Positive Solutions for Resonant (p, q)-equations with convection

Proposition 2.1

The operator A_r(⋅) is bounded (that is, maps bounded sets to bounded sets), continuous, strictly monotone (hence maximal monotone too) and of type (S)₊, that is

For x ∈ ℝ. we set x^± = max{±x, 0}. Then for u ∈ W01,p (Ω), we define u^±(z) = u(z)^± for all z ∈ Ω. We have

If u ∈ W01,p (Ω), then

If X is a Banach space and φ ∈ C¹(X, ℝ), then we set

We say that φ(⋅) satisfies the “C-condition”, if the following property holds:

“Every sequence {u_n} ⊆ X such that {φ(u_n)}_n≥1 ⊆ ℝ is bounded and (1 + ∥u_n∥_X)φ′(u_n) → 0 in X^* as n → ∞, admits a strongly convergent subsequence.”

Next, we recall some basic facts about the spectrum of (−Δ_p, W01,p (Ω)). So, we consider the following nonlinear eigenvalue problem

We say that λ̂ ∈ ℝ is an “eigenvalue” of (−Δ_p, W01,p (Ω)), if problem (2.1) admits a nontrivial solution û, known as an “eigenfunction” corresponding to λ̂. By σ̂(p) we denote the set of eigenvalues (the “spectrum”) of (−Δ_p, W01,p (Ω)). There is a smallest eigenvalue denoted by λ̂₁(p) which has the following properties:

1. λ̂₁(p) > 0 and it is isolated (that is, there exists ε > 0 such that σ̂(p)⋂(λ̂₁(p), λ̂₁(p) + ϵ) = ∅).

2. λ̂₁(p) is simple (that is, if û, v̂ ∈ W01,p (Ω) are eigenfunctions for λ̂₁(p), then we have û = θv̂ for some θ ∈ ℝ∖{0}).

3. λ^1(p)=inf∥Du∥pp∥u∥pp:u∈W01,p(Ω),u≠0. (2.2)

The infimum in (2.2) is realized on the one dimensional eigenspace corresponding to λ̂₁(p). By û₁(p) we denote the L^p-normalized (that is, ∥û₁(p)∥_p = 1), positive eigenfunction corresponding to λ̂₁(ρ). The nonlinear regularity and the nonlinear maximum principle (see, for example, Gasiński-Papageorgiou [21], pp 737-738), imply that û₁ ∈ intC₊. Since σ̂(p) ⊆ (0, +∞) is closed and λ̂₁(p) is isolated, the second eigenvalue is well-defined by

Using the Ljusternik-Schnirelmann minimax scheme, we produce a whole strictly increasing sequence {λ̂_k(p)}_k∈ℕ of eigenvalues such that λ̂_k(p) → +∞. These eigenvalues are known as “variational eigenvalues” of (−Δ_p, W01,p (Ω)). We do not know if the variational eigenvalues exhaust σ̂(p). This is the case if p = 2 (linear eigenvalue problem) or if N = 1 (ordinary differential equations). By the nonlinear regularity theory, we know that every eigenfunction û ∈ C01 (Ω). Moreover, if λ̂ ∈ σ̂(p)∖{λ̂₁(p)}, then every eigenfunction corresponding to λ̂ is nodal (that is, sign changing).

We will also use a weighted version of the eigenvalue problem (2.1). So, let m ∈ L^∞(Ω), m(z) ≥ 0 for a.a. z ∈ Ω, m ≠ 0. We consider the following nonlinear eigenvalue problem

The same results hold for the eigenvalues λ̂(p, m) of problem (2.3). In this case the Rayleigh quotient in the variational characterization of λ̂₁(p, m) > 0, is given by

Then using this fact, we can easily prove the following monotonicity property of the map m → λ̂₁(p, m).

Proposition 2.2

If m, m͡ ∈ L^∞(Ω), 0 ≤ m(z) ≤ m͡(z) for a.a. z ∈ Ω, m ≠ 0, m ≠ m͡, then λ̂₁(p, m͡) < λ̂₁(p, m).

Remark 2.1

Evidently, if m(z) = 1 for a.a. z ∈ Ω, then λ̂₁(p, m) = λ̂₁(p).

Now, we introduce the hypotheses on the data of (E_λ).

H₀ : r ∈ L^∞(Ω), r(z) ≥ 0 for a.a. z ∈ Ω, r ≠ 0.

