Lane-Emden equations perturbed by nonhomogeneous potential in the super critical case

Theorem 1.1

Assume that p_c is given by (1.5), p∈[pc,N+2N−2), the potential V ∈ C¹(ℝ^N ∖ {0}) satisfies that

where ρ0=1N(N−2).

Then there exists ν^* > 0 such that for ν ∈ (ν^*, +∞), problem (1.1) has a ν-fast decaying solution u_ν, which has singularity at the origin as

where c_p is given in (1.4).

Unlike the case of p∈(NN−2,pc), the Schauder fixed point theorem fails to build the lower bound for problem (1.1), due to lack the stability of fast decaying solution w_k of (1.3) for p∈[pc,N+2N−2). Note that

with μ > 0 is the extremal solution of −ΔUμ=UμN+2N−2−pUμp in ℝ^N, which is possible to provide a sub fast decaying solution for (1.1) if UμN+2N−2−p≤1 and U_μ < w_p, the slow decaying solution of (1.3). Based on this observation, we construct pairs sub-super solution (U_μ^*, w_k) with μ∗=(N(N−2))−12 and k ≥ k^* for some k^* > 0 and iterating procedure with initial data w_k could be applied to approach a sequence of fast decaying.

Our final interest is to study the limit of {u_ν}_ν as ν → +∞ and the result states as follows.

Theorem 1.2

Assume that p_c is given by (1.5), p∈[pc,N+2N−2), V ∈ C¹(ℝ^N ∖ {0}) satisfies (1.6) and for some γ > 1, α ∈ ((N − 2)p − N − 2, 0)

Let u_ν be a ν-fast decaying solution of problem (1.1) with ν ∈ (ν^*, +∞) derived by Theorem 1.1. Then the limit of {u_ν}_ν as ν → +∞ exists, denoting u_∞ = limν→+∞ u_ν, and u_∞ is a solution of (1.1) verifying (1.7) and

where c > 1.

The rest of this paper is organized as follows. In Section 2, we show qualitative properties of the solutions to elliptic problem with homogeneous potential and some basic estimates. Section 3 is devoted to build fast decaying solutions of (1.1) by iteration method. Section 4 is devoted to the slow decaying solution as the limit of fast decaying solutions.

Proof

Proof of Theorem 1.1. Our proof divides into five steps.

1. Existence by iteration method. We initiate from v₀ := w_k, denote by v_n iteratively the unique solution of

   vn=Γ∗(Vvn−1p)inRN∖{0}, (3.1)

   that is,

   −Δvn=Vvn−1pinRN∖{0},lim|x|→0vn(x)|x|N−2=0.

   As v0=Γ∗(v0p) in ℝ^N ∖ {0} and V ≤ 1, we have that

   v1≤v0inRN∖{0}.

   Inductively, we can deduce that v_n ≤ v_n−1 in ℝ^N ∖ {0}. Thus, the sequence {v_n}_n is decreasing.

   Now we show that U_μ^* is a lower bound for {v_n}_n for k ∈ [k^*, +∞). From (2.7) and the assumption that

   Vp≤V≤1,

   we obtain that U_μ^* is a sub solution of (1.1), i.e.

   −ΔUμ∗=VpUμ∗p≤VUμ∗p≤VwkpinRN∖{0},

   i.e. Uμ∗≥Γ∗(VUμ∗p), then

   v1≥Uμ∗inRN∖{0}.

   Inductively, we see that for any n ∈ ℕ, we have that

   vn≥Uμ∗inRN∖{0},

   so {v_n}_n has a lower barrier U_μ^*. Therefore, the sequence {v_n}_n converges. Denote u_νk = lim_n→∞ v_n, then for any compact set K in ℝ^N ∖ {0}, u_νk verifies the equation

   −Δu=VupinK,

   and then u_νk is a classical solution of (1.1) verifying

   Uμ∗≤uνk≤wkinRN∖{0}.

   Here we let

   νk(V)=cN∫RNVuνkpdx,

   we also replace ν_k by ν_k(V) if it is not confusing. Then

   k≤νk≤ν~k≤k+cδ0kpandlim|x|→+∞uνk(x)|x|N−2=νk

   hold by Lemma 2.2. Here and in what follows, we always denote u_νk as the ν_k-fast decaying solution of (1.1) derived by the sequence v_n defined in (3.1) with initial value w_k.

