016/j.aml.2012.05.007
![]() |
[26] |
A. Alvino, V. Ferone, G. Trombetti, Moser-Type Inequalities in Lorentz Spaces, Potential Anal., 5 (1996), 273–299. https://doi.org/10.1007/BF00282364 doi: 10.1007/BF00282364
![]() |
[27] |
H. Brézis, S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Comm. Partial Differ. Equations, 5 (1980), 773–789. https://doi.org/10.1080/03605308008820154 doi: 10.1080/03605308008820154
![]() |
[28] |
D. Cassani, C. Tarsi, A Moser-type inequalities in Lorentz-Sobolev spaces for unbounded domains in RN, Asymptotic Anal., 64 (2009), 29–51. https://doi.org/10.3233/ASY-2009-0934 doi: 10.3233/ASY-2009-0934
![]() |
[29] |
G. Lu, H. Tang, Sharp singular Trudinger-Moser inequalities in Lorentz-Sobolev spaces, Adv. Nonlinear Stud., 16 (2016), 581–601. https://doi.org/10.1515/ans-2015-5046 doi: 10.1515/ans-2015-5046
![]() |
[30] |
Y. R. S. Leuyacc, Supercritical Trudinger-Moser inequalities with logarithmic weights in dimension two, AIMS Math., 8 (2023), 18354–18372. https://doi.org/10.3934/math.2023933 doi: 10.3934/math.2023933
![]() |
[31] |
Q. A. Ngô, V. H. Nguyen, Supercritical Moser-Trudinger inequalities and related elliptic problems, Calc. Var. Partial Differ. Equations, 59 (2020), 69. https://doi.org/10.1007/s00526-020-1705-y doi: 10.1007/s00526-020-1705-y
![]() |
[32] |
S. Aouaoui, A new Trudinger-Moser type inequality and an application to some elliptic equation with doubly exponential nonlinearity in the whole space R2, Arch. Math., 114 (2020), 199–214. https://doi.org/10.1007/s00013-019-01386-7 doi: 10.1007/s00013-019-01386-7
![]() |
[33] | O. Kavian, Introduction à la théorie des Points Critiques et Applications aux Problèmes Elliptiques, Springer-Verlag, Paris, 1993. |
[34] |
F. S. B. Albuquerque, C. O. Alves, E. S. Medeiros, Nonlinear Schrödinger equation with unbounded or decaying radial potentials involving exponential critical growth in R2, J. Math. Anal. Appl., 409 (2014), 1021–1031. https://doi.org/10.1016/j.jmaa.2013.07.005. doi: 10.1016/j.jmaa.2013.07.005
![]() |
[35] |
B. Ruf, A sharp Trudinger-Moser type inequality for unbounded domains in R2, J. Funct. Anal., 219 (2005), 340–367. https://doi.org/10.1016/j.jfa.2004.06.013 doi: 10.1016/j.jfa.2004.06.013
![]() |
[36] | B. Ruf, F. Sani, Ground states for elliptic equations with exponential critical growth, in Geometric Properties for Parabolic and Elliptic PDE's (eds. R. Magnanini, S. Sakaguchi, A. Alvino), Springer INdAM Series, Springer, Milano, (2013), 321–349. |
[37] |
D. Cassani, C. Tarsi, Existence of solitary waves for supercritical Schrödinger systems in dimension two, Calc. Var. Partial Differ. Equations, 54 (2015), 1673–1704. https://doi.org/10.1007/s00526-015-0840-3 doi: 10.1007/s00526-015-0840-3
![]() |
[38] |
Y. R. S. Leuyacc, S. H. M. Soares, On a Hamiltonian system with critical exponential growth, Milan J. Math., 87 (2019), 105–140. https://doi.org/10.1007/s00032-019-00294-3 doi: 10.1007/s00032-019-00294-3
![]() |
[39] |
S. H. M. Soares, Y. R. S. Leuyacc, Singular Hamiltonian elliptic systems with critical exponential growth in dimension two, Math. Nachr., 292 (2019), 137–158. https://doi.org/10.1007/s00032-019-00294-3 doi: 10.1007/s00032-019-00294-3
![]() |
[40] | H. Zhao, Y. Guo, Y. Shen, Singular type trudinger-moser inequalities with logarithmic weights and the existence of extremals, Mediterr. J. Math., 21 (2024), 50. https://doi.org/0.1007/s00009-023-02582-0 |
[41] |
Y. R. S. Leuyacc, A class of Schrödinger elliptic equations involving supercritical exponential growth, Boundary Value Probl., 39 (2023), 17. https://doi.org/10.1186/s13661-023-01725-2. doi: 10.1186/s13661-023-01725-2
![]() |
[42] |
W. A. Strauss, Existence of solitary waves in higher dimension, Commun. Math. Phys., 55 (1977), 149–162. https://doi.org/10.1007/BF01626517 doi: 10.1007/BF01626517
![]() |
[43] | P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, American Mathematical Society, Providence, RI, 1986. |
[44] | M. Willem, Minimax Theorems, Boston: Birkhäuser, 1996. |