H₁ : f : Ω × ℝ → ℝ is a Carathéodory function such that f(z, 0) = 0 for a.a. z ∈ Ω and

1. 0 ≤ f(z, x) ≤ a(z)(1 + x^p−1) for a.a. z ∈ Ω, all x ≥ 0, with a ∈ L^∞(Ω);

2. there exists m ∈ ℕ, m ≥ 2 such that

   limx→+∞f(z,x)xp−1=λ^m(p)uniformly for a.a. z∈Ω;

3. if F(z, x) = ∫0x f(z, s)ds, then there exists θ ∈ (q, p) such that

   0<c0≤lim infx→+∞pF(z,x)−f(z,x)xxθuniformly for a.a. z∈Ω;

4. limx→0+f(z,x)xq−1=0, uniformly for a.a. z ∈ Ω.

Remark 2.2

Since our goal is to find positive solutions and the above hypotheses concern the positive semi-axis ℝ₊ = [0, +∞), without any loss of generality, we may assume that

Hypothesis H₁(ii) implies that the problem is resonant with respect to a nonprincipal variational eigenvalue λ̂_m(p).

As we already mentioned in the Introduction eventually our proof will be topological and to reach that point we will use the frozen variable method.

The topological tool which we will use is the so-called “Leray-Schauder Alternative Principle” (see Papageorgiou-Rădulescu-Repovš [22], Proposition 3.2.22, p.198). So, let X be a Banach space and ξ : X → X a map. We say that ξ(⋅) is “compact”, if it is continuous and maps bounded sets to relatively compact sets. The Leray-Schauder Alternative Principle asserts the following:

Theorem 2.3

If X is a Banach space, ξ : X → X is compact and

then the following alternative holds

1. D(ξ) is unbounded; or

2. ξ(⋅) has a fixed point.

We said in the Introduction that in order to solve the frozen problem using the critical point theory, we will need to use the solution of an auxiliary double phase Dirichlet problem. This is the following parametric purely singular problem

From Proposition 11 of Papageorgiou-Rădulescu-Repovš [4], we have the following result concerning problem (Q_λ).

Proposition 2.3

For every λ > 0, problem (Q_λ) has a unique positive solution u_λ ∈ int C₊.

Consider the Banach space C₀(Ω) = {u ∈ C(Ω) : u∣_∂Ω = 0}. This is an ordered Banach space with positive (order) cone K₊ = {u ∈ C₀(Ω) : u(z) ≥ 0 for all z ∈ Ω}. This cone has a nonempty interior given by

with d̂(z) = d(z, ∂ Ω) for all z ∈ Ω̄. According to Lemma 14.16, p. 355, of Gilbarg-Trudinger [23], there exists δ₀ > 0 such that d̂ ∈ C²(Ω_δ0), where Ω_δ0 = {z ∈ Ω̄ : d(z, ∂ Ω) < δ₀}. It follows that d̂ ∈ intC₊ and so from Proposition 4.1.22, p.274, of Papageorgiou-Rădulescu-Repovš [22], we can find 0 < c₁ < c₂ such that

Let s > N and consider u^1(p)1s∈K+. Then using Proposition 4.1.22, p. 274, of Papageorgiou-Rădulescu-Repovš [22], we know that we can find c₃ > 0 such that

From the Lemma in Lazer-McKenna [24], we have u^1(p)−ηs∈Ls(Ω),s>N. So, we have

In the next section, this fact will help us bypass the singular term and use variational tools on the “frozen” problem.

Proposition 3.1

If hypotheses H₀, H₁ hold and λ > 0, then the function ψ^vλ (⋅) satisfies the C-condition.

Proof

We consider a sequence {u_n}_n≥1 ⊆ W01,p (Ω) such that

From (3.8) we have

In (3.9) we test with h=−un−∈W01,p(Ω) and obtain

Next, we show that {un+}n≥1⊆W01,p(Ω) is bounded. Suppose that this is not true. Passing to a subsequence if necessary, we may assume that

We set yn=un+∥un+∥ for all n ∈ ℕ. Then ∥y_n∥ = 1, y_n ≥ 0 for all n ∈ ℕ. So, we may assume that

From (3.9) and (3.10), we have

From (2.5),(3.6) and hypothesis H₁(i), we see that

In (3.13) we choose h=yn−y∈W01,p(Ω), pass to the limit as n → ∞ and use (3.11) (recall q < p), (3.12),(3.14), we obtain

From (3.14) and hypothesis H₁(ii) and if we pass to a subsequence if necessary, we have

We return to (3.13), pass to the limit as n → ∞ and use (3.11) (recall q < p), (3.15), and (3.16). We obtain

a contradiction to (3.15).