2. the mapping k ↦ ν_k is increasing and V ↦ ν_k(V) is increasing in the sense that ν_k(V₁) ≥ ν_k(V₂) if V₁ ≥ V₂. For k^* ≤ k₁ < k₂, by the increasing monotonicity of k ↦ w_k, we have that w_k1 < w_k2. Let {v_n,ki} be sequence of (3.1) with the initial data v_i,0 = w_ki, here i = 1,2.

   Let

   νi,n=lim|x|→+∞vi,n(x)|x|N−2,i=1,2,n=1,2,3,⋯.

   We see that

   ν1,1=cN∫RNVwk1pdx≥cN∫RNVwk2pdx=ν2,1

   and

   v1,1≥v2,1inRN∖{0}.

   Inductively, we have that for any n ∈ ℕ,

   ν1,n≥ν2,n,

   which implies that the limit u_νk1 of {v_n,k1} and the limit u_νk2 of {v_n,k2} as n → +∞ verifies that

   lim|x|→+∞uνk2(x)|x|N−2≥lim|x|→+∞uνk1(x)|x|N−2,

   that is,

   νk2≥νk1.

   As a conclusion, for any k ∈ [k^*, +∞), there exists a ν^* := ν_k^* > 0 such that problem (1.1) has a solution u_νk such that

   lim|x|→+∞uνk(x)|x|N−2=νk.

   We conclude u_νk1 ≥ u_νk2 in ℝ^N ∖ {0} for k^* ≤ k₂ ≤ k₁ < +∞ by v_1,n ≥ v_2,n in ℝ^N ∖ {0} for any n ∈ ℕ. That means that the mapping k ↦ ν_k is increasing.

   Similarly, we can obtain that the mapping: V → ν_k(V) is increasing.

3. we prove that the mapping k ∈ (k^*, +∞) ↦ ν_k is continuous. For 0 < k₁ < k₂, we have that w_k1 < w_k2. Let {v_n,ki} be sequence of (3.1) with the initial data v_0,i = w_ki, here i = 1, 2.

   Let

   νn,i=lim|x|→+∞vn,ki(x)|x|N−2,i=1,2,n=1,2,3,⋯.

   We see that

   ν1,1=cN∫RNVwk1pdx<cN∫RNVwk2pdx=ν1,2

   and

   0<ν1,2−ν1,1=cN∫RNV(wk2p−wk1p)dx≤cN∫RN(wk2p−wk1p)dx=k2−k1.

   Inductively, we have that for any n ∈ ℕ,

   0≤νn,2−νn,1≤k2−k1,

   which implies that the limit u_νk1 of {v_n,k1} and the limit u_νk2 of {v_n,k2} as n → +∞ verifies that

   0≤lim|x|→+∞uνk2(x)|x|N−2−lim|x|→+∞uνk1(x)|x|N−2≤k2−k1,

   that is,

   0≤νk2−νk1≤k2−k1.

   As a conclusion, k → ν_k is increasing and continuous.

   Let

   ν∞=limk→+∞νk,

   then we have that for any k ∈ [k^*, +∞), there exists ν ∈ [ν^*, ν_∞) (ν = ν^* if ν_∞ = ν^*) such that problem (1.1) has a solution u_νk such that

   lim|x|→+∞uνk(x)|x|N−2=νk.

4. we prove that ν_∞ = +∞. By the increasing monotonicity of the mapping: V → ν_k(V), we only have to prove that ν_k(V_p) → +∞ as k → +∞ for μ > μ^#.

   By contradiction, we may assume that

   ν∞(Vp)<+∞, (3.2)

   where we recall Vp=Uμ∗N+2N−2−p in ℝ^N. Now fix ν̄ ∈ [ν^*, ν_∞(V_p)], then there exist α₁ = p(N − 2) − N − 2 ∈ (−2, 0) and l₁ > 1 such that

   ν¯l1N−2−2+α1p−1>ν∞(Vp)

   and denote

   ψ1(x)=l1−2+α1p−1uν¯(l1−1x),∀x∈RN∖{0}.

   Let k̄ be the number such that ν_k̄ = ν̄. By direct computation, we have that

   ψ1(x)⩽l1−2+α1p−1wk¯(l1−1x)),∀x∈RN∖{0}

   and

   −Δψ1=Vl1ψ1pinRN∖{0},

   where Vl1(x):=l1α1Vp(l1−1x)≤Vp(x) by the decreasing monotonicity of V_p.

