Nonlinear Dirichlet problems with the combined effects of singular and convection terms

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

NONLINEAR DIRICHLET PROBLEMS WITH THE COMBINED EFFECTS OF SINGULAR AND CONVECTION TERMS

YUNRU BAI, LESZEK GASI ´NSKI, NIKOLAOS S. PAPAGEORGIOU

Abstract. We consider a nonlinear Dirichlet elliptic problem driven by the p-Laplacian. In the reaction term of the equation we have the combined effects of a singular term and a convection term. Using a topological approach based on the fixed point theory (the Leray-Schauder alternative principle), we prove the existence of a positive smooth solution.

1. Introduction

Let Ω ⊆ R^N be a bounded domain with a C²-boundary ∂Ω. In this article we study the nonlinear Dirichlet problem

−∆pu(z) = u(z)^−γ+ f (z, u(z), Du(z)) in Ω,

u|∂Ω= 0, u > 0, (1.1)

where 1 < p < +∞ and 0 < γ < 1. In this problem ∆_p denotes the p-Laplace differential operator defined by

∆_pu = div(|Du|^p−2Du) ∀u ∈ W₀^1,p(Ω).

In the right-hand side of (1.1) (the reaction of the problem), we have the combined effects of a singular term u^−γ (0 < γ < 1) and of a convection term f (z, u, Du).

The convection term f is a Carath´eodory function, that is, for all (x, y) ∈ R × R^N, z 7→ f (z, x, y) is measurable and for a.a. z ∈ Ω, (x, y) 7→ f (z, x, y) is continuous. We assume that f (z, ·, y) exhibits (p − 1)-linear growth near +∞ and we have nonuni- form non-resonance with respect to the principal eigenvalue of (−∆p, W₀^1,p(Ω)). We look for positive solutions. The dependence of the gradient Du of the perturbation f , removes from consideration a variational approach directly on the equation. In- stead our method of proof is topological based on fixed point theory. More precisely, we employ the Leray-Schauder alternative principle. This leads to the existence of a positive smooth solution for problem (1.1).

In the past, singular problems and problems with convection, were investigated mostly separately. For singular problems, we mention the following works: Bai- Gasi´nski-Papageorgiou [2], Gasi´nski-Papageorgiou [9], Giacomoni-Schindler-Tak´aˇc [13], Hirano-Saccon-Shioji [17], Papageorgiou-R˘adulescu [24], Papageorgiou-R˘adu- lescu-Repovˇs [25], Papageorgiou-Smyrlis [27, 28], Perera-Zhang [29], Sun-WuLong

2010 Mathematics Subject Classification. 35J92, 35P30.

Key words and phrases. Singular term; convection term; strong comparison;

Leray-Schauder alternative principle.

c

2019 Texas State University.

Submitted November 12, 2018. Published May 3, 2019.

1

[32]. For problems with convection, we mention the following works Bai-Gasi´nski- Papageorgiou [1], Faraci-Motreanu-Puglisi [3], de Figueiredo-Girardi-Matzeu [4], Gasi´nski-Papageorgiou [12], Girardi-Matzeu [14], Huy-Quan-Khanh [19], Papageor- giou-R˘adulescu-Repovˇs [26], Ruiz [30].

2. Preliminaries and hypotheses

If V and W are two Banach spaces, a map h : V → W is said to be “compact”

if it is continuous and maps bounded sets in V onto relatively compact sets in W . As we already mentioned in the Introduction, we will use the Leray-Schauder alternative principle which we recall below (see e.g., Gasi´nski-Papageorgiou [7, p.

827]).

Theorem 2.1. If X is a Banach space and h : X → X is compact, then exactly one of the following holds:

(a) h has a fixed point;

(b) the set K = {x ∈ X : x = th(x), 0 < t < 1} is unbounded.

In the analysis of problem (1.1) we will use the Sobolev space W₀^1,p(Ω) and the Banach space

C₀¹(Ω) = {u ∈ C¹(Ω) : u|∂Ω= 0}.

