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Research article

Elliptic equations in R2 involving supercritical exponential growth

  • Received: 16 July 2024 Revised: 09 September 2024 Accepted: 10 September 2024 Published: 18 September 2024
  • In this work, we investigated the existence of nontrivial weak solutions for the equation

    div(w(x)u) = f(x,u),xR2,

    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/(1β)), 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

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  • In this work, we investigated the existence of nontrivial weak solutions for the equation

    div(w(x)u) = f(x,u),xR2,

    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/(1β)), 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 ΩRN 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|q1), with 1<q2=2NN2 and N3. Some pioneering results considering the above nonlinearity in a bounded domain ΩRN 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|q1), with 1<q2=2NN2 in N3, 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 xRN, 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|p2u). 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 q1. Therefore, there is no restriction on (1.1) for the values q>1 in |f(x,u)|c(1+|u|q1). 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|2L1(Ω),for alluH10(Ω)  and  α>0. (1.2)

    Furthermore, Moser [21] showed that there exists a positive constant C=C(α,Ω) such that

    supuH10(Ω)u21Ωeα|u|2dx{C,α4π,+,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 C0(Ω) with respect to the norm

    uΩ,w:=uH10,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 0β<1, the mentioned authors proved that

    B1eα|u|21βdx<+,  for all  uH10,rad(B1,w)  and for all  α>0. (1.4)

    Furthermore, setting αβ=2[2π(1β)]11β, there exists C=C(α,β)>0 such that

    supuH10,rad(B1,w)uB1,w1B1eα|u|21βdx{C,ααβ,+,α>αβ. (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)clnr,near to 0.

    (h3) There exists γ(0,1) such that

    h(r)2γπln(1r)lnr,near to 1.

    In [31], it was shown that

    B1exp((α+h(|x|))|u|2)dx<+,for all  uH10,rad(B1) and for all α>0. (1.6)

    Furthermore, there exists C=C(α,h)>0 such that

    supuH10,rad(B1)uB1,I1B1exp((α+h(|x|))|u|2)dx{C,α4π,=+,α>4π. (1.7)

    Let us consider

    (h3)There exist γ(0,1) such that

    h(r)γαβln(1r)lnr,near to 1.

    The next proposition combines the above results.

    Proposition 1.1 (See [30]). Assume that h satisfies (h1), (h2), and (h3), 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/(1β))dx<+,for all uH10,rad(B1,w)and for allα>0.

    Furthermore, there exists C=C(α,h)>0 such that

    supuH10,rad(B1,w)uB1,w1B1exp((α+h(|x|))|u|2/(1β))dx{C,α<αβ,+,α>αβ.

    We point out that conditions (h3) or (h3) allow the function h(r)+ as r1, 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/(1β))={+,α<α0,0,α>α0,

    uniformly on xR2. The above limit implies that f(x,s)=g(x,s)exp((α0+h(|x|))|s|2/(1β)), where

    lims+g(x,s)exp((α+h(|x|))|s|2/(1β))=0,uniformly on  xR2, 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 0β<1 and a>2. On h, we assume that

    (h3) h(r)>0 for r[1,+). Moreover, there exist c>0 and ξ<a/(1β)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/(1β))1]dx<+,for all uH1rad(R2,w)andα>0. (1.9)

    Moreover, if α<αβ, there exists C>0 satisfying

    supuR2,w1R2exp[((α+h(|x|))|u|2/(1β))1]dxC. (1.10)

    If α>αβ, it holds that

    supuR2,w1R2exp[((α+h(|x|))|u|2/(1β))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),xR2. (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×RR 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 xR2 and s0. (H_2) The following limit holds:

    lims0f(x,s)s=0,uniformly on xR2.

    (H3) There exists a constant μ>2 such that

    0<μF(x,s):=μs0f(x,t)sf(x,s),for allxR2  and for all  s>0.

    (H4) There exists a constant α0>0 such that

    lims+f(x,s)exp((α+h(|x|))|s|2/(1β))={+,α<α0,0,α>α0,

    uniformly on xR2, where h satisfies (h1)(h3).

    (H5) There exist constants p>2 and Cp>0 such that

    f(x,s)Cpsp1,for alls0  and for all  xR2,

    where

    Cp>Spp(α0αβ)(1β)(p2)/2(121p)(p2)/2(121μ)(p2)/2

    and

    Sp:=inf0uH1rad(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,vE=R2w(x)uvdx,for all  u, vE,

    which induces the norm

    u:=uE=(R2w(x)|u|2dx)1/2.

    Additionally, E denotes the dual space of E equipped with its standard norm. We define uE to be a weak solution of (1.12) if

    R2w(x)uϕdx=R2f(x,u)ϕdx,for all ϕE. (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:ER defined as:

    J(u)=12R2w(x)|u|2dxR2F(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)uϕdxR2f(x,u)ϕdx,for all  u, ϕE.