   Note that wl1N−2−2+α1p−1k¯(x)=l1−2+α1p−1wk¯(l1−1x) and then we may initiate the iteration (3.1) with v0=wl1N−2−2+α1p−1k¯ and ψ₁ is a lower bound, so we have a solution uνl1N−2−2+α1p−1k¯ of (1.1) such that

   ψ2≤uνl1N−2−2+α1p−1k¯≤wl1N−2−2+α1p−1k¯,

   it yields

   νl1N−2−2+α1p−1k¯>ν¯l1N−2−2+α1p−1>ν∞(Vp),

   which contradicts (3.2). Thus, we have that ν_∞(V_p) = +∞.

   Finally, we prove (1.7). From Theorem 2.1, we only have to rule out the case that u_νk is removable at the origin. If u_νk is removable at the origin. then u_νk is a bounded classical sub solution of

   −Δu=upinRN.

   Since u_νk ≤ w_k and u_νk is bounded at the origin, there exists ∣x₁∣ > 0 small enough such that

   wk(x)≤wp(x+x1),∀x∈RN∖B1(0)

   and

   uνk(x)≤wp(x+x1),∀x∈B1(0).

   Now consider the sequence vn=Γ∗(vn−1p) with the initial data v₀ = u_νk, and this sequence is increasing and controlled by function w_k and w_p(x₁ + ⋅), the limit of {v_n} will be a bounded positive classical solution of

   −Δu=upinRN. (3.3)

   This contradicts [16, Theorem 1.1], which says the nonexistence of bounded positive solution of (3.3). We complete the proof.□

Theorem 4.1

Assume that p_c is given by (1.5), p∈[pc,N+2N−2), V ∈ C¹(ℝ^N ∖ {0}) is radially symmetric, decreasing with respect to ∣x∣, satisfies (1.6) and (1.8).

Let u_ν be a ν-fast decaying solution of problem (1.1) with ν ∈ (ν^*, +∞) derived by Theorem 1.1. Then the limit of {u_ν}_ν as ν → +∞ exists, denoting u_∞ = limν→+∞ u_ν, and u_∞ is a solution of (1.1) verifying (1.7) and (1.9).

Proof

Recall the mapping ν ∈ (ν^*, +∞) ↦ u_ν is increasing, where u_ν is a ν-fast decaying solution of problem (1.1), so our aim is to show the existence of sequence {u_ν}_ν as ν → +∞. To this end, we show a priori estimates for {u_ν}_ν.

1. Radial symmetry. We recall that the solution u_ν is approaching by sequence vn=Γ∗(Vvn−1p) with initial data v₀ = w_k. Note that w_k and V are radially symmetric and decreasing in r = ∣x∣, so is {v_n} for any n ∈ ℕ. Therefore, u_ν is radially symmetric and decreasing in r = ∣x∣.

2. Uniform estimates. It is standard to show that u_ν is a very weak solution of (1.1) in the distributional sense that uν∈Lloc1(RN)∩Llocp(RN,Vdx) satisfies the identity

   ∫RNuν(−Δ)ξdx=∫RNVuνpξdx,∀ξ∈Cc∞(RN). (4.1)

   We claim that there exists c > 0 independent of ν such that

   ∥uν∥Lloc1(RN)≤cand∥uν∥Llocp(RN,Vdx)≤c.

   Indeed, we recall that

   U1(x)=cN(1+|x|2)N−22,∀x∈RN,

   then

   −ΔU1=U1N+2N−2inRN,

   where cN=(N(N−2))N−24.

   For ϵ∈(0,14), we denote

   ηϵ(x)=η0(ϵ|x|),∀x∈RN, (4.2)

   where η₀ : [0, +∞) → [0, 1] is a smooth increasing function such that

   η0(t)=0,∀t≥2andη0(t)=1,∀t∈[0,1].

   Take U1ηϵ2 as a test function of (4.1), then by Hölder inequality, we have that

   ∫RNVuνpU1ηϵ2dx=∫RNuν(−Δ)(U1ηϵ2)dx=∫RNuν(U12∗−1ηϵ2+4ηϵ∇U1⋅∇ηϵ+U1(−Δ)(ηϵ2))dx≤∫RNVuνpU1ηϵ2dx1p∫B2ϵV1−pU11−p+(2∗−1)pp−1dx+ϵ∥η0∥C1(R)∫B2ϵ(0)∖B1ϵ(0)V1−pU11−p+(2∗−1)pp−1|∇U1|p−1dx+ϵ2∥η0∥C2(R)∫B2ϵ(0)∖B1ϵ(0)V1−pU1p−1dx1−1p=c0∫RNVuνpU1ηϵ2dx1p,

   where c₀ > 0 depends on ϵ and V, but it is independent of ν. Then we have that

   cN(1+1ϵ2)−N−22∫B1ϵ(0)Vuνpdx≤∫B1ϵ(0)VuνpU1dx≤∫RNVuνpU1ηϵ2dx≤c0pp−1,

   that is,

   ∥uν∥Llocp(RN,Vdx)≤c

   for some c > 0 independent of ν.