By k·k we denote the norm of the Sobolev space W₀^1,p(Ω). On account of Poincar´e’s inequality, we can have

kuk = kDukp ∀u ∈ W₀^1,p(Ω).

The Banach space C₀¹(Ω) is an ordered Banach space with positive (order) cone C+= {u ∈ C₀¹(Ω) : u(z) > 0 for all z ∈ Ω}.

This cone has a nonempty interior given by

int C+=u ∈ C+: u(z) > 0 ∀z ∈ Ω, ∂u

∂n|∂Ω< 0 .

Here _∂n^∂u denotes the normal derivative of u, that is ^∂u_∂n = (Du, n)_RN with n(·) being the outward unit normal on ∂Ω.

We know that W₀^1,p(Ω)^∗ = W^−1,p0(Ω) (where _p¹+_p¹0 = 1). Let A : W₀^1,p(Ω) → W^−1,p0(Ω) be the nonlinear operator defined by

hA(u), hi = Z

Ω

|Du|^p−2(Du, Dh)_RNdz ∀u, h ∈ W₀^1,p(Ω).

This operator has the following properties (see Gasi´nski-Papageorgiou [11, Problem 2.192, p.279] or [8, Lemma 3.2]).

Proposition 2.2. The map A : W₀^1,p(Ω) → W^−1,p0(Ω) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal mono- tone too) and of type (S)₊; that is,

“if un→ u weakly in W₀^1,p(Ω) and lim sup_n→+∞hA(un), un− ui 6 0, then u_n→ u in W₀^1,p(Ω).”

Consider the nonlinear eigenvalue problem

−∆pu(z) = bλ|u(z)|^p−2u(z) in Ω,

u|∂Ω= 0. (2.1)

This problem has a smallest eigenvalue bλ₁, which has the following properties:

• bλ₁> 0 and is isolated (that is, ifσ(p) is the spectrum of (2.1), we can findb ε > 0 such that (bλ₁, bλ₁+ ε) ∩σ(p) = ∅).b

• bλ₁is simple (that is, ifbu,bv ∈ W₀^1,p(Ω) are eigenfunctions corresponding to bλ1, then bu = ξbv for some ξ ∈ R \ {0}).

• We have

bλ1= infnkDuk^p_p kuk^pp

: u ∈ W₀^1,p(Ω), u 6= 0o

. (2.2)

The infimum in (2.2) is realized on the corresponding one-dimensional eigenspace.

The nonlinear regularity theory of Lieberman [21], implies that ifu is an eigen-b value of (2.1), then bu ∈ C₀¹(Ω). The above properties of bλ1 imply that the eigen- functions corresponding to bλ1 do not change sign.

By bu1 we denote the positive, L^p-normalized (that is, kub1kp = 1) eigenfunction corresponding to bλ₁> 0. From the nonlinear maximum principle (see e.g., Gasi´nski- Papageorgiou [7, p. 738]), we have thatub₁∈ int C+. Using these properties, we can easily prove the following result (see Filippakis-Gasi´nski-Papageorgiou [5, Lemma 3.2] or Motreanu-Motreanu-Papageorgiou [23, p. 305]).

Lemma 2.3. Let ϑ ∈ L^∞(Ω), ϑ(z) 6 bλ1 for a.a. z ∈ Ω and the inequality is strict on a set of positive measure, then there exists c0> 0 such that

kDuk^p_p− Z

Ω

ϑ(z)|u|^pdz > c⁰kuk^p ∀u ∈ W₀^1,p(Ω).