Therefore, {un+}n≥1⊆W01,p(Ω) is bounded and so from (3.10), it follows that

We may assume that

In (3.9) we choose h = u_n − u ∈ W01,p (Ω), pass to the limit as n → ∞ and use (3.17), we have

This proves that the functional ψ^vλ (⋅) satisfies the C-condition.□

The next proposition will help us satisfy the mountain pass geometry for the functional ψ^vλ (⋅).

Proposition 3.2

If Hypotheses H₀, H₁ hold and λ > 0, then ψ^vλ (tû₁(p)) → −∞ as t → +∞.

Proof

We have

By hypothesis H₁(iii), we can find M > 0 and c₈ ∈ (0,c₀) such that

Using (3.19) in (3.18), we have

Hypothesis H₁(ii) implies that

In (3.20), we let t → +∞ and use (3.21). We obtain

For t > 0, we have

Recall that û₁(p) ∈ int C₊. So, we can find t ≥ 1 big such that ū_λ ≤ tû₁(p) (see [22], p.274). Then from (3.6) and since m ≥ 2, we have

for some c₁₀ > 0 (see (3.22), using Fatou′s Lemma and recall q < θ),

□

Remark 3.1

The above proof reveals that the resonance at λ̂_m(p) occurs from the right of the eigenvalue in the sense that

Proposition 3.3

If hypotheses H₀, H₁ hold and λ > 0, then Kψ^vλ ⊆ [ū_λ) ⋂ int C₊.

Proof

If Kψ^vλ = ∅, then the result is trivially true. So, suppose that Kψ^vλ ≠ ∅ and let u ∈ Kψ^vλ .We have

In (3.23), we choose h = (ū_λ − u)⁺ ∈ W01,p (Ω). We have

From (3.24), (3.6), and (3.23), it follows that

Theorem 7.1, p.286, of Ladyzhenskaya-Uraltseva [29], implies that u ∈ L^∞(Ω). Let

On account of (3.24), (2.5) and hypotheses H₀, H₁(i), we have that

We consider the following linear Dirichlet problem

Theorem 9.15, p.241, of Gilbarg-Trudinger [23] implies that problem (3.26) admits a unique solution w ∈ W02,s (Ω). The Sobolev embedding theorem says that

Let σ(z) = Dw(z). Then σ ∈ C^0,α(Ω̄,ℝ^N). Using this function, we can rewrite (3.25) as

Then the nonlinear regularity theory of Lieberman [27] and (3.24) imply that u ∈ intC₊. So, we conclude that

□

Next, we are ready to show the nonemptiness of Svλ when λ > 0 is small.

Proposition 3.4

If hypotheses H₀, H₁ hold and v ∈ C01 (Ω̄), then there exists λ^* > 0 such that for all λ ∈ (0,λ^*), we have

Proof

Let u ∈ W01,p (Ω) with ∥u∥ ≤ 1. We have

On account of hypotheses H₁(i), (iv), we see that given ε > 0, we can find c₁₁ > 0 such that

Returning to (3.27) and using (3.28) with ε > 0 small, we obtain from the Poincaré inequality and ∥u∥ ≤ 1

Let m=inf∥Dv∥qq:v∈W01,p(Ω),∥v∥=∥Dv∥p=1. Obviously, m > 0. We choose ρ ∈ (0,1) such that for all v ∈ W01,p (Ω) with ∥v∥ = 1 and u = ρv

and then choose λ^* > 0 small so that λ c₁₃ρ^1−η < η̂₀ for all λ ∈ (0,λ^*). From (3.29), it follows that

Then (3.30) together with Propositions 3.1 and 3.2, permit the use of the mountain pass theorem. So, we can find u^vλ∈W01,p(Ω) such that

□

The next proposition suggests a canonical way to choose a solution from Svλ as v ∈ C01 (Ω̄) moves.

Proposition 3.5

If hypotheses H₀, H₁ hold, v ∈ C01 (Ω̄) and λ ∈ (0,λ^*), then Svλ admits a smallest element u~vλ ∈ int C₊, that is

Proof

From Proposition 19 of Papageorgiou-Rădulescu-Repovš [4], we know that the solution set Svλ is downward directed (that is given u₁, u₂ ∈