    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/(1β)) for 0β<1 (see [14,16]), exp((α+h(|x|))|s|2) (see [31,40]), or exp(α+h(|x|)|s|2/(1β)) (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/(1β)) 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×RR defined by

    f(x,s)={Asp1+p(1+|x|ξ)sp1exp((1+|x|ξ)sp),s00,s<0,

    for some positive constants a>2, 0<β<1, 0<ξ<a/(1β)2, p=2/(1β), 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|)1β22π(1β)u,if0<|x|<1,12πa|x|a/2u,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 uE with u1 and α<αβ. By Lemma 2.1, we have

    R2B1[exp((α+h(|x|))|u|2/(1β))1]dx=+k=11k!R2B1[α+h(|x|)]k|u|2k/(1β)dx+k=11k!R2B1[α+h(|x|)]k|x|ak1βdx+k=12kαkk!R2B11|x|αk1βdx++k=12kk!R2B1hk(|x|)|x|ak1βdx. (2.1)

    Since a>2(1β), there exists C1>0 such that

    R2B11|x|ak1βdxR2B11|x|a1βdx=C1,for allk1. (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)(1β), we can get C2>0 such that

    R2BR0hk(|x|)|x|ak1βdxR2BR0ck1|x|(a1βξ)kdxck1R2BR01|x|a1βξdx=C2,for all  k1. (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

    BR0B1hk(|x|)|x|αk1βdxBR0B1ck2|x|αk1βdxck2BR0B11|x|a1βdx=C3,for all  k1. (2.4)

    Replacing (2.2)–(2.4) in (2.1), one has

    R2B1[exp((α+h(|x|))|u|2/(1β))1]dxC1e2α+(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, vH10,rad(B1,w) for each uE. Moreover, using the fact that u1, we have that vH10,rad(B1,w)1. Taking ϵ>0 sufficiently small satisfying α(1+ϵ)<αβ, we can find Cϵ>0 such that

    |u(x)|2/(1β)(1+ϵ)|v(x)|2/(1β)+Cϵ|u(e)|2/(1β).

    Then,

    B1[exp((α+h(|x|))|u|2/(1β))1]dxB1exp((α+h(|x|))|u|2/(1β))dxB1exp((α+h(|x|))((1+ϵ)|v(x)|2/(1β)+Cϵ|u(e)|2/(1β)))dxsup|x|1exp((α+h(|x|))Cϵ|u(e)|2/(1β))B1exp(((1+ϵ)α+(1+ϵ)h(|x|))|v(x)|2/(1β))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/(1β))C4.

    Therefore,

    B1[exp((α+h(|x|))|u|2/(1β))1]dxC4B1exp(αβ+(1+ϵ)h(|x|))|v(x)|2/(1β))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 vH10,rad(B1,w), we can find C5>0 such that

    B1[exp((α+h(|x|))|u|2/(1β))1]dxC4supvH10,rad(B1,w)vB1,w1B1exp((αβ+hϵ(|x|))|v(x)|2/(1β))dxC5. (2.7)

    Using the above inequality and (2.5), we obtain C>0, independent of the election of uE, satisfying

    R2[exp((α+h(|x|))|u|2/(1β))1]dxC.

    Therefore, the inequalities (1.9) and (1.10) follow. Moreover, we consider the sequence (ψk)E defined as

    ψk(x)=(1αβ)(1β)/2{k21βln(1|x|2)1β,0|x|ek/2,k1β2,ek/2|x|1,0,|x|>1.

    Note that ψk=1 for each k1, and for α>αβ, it follows that

    R2exp((α+h(|x|))|ψk|2/(1β))dxB1exp(α|ψk|2/(1β))dx2π1ek/2exp(ααβk)rdr.

    Consequently,

    R2exp((α+h(|x|))|ψk|2/(1β))dxπek(ααβ1)(ek1)+,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/(1β)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 alluE 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/(1β))1],for all(x,s)R2×R.

    By the Cauchy-Schwarz inequality and the inequality (ew1)2e2w1 for all w0, we obtain

    R2F(x,u)dxϵu22+cuq2q(R2[exp((4α0+2h(|x|))|u|2/(1β))1]dx)1/2. (3.1)

    Using Lemma 2.1, for u in E with u1, one has

    |u(x)|12πa|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/(1β)4α0(2πa)1/(1β)|x|a1β+2c1(2πa)1/(1β)|x|a1βξC1|x|η,for all|x|R0,

    where η=min{a/(1β)ξ,a/(1β)}>2, which implies the existence of C2>0 such that

    R2BR0[exp((4α0+2h(|x|))|u|2/(1β))1]dx2π+R0r(exp(C1rη)1)dr=C2. (3.2)

    Let h0=max0rR0h(r). Using Theorem 1.2, we can get C3>0 such that

    BR0[exp((4α0+2h(|x|))|u|2/(1β))1]drBR0[exp((4α0+2h0)|u|2/(1β))1]dxBR0[exp((4α0+2h0)u2/(1β)(|u|u)2/(1β))1]dxC3, (3.3)

    provided that uρ1 for some ρ1>0 such that (4α0+2h0)ρ2/(1β)1<αβ. From (3.1)–(3.3), and Lemma 2.2, there exists C>0 such that

    R2F(x,u)dxϵCu2+Cuq,

    provided that uρ0 for some 0<ρ0min{1,ρ1}. Then,

    J(u)12u2R2F(x,u)dx(12ϵC)u2Cuq.