For x ∈ R, we set x^± = max{±x, 0}. Then given u ∈ W₀^1,p(Ω), we set u^±(·) = u(·)^±. We know that

u^±∈ W₀^1,p(Ω), u = u⁺− u⁻, |u| = u⁺+ u⁻. The hypotheses on the perturbation term f are the following:

(H1) f : Ω × R × R^N → R is a Carath´eodory function such that f(z, 0, y) = 0 for a.a. z ∈ Ω, all y ∈ R^N, f (z, x, y) = f₀(z, y) for a.a. z ∈ Ω, all x 6 0, all y ∈ R^N with f₀being a Carath´eodory function such that f₀> 0 and (i) we have

f (z, x, y) 6 a(z) + ϑ(z)x^p−1+ c|y|^p−1 for a.a. z ∈ Ω, all x > 0, y ∈ R^N, with a, ϑ ∈ L^∞(Ω), 0 < c < bλ^1/p₁ , ϑ(z) 6 (1 − ^c

bλ^1/p₁ )bλ₁ a.e. on Ω and the last inequality is strict on a set of positive measure;

(ii) there exists δ0> 0 such that for all δ ∈ (0, δ0) there exists cδ> 0 such that 0 < cδ6 f (z, x, y) for a.a. z ∈ Ω, all 0 < δ 6 x 6 δ⁰, y ∈ R^N;

(iii) for every % > 0, there exists bξ%> 0 such that for a.a. z ∈ Ω, all |y| 6 %, the map x 7→ f (z, x, y) + bξ_px^p−1 is nondecreasing on [0, %];

(iv) for a.a. z ∈ Ω, all x > 0, y ∈ R^N and t ∈ (0, 1), we have f z,¹_tx, y) 6t^p−11 f (z, x, y).

Remark 2.4. Hypothesis (H1)(i) implies that asymptotically at +∞ we may have nonuniform non-resonance with respect to the principal eigenvalue bλ1 > 0. Hy- pothesis H(f )(iv) is satisfied if for a.a. z ∈ Ω, all y ∈ R^N, the function

x 7→ f (z, x, y) x^p−1 is non-increasing on (0, +∞).

Example 2.5. The following function satisfies hypotheses (H1). For the sake of simplicity we drop the z-dependence.

f (x, y) =







0 if x < 0,

ϑ(xb ^p−1− x^{τ −1}) + η|y|^p−1 if 0 6 x 6 1, ϑ[x^p−1− x^q−1] + η|y|^p−1 if 1 < x,

∀y ∈ R^N,

with 0 < η < bλ^1/p₁ , 0 < ϑ < (1 − ^η

bλ^1/p₁ )bλ₁, bϑ > 0, 1 < q < p < τ < +∞.

3. Positive solutions We start by considering the purely singular problem

−∆pu(z) = u(z)^−γ in Ω,

u|∂Ω= 0, u > 0. (3.1)

From Papageorgiou-Smyrlis [28, Proposition 5], we have the following result.

Proposition 3.1. Problem (3.1) admits a unique positive solution u ∈ int C+. Let δ0 > 0 be as postulated by hypothesis (H1)(ii). We choose t ∈ (0, 1) small such that

u = tu 6 δe ⁰. (3.2)

For every y ∈ W₀^1,p(Ω), we have

−∆peu(z) = t^p−1[−∆pu(z)] = t^p−1u(z)^−γ= t^p−1+γu(z)e ^−γ

<eu(z)^−γ+ f (z,eu(z), Dy(z)) for a.a. z ∈ Ω, (3.3) (see (3.2) and hypothesis (H1)(ii)).

Given v ∈ C₀¹(Ω), we consider the nonlinear Dirichlet problem

−∆pu(z) = u(z)^−γ+ f (z, u(z), Dv(z)) in Ω,

u|∂Ω= 0, u > 0, (3.4)

Proposition 3.2. If hypotheses (H1) hold and v ∈ C₀¹(Ω), then problem (3.4) admits a positive solution uv ∈ int C+ andu 6 ue ^v.

Proof. We consider the following truncation of the reaction in problem (1.1),

fb_v(z, x) = (

eu(z)^−γ+ f (z,u(z), Dv(z))e if x 6u(z),e

x^−γ+ f (z, x, Dv(z)) ifeu(z) < x. (3.5) Evidently this is a Carath´eodory function.