   Furthermore,

   ∫RNuνU12∗−1ηϵ2dx≤c0∫RNVuνpU1ηϵ2dx1p≤c0pp−1

   and

   ∥uν∥Lloc1(RN)≤c.

3. the limit of {u_ν}_ν. From Theorem 1.1, the mapping ν ∈ [ν^*, ∞) ↦ u_ν is increasing and uniformly bounded in Lloc1(RN)∩Llocp(RN,Vdx), so there exists u∞∈Lloc1(RN)∩Llocp(RN,Vdx) such that

   uν→u∞asν→+∞a.e.inΩandinLloc1(RN)∩Llocp(RN,Vdx).

   It is known in [20] that u_ν is also a weak solution of (1.1), i.e.

   ∫RNuν(−Δ)ξdx=∫RNVuνpξdx,∀ξ∈Cc∞(RN). (4.3)

   Passing to the limit of (4.3), we obtain that u_∞ is a weak solution of (1.1) in the sense of (4.1).

   Note that u_ν is radially symmetric and decreasing with respect to ∣x∣, so is u_∞. Then we have that u_∞ ∈ Lloc∞ (ℝ^N ∖ {0}) and then Vu∞p is in Lloc∞ (ℝ^N ∖ {0}). By standard regularity results, we have that u is a classical solution of (1.1).

   Since u_ν verifies (1.7) at the origin for any ν > 0 and u_∞ is the limit of an increasing sequence {u_ν}_ν, then we have that

   lim inf|x|→0u∞(x)|x|2p−1≥cp. (4.4)

   Next we claim

   lim sup|x|→0u∞(x)|x|2p−1≤cp, (4.5)

   Indeed, from [22, Theorem 2.1], there exists c > 0 such that

   u∞(x)≤c|x|−2p−1,∀x∈B1(0)∖{0},

   and by [2, Theorem B], then we have that (4.5) and (1.7) hold true for u_∞.

   Finally, we claim (1.9). As in [15], we denote

   v(x)=u∞(x|x|2)forallx∈RN∖{0}.

   By direct computation, we have that

   ∇v(x)=∇u∞x|x|21|x|2−2∇u∞(x|x|2)⋅xx|x|4

   and

   Δv(x)=1|x|4Δu∞(x|x|2)+2(2−N)|x|4(∇u∞(x|x|2)⋅x).

   Let u^♯(x) = ∣x∣^2−Nv(x), then for x ∈ ℝ^N ∖ {0}, we obtain that

   −Δu♯(x)=−Δv(x)|x|2−N−2∇v(x)⋅(∇|x|2−N)=|x|−2−N(−Δ)u(x|x|2)=V♯(x)u♯(x)p,

   where

   V♯(x)=|x|−2−N+p(N−2)Vp(x|x|2).

   Direct computation implies that

   −Δu♯(x)=V♯(x)u♯(x)p,∀x∈RN∖{0},

   where

   V♯(x)∼|x|ϱas|x|→0withϱ=−α1−4+(p−1)(N−2)

   and α₁ = (N − 2)p − N − 2 ∈ (−2, 0).

   Now claim that p∈(N+ϱN−2,N+2+ϱN−2). Note that p>N+ϱN−2 follows by the fact p>NN−2 and p<N+2+ϱN−2 is equivalent to (N − 2)p − N − 2 < −4 + (p − 1)(N − 2) − α₁, which is true thanks to α ≤ 0. Then by [22, Theorem 2.1] and [16, Theorem 3.3], we have that

   1c|x|−2+ϱp−1≤u♯(x)≤c|x|−2+ϱp−1,∀x∈B1(0)∖{0},

   where c > 1 and

   −2+ϱp−1=−(N−2)+2+α1p−1.

   By Kelvin transformation, we turn back that

   1c|x|−2+α1p−1≤u∞(x)≤c|x|−2+α1p−1,∀x∈RN∖B1(0)

   for some c > 1.□