    Note that we can assume that ϵ>0 satisfies (1/2ϵC)1/4. Consequently, it is possible to choose ρ>0 and σ>0 with 0<ρρ0 such that J(u)σ>0, for all uE with u=ρ.

    The next lemma follows the same lines as [30, Lemma 3.3].

    Lemma 3.2. Suppose that (H1)(H2) hold. If e00 in E, then there exists t>0 large enough such that e=te0 satisfies

    J(e)<0ande>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)candJ(un)E0. (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=(121μ)un21μR2(μF(x,un)f(x,un)un)dx(121μ)un2.

    Using (4.1), we have

    J(un)=c+on(1)andJ(un)E=on(1).

    Therefore, for n sufficiently large, we obtain

    (121μ)un2c+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<(121μ)(αβα0)1β.

    Proof. Take a Palais-Smale sequence (un)E for J at the level c of J. Using Lemma 4.1, we can find uE, up to a subsequence, such that unu weakly in E. Setting vn:=unu, we have that vn0 weakly in E. Then,

    R2w(x)unvndxR2f(x,un)vndx=J(un)vn=on(1)

    and

    R2w(x)unvndx=un2u2+on(1).

    Therefore,

    un2u2=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

    (121μ)un2=c+on(1)<(121μ)(αβα0)1β+on(1).

    Thus, without loss of generality, we can find δ>0 such that

    un2/(1β)αβα0δ,for all nN. (4.3)

    Now, take m>1 and ϵ>0 such that

    m(α0+2ϵ)(αβα0δ)<αβ. (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/(1β))1],for all(x,s)R2×R.

    By the Hölder inequality with 1/m+1/m=1 and the identity (er1)merm1 for all r0, we obtain

    R2|f(x,un)vn|dxϵun2vn2+Cϵvnm(R2[exp(m(α0+ϵ+h(|x|))|un|2/(1β))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/(1β))1]dxBr1[exp(m(α0+2ϵ)un2/(1β)(|un|un)2/(1β))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/(1β))1]dxBr1[exp(αβ(|un|un)2/(1β))1]dxC1. (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/(1β)C2|x|η,for alln1and|x|r2,

    where η=min{a/(1β)ξ,a/(1β)}>2, which implies the existence of C3>0 such that

    R2Br2[exp(m(α0+ϵ+h(|x|))|un|2/(1β))1]dx2π+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 alln1.

    Additionally, since h is continuous, there exists C3>0 such that

    Br2Br1[exp(m(α0+ϵ+h(|x|))|un|2/(1β))1]dxC3. (4.8)

    Using (4.6)–(4.8), the integral on the right-hand side of (4.5) is bounded. Moreover, by the compact embeddings EL2(R2) and ELm(R2), and the weakly convergence vn0 in E, up to a subsequence, we obtain

    R2|f(x,un)vn|dxϵu2vn2+Cvnm0,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)1/2Sp.

    Therefore, (uk) is bounded in E. Thus, we can assume that there exists some upE such that ukup weakly in E, ukup strongly in Lp(R2), and uk(x)up(x) almost everywhere in R2. Hence, upp=1 and uplim infk+uk=Sp. Noticing that Spup, and taking the absolute value of the functions, we can guarantee that up0. Thus there exists upE such that u(x)0 in R2 with upp=1 satisfying

    Sp=inf0uH1rad(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 dσ, where σ is given by Lemma 3.1, and d>0 is given by

    d=infγΓmaxt[0,1]J(γ(t)),

    and

    Γ={γC([0,1],E):γ(0)=0,γ(1)=e}.

    From (H5), we get

    J(tup)=t22up2R2F(x,tup)dxt22up2CptppR2|up|pdx.

    By the assumption on Cp, we obtain

    supt0J(tup)maxt0{t2S2p2Cptpp}=(p2)S2p/(p2)p2pC2/(p2)p<(121μ)(αβα0)1β. (5.1)

    Note that e=t0up with t0>0 is given by Lemma 3.2. Consider γ0Γ defined by γ0(t)=tt0up. By (5.1), we get

    d=infγΓmaxt[0,1]J(γ(t))maxt[0,1]J(γ0(t))maxt[0,1]J(tt0up)maxt0J(tup)<(121μ)(αβα0)1β.

    Using Lemma 4.2, the sequence (un), up to a sequence, is convergent, that is, we can get uE such that unu 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)=dσ, 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.



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