Sinceu,e bu1∈ int C+, on account of [22, Proposition 2.1], we can find c1> 0 such thatub16 c¹eu^p0, so

ub^1/p

0

1 6 c^1/p

0

1 u,e thus

eu^−γ6 c2ub^−γ/p₁ ⁰, for some c2> 0.

Using a Lemma in Lazer-McKenna [20], we have thatub^−γ/p₁ ⁰ ∈ L^p0(Ω). Therefore

ue^−γ∈ L^p0(Ω). (3.6)

We set

Fbv(z, x) = Z x

0

fbv(z, s) ds and consider the functionalϕbv: W₀^1,p(Ω) → R defined by

ϕbv(u) = 1

pkDuk^p_p− Z

Ω

Fbv(z, u) dz ∀u ∈ W₀^1,p(Ω).

From hypothesis (H1)(i) and (3.6), we infer that ϕbv ∈ C¹(W₀^1,p(Ω)) (see also Papageorgiou-Smyrlis [28, Proposition 3]).

Claim. ϕbv is coercive.

Clearly it suffices to check when u(z) >eu(z). We have

Fbv(z, u(z)) = Z u(z)

0

fbv(z, x) dx

= Z u(z)_e

0

fbv(z, x) dz + Z u(z)

eu(z)

fbv(z, x) dx 6 (eu(z)^−γ+ f (z,u(z), Dv(z)))e u(z)e

+ Z u(z)

u(z)e

(eu(z)^−γ+ba(z) + ϑ(z)x^p−1) dx 6ba0(z) +1

pϑ(z)|u(x)|^p withba ∈ L^∞(Ω),ba0∈ L^p0(Ω). Therefore

ϕb_v(u) =1

pkDuk^p_p− Z

Ω

Fb_v(z, u(z)) dz

>1 p

kDuk^p_p− Z

Ω

ϑ(z)|u|^pdz

−bc₁

>bc₂kDuk^p_p−bc₁,

for some bc1,bc2 > 0 (see Lemma 2.3). Thus ϕbv is coercive and so the Claim is proved.

From (3.6) and the Sobolev embedding theorem, we see thatϕbv is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find u_v∈ W₀^1,p(Ω) such that

ϕb_v(u_v) = inf

u∈W₀^1,p(Ω)ϕb_v(u),

soϕb⁰_v(uv) = 0 and thus hA(u_v), hi =

Z

Ω

fb_v(z, u_v)h dz ∀h ∈ W₀^1,p(Ω). (3.7) In (3.7) we choose h = (u − ue _v)⁺∈ W₀^1,p(Ω). Then

hA(u_v), (eu − u_v)⁺i = Z

Ω

(ue^−γ+ f (z,eu, Dv))(u − ue _v)⁺dz

> hA(eu), (u − ue v)⁺i (see (3.5) and (3.3) with y = v), so

hA(eu) − A(u_v), (u − ue _v)⁺i 6 0, and

eu 6 uv. (3.8)

From (3.8), (3.5) and (3.7), we infer that

−∆puv(z) = uv(z)^−γ+ f (z, uv(z), Dv(z)) in Ω,

u_v|∂Ω= 0. (3.9)

Then from (3.7) and Giacomoni-Schindler-Tak´aˇc [13, Theorem B.1] we get that

u_v∈ int C₊ (see (3.8)). 

Given v ∈ C₀¹(Ω), let

S_v= {u ∈ W₀^1,p(Ω) : u is a solution of (3.4),u 6 u}.e From Proposition 3.2 we know that

∅ 6= Sv⊆ int C+.

In the next proposition we prove a useful property of the elements of Sv.

Proposition 3.3. If hypotheses (H1) hold, v ∈ C₀¹(Ω) and u ∈ S_v, then u −u ∈e int C₊.

Proof. We know that u ∈ int C+. Let % = kuk_C1

0(Ω)and let eξ%> 0 be as postulated by hypothesis (H1)(iii). We have

−∆pu(z) + be ξpu(z)e ^p−1−eu(z)^−γ

< f (z,u(z), Dv(z)) + be ξ_pu(z)e ^p−1 6 f (z, u(z), Dv(z)) + bξ_pu(z)^p−1

= −∆_pu(z) + bξ_pu(z)^p−1− u(z)^−γ for a.a. z ∈ Ω (3.10) (see (3.3) with y = v, hypothesis (H1)(iii), recall thateu 6 u and see (3.9)).

We know that

−∆peu(z) + bξpu(z)e ^p−1= t^p−1(−∆pu(z) + bξpu(z)^p−1)

= t^p−1(u(z)^−γ+ bξpu(z)^p−1)

= t^p−1+γ(tu(z))^−γ(1 + bξpu(z)^p−1+γ)

<u(z)e ^−γ for a.a. z ∈ Ω

for t ∈ (0, 1) sufficiently small (as u ∈ L^∞(Ω) and see Proposition 3.1), so

− ∆peu(z) + bξpu(z)e ^p−1−u(z)e ^−γ< 0 for a.a. z ∈ Ω. (3.11) Sinceu ∈ int Ce ₊, for K ⊆ Ω compact, we have

0 < δK 6eu(z) ∀z ∈ K.

Then hypothesis (H1)(ii) implies that there exists c_K = c_δK > 0 such that

0 < cK 6 f (z,eu(z), Dv(z)) for a.a. z ∈ K. (3.12) From (3.10), (3.11), (3.12) and Papageorgiou-Smyrlis [28, Proposition 4] (the strong comparison principle), we have that u −u ∈ int Ce +.  Next we show that the set S_vhas a smallest element, that is there existsbu_v ∈ Sv

such thatub_v6 u for all u ∈ Sv.

Proposition 3.4. If hypotheses (H1) hold and v ∈ C₀¹(Ω), then there existsbuv ∈ Sv

such that buv6 u for all u ∈ S^v.

Proof. From Filippakis-Papageorgiou [6] we know that S_v is downward directed (that is, if u,u ∈ Sb _v, then there exists y ∈ S_v such that y 6 u, y 6bu). Invoking Hu-Papageorgiou [18, Lemma 3.10, p. 178], we can find a decreasing sequence {u_n}_n>1⊆ S_v such that

inf S_v = inf

n>1u_n. We have

hA(un), hi = Z

Ω

(u^−γ_n + f (z, un, Dv))h dz ∀h ∈ W₀^1,p(Ω), n > 1. (3.13) Let h = un∈ W₀^1,p(Ω) in (3.13). Then

kDunk^p_p= Z

Ω

(u^1−γ_n + f (z, un, Dv)un) dz, so

kDunk^p_p6 c³ ∀n > 1,

for some c3 > 0. Here we used that 0 6 uⁿ 6 u¹ ∈ int C+ for all n > 1 and Hewitt-Stromberg [16, Theorem 13.17, p. 196] and hypothesis (H1)(i). It follows that the sequence {u_n}_n>1⊆ W₀^1,p(Ω) is bounded. So, passing to a subsequence if necessary, we may assume that

un→buv weakly in W₀^1,p(Ω) and un→ubv in L^p(Ω). (3.14) In (3.13) we choose h = un−ubv∈ W₀^1,p(Ω), pass to the limit as n → +∞ and use (3.14) and (3.6). Then

n→+∞lim hA(un), un−ubvi = 0, so

un→ubv in W₀^1,p(Ω) (3.15) (see Proposition 2.2).

If in (3.13) we pass to the limit as n → +∞ and use (3.15), then we obtain hA(buv), hi =

Z

Ω

(bu^−γ_v + f (z,buv, Dv))h dz ∀h ∈ W₀^1,p(Ω),

soub_v∈ Sv ⊆ int C+ andub_v= inf S_v. 

We define a map g : C₀¹(Ω) → C₀¹(Ω) by setting g(v) =bu_v.

This map is well-defined and clearly a fixed point of g is a solution of (1.1). To produce a fixed point of g, we will use the Leray-Schauder alternative principle (see Theorem 2.1). To this end, we need to show that the minimal solution map g is compact (that is, g is continuous and maps bounded sets to relatively compact sets). The next lemma will be useful in this respect.

Lemma 3.5. If hypotheses (H1) hold, {v_n}_n>1 ⊆ C₀¹(Ω), v_n → v in C₀¹(Ω) and u ∈ Sv, then we can find un ∈ Sv_n for n > 1 such that uⁿ→ u in C₀¹(Ω).

Proof. We start by considering the nonlinear Dirichlet problem

−∆py(z) = u(z)^−γ+ f (z, u(z), Dvn(z)) in Ω,

y|_∂Ω= 0, (3.16)

for n > 1. As in the proof of Proposition 3.2, using Marano-Papageorgiou [22, Proposition 2.1] and a Lemma by Lazer-McKenna [20], we have that u^−γ ∈ L^q(Ω) with q > N . We set

k_n(z) = u(z)^−γ+ f (z, u(z), Dv_n(z)).

Then hypothesis (H1)(i) implies that

k_n ∈ L^q(Ω), k_n> 0, k_n6≡ 0, kknkq 6 c4 ∀n > 1,

for some c4> 0. Hence problem (3.16) has a unique solution y_n⁰∈ W₀^1,p(Ω), y_n⁰ > 0, y_n⁰ 6≡ 0 and using Guedda-V´eron [15, Proposition 1.3], we have

y_n⁰∈ L^∞(Ω), ky_n⁰k∞6 c⁵ ∀n > 1, (3.17) for some c5> 0. Consider the linear Dirichlet problem

−∆w(z) = kn(z) in Ω, w|∂Ω= 0

for all n > 1. Standard regularity theory (see e.g., Struwe [31, p. 218]), implies that this problem has a unique solution w_n such that

w_n∈ W₀^2,q(Ω) ⊆ C₀^1,α(Ω) = C^1,α(Ω) ∩ C₀¹(Ω), kwnk_C^1,α

0 (Ω)6 c6 ∀n > 1, with α = q − ^N_q > 0 and for some c6> 0. We put σn(z) = ∇wn(z) for all z ∈ Ω and all n > 1. Evidently σⁿ ∈ C^α(Ω) for all n > 1. Then from (3.16) we see that y_n⁰ satisfies

− div |∇y⁰_n(z)|^p−2∇y_n⁰(z) − σ_n(z)
= 0 in Ω, y⁰_n|∂Ω= 0,

for n > 1. Invoking Lieberman [21, Theorem 1] (see also Guedda-V´eron [15, Corol- lary 1.1]) and using (3.17), we infer that there exists β ∈ (0, 1) and c7 > 0 such that

y⁰_n∈ C₀^1,β(Ω) ∩ int C+, ky⁰_nk_C1,β

0 (Ω)6 c⁷ ∀n > 1. (3.18) Recall that C₀^1,β(Ω) is embedded compactly in C₀¹(Ω). So, from (3.18) it follows that there exists a subsequence {y⁰_nk}_k>1of {y_n⁰}_n>1such that

y_n⁰

k → y⁰ in C₀¹(Ω) as k → +∞, (3.19)

with y⁰> 0. Note that

kn → k in L^q(Ω), (3.20)

with k(z) = u(z)^−γ+ f (z, u(z), Dv(z)). From (3.16), (3.19), (3.20), in the limit as n → +∞, we have

−∆_py⁰(z) = k(z) in Ω,

y⁰|_∂Ω= 0. (3.21)

This problem has a unique solution y⁰∈ C₀¹(Ω). On the other hand, since u ∈ Sv, from (3.20) it follows that u also solves (3.21). Hence y⁰ = u. It follows that for the original sequence we have

y_n⁰→ u in C₀¹(Ω). (3.22)

Next we consider the nonlinear Dirichlet problem

−∆py(z) = y⁰_n(z)^−γ+ f (z, y⁰_n(z), Dvn(z)) in Ω, y⁰_n|∂Ω= 0,

for n > 1. Again this problem has a unique solution yn¹ ∈ int C+ for n > 1 and as above (see (3.22)), we have

y_n¹→ u in C₀¹(Ω).

Continuing this way, we generate a sequence {y_n^k}_n>1 ⊆ int C+ for all k > 1 such that

−∆py_n^k(z) = y^k−1_n (z)^−γ+ f (z, y^k−1_n (z), Dvn(z)) in Ω,

y^k_n|∂Ω= 0, (3.23)

for k, n > 1 and

y^k_n→ u in C₀¹(Ω) as n → +∞ ∀k > 1. (3.24) As before from (3.23) and Lieberman [21, Theorem 1 ], we know that {y^k_n}_k>1⊆ C₀¹(Ω) is relatively compact.

So, we can find a subsequence {y^k_n^m}_m>1of {y_n^k}_k>1 such that y_n^km →yb_n in C₀¹(Ω) as m → +∞ ∀n > 1.

From (3.23) in the limit as m → +∞, we obtain

−∆pbyn(z) =ybn(z)^−γ+ f (z,ybn(z), Dvn(z)) in Ω,

ybn|∂Ω= 0, (3.25)

for n > 1.

From (3.25) we have kDybnk^p_p=

Z

Ωyb_n^1−γdz + Z

Ω

f (z,ybn, Dvn)ybndz 6bc3+ Z

Ω

ϑ(z)by_n^pdz for somebc₃> 0, so

kDbynk^p_p− Z

Ω

ϑ(z)yb_n^pdz 6bc3

and hence the sequence {by_n}_n>1⊆ W₀^1,p(Ω) is bounded (by Lemma 2.3).

From this and Lieberman [21, Theorem 1], it follows that the sequence {by_n}_n>1⊆ C₀¹(Ω) is relatively compact. Passing to a subsequence if necessary, we may assume that

by_n→ub in C₀¹(Ω).

By the double limit lemma (see e.g., Gasi´nski-Papageorgiou [10, Problem 1.175, p.

61]), we have

y^k_n^m(n)→ub in C₀¹(Ω) as n → +∞.

Ifbu 6= u, then 0 < ε06 ku −ukb _C1 0(Ω), so 0 < ε0

2 6 ku − yn^km(n)k_C1

0(Ω) ∀n > n0, a contradiction (see (3.24)). So, we have

by_n→ u in C₀¹(Ω) as n → +∞.

Recall that u −u ∈ int Ce ₊ (see Proposition 3.3). So, it follows that ybn−u ∈ int Ce + ∀n > n0,

andybn ∈ Sv_n ∀n > n0 (see (3.25)). 

Using this lemma, we can show that the minimal solution map is compact.

Proposition 3.6. If hypotheses (H1) hold, then the minimal solution map g : C₀¹(Ω) → C₀¹(Ω) defined by g(v) =bu_v is compact.

Proof. First we show that g is continuous. To this end let vn → v in C₀¹(Ω). We setubn=buv_n = g(vn) for all n > 1. We have

−∆pbun(z) =bun(z)^−γ+ f (z,bun(z), Dvn(z)) in Ω,

bu_n|_∂Ω= 0, (3.26)

for n > 1.

As in the proof of Lemma 3.5, using Guedda-V´eron [15, Proposition 1.3] and Lieberman [21, Theorem 1], we have that the sequence {bun}_n>1 ⊆ C₀¹(Ω) is rel- atively compact (see also Giacomoni-Schindler-Tak´aˇc [13, Theorem B.1]). So, we may assume that

bun →bu0 in C₀¹(Ω) as n → +∞. (3.27) Passing to the limit as n → +∞ in (3.26) and using (3.27), we obtain that

ub0∈ Sv. (3.28)

From Lemma 3.5, we know that we can find u_n∈ S_vn for n > 1 such that

un →bu =ubv= g(v) in C₀¹(Ω) as n → +∞. (3.29) We haveub_n6 un ∀n > 1, so

bu06bu = g(v)

(see (3.27) and (3.29)). Sinceub0∈ Sv (see (3.28)), we conclude that bu0= g(v) =bu.

Therefore for the original sequence we haveubn→u in Cb ₀¹(Ω); thus g is continuous.

Also, if B ⊆ C₀¹(Ω) is bounded, then as before via the results by Guedda-V´eron [15] and Lieberman [21], we obtain that g(B) ⊆ C₀¹(Ω) is relatively compact and

thus g is compact. 

Now we can employ the Leray-Schauder alternative principle (see Theorem 2.1) to produce a positive solution to problem (1.1).

Theorem 3.7. If hypotheses (H1) hold, then problem (1.1) admits a positive solu- tionbu₀∈ int C+.

Proof. From Proposition 3.6 we know that the minimal solution map g : C₀¹(Ω) → C₀¹(Ω) is compact. Let K ⊆ C₀¹(Ω) be the set

K = {u ∈ C₀¹(Ω) : u = tg(u), 0 < t < 1}.

If u ∈ K, then ¹_tu = g(u), so

− ∆pu(z) = t^p−1 t^γ

u(z)^γ + f (z,1

tu(z), Du(z))

a.e. in Ω. (3.30) Hypothesis (H1)(iv) implies that

f (z,¹_tu(z), Du(z)) 6 1

t^p−1f (z, u(z), Du(z)) for a.a. z ∈ Ω. (3.31) Returning to (3.30) and using (3.31) and hypothesis (H1)(i), we have

−∆_pu(z) 6 t^p+γ−1

u(z)^γ + f (z, u(z), Du(z))

6 1

eu(z)^γ + a(z) + ϑ(z)u(z)^p−1+ c|Du(z)|^p−1,

(3.32)

for a.a. z ∈ Ω, so

kDuk^p_p 6bc₄+ Z

Ω

ϑ(z)u^pdz + c Z

Ω

|Du|^p−1u dz 6bc4+

Z

Ω

ϑ(z)u^pdz + ckDuk^p−1_p kukp

6bc4+ Z

Ω

ϑ(z)u^pdz + c bλ^1/p₁

kDuk^p_p, for somebc4> 0 (by H¨older’s inequality and using (2.2)), thus

 1 − c

bλ^1/p₁

kDuk^p_p− Z

Ω

ϑ(z)u^pdz ≤bc4, hence, by Lemma 2.3, we have

bc₅kDuk^p_p6bc₄,

for somebc5> 0. This proves the boundedness of K ⊆ W₀^1,p(Ω).

Invoking Theorem 2.1 (the Leray-Schauder alternative principle), we can find ub0∈ C₀¹(Ω) such that

ub0= g(bu0) ∈ S

ub₀ ⊆ int C+.

This is a positive solution of (1.1). 

Remark 3.8. It will be interesting to know if we can have multiplicity of positive solutions (for example a pair of positive solutions). For purely singular elliptic problem such a result was proved by Papageorgiou-R˘adulescu-Repovˇs [25]. Also another interesting open problem is whether we can treat resonant equations.

Acknowledgments. L. Gasi´nski was supported by the National Science Center of Poland under Project No. 2015/19/B/ST1/01169, and the H2020-MSCA-RISE- 2018 Research and Innovation Staff Exchange Scheme Fellowship within the Project no. 823731CONMECH.

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Yunru Bai

Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Lojasiewicza 6, 30-348 Krak´ow, Poland

Email address: yunrubai@163.com Leszek Gasi´nski

Department of Mathematics, Pedagogical University of Cracow, Podchorazych 2, 30- 084 Cracow, Poland

Email address: leszek.gasinski@up.krakow.pl Nikolaos S. Papageorgiou

National Technical University, Department of Mathematics, Zografou Campus, Athens 15780, Greece

Email address: npapg@math.ntua.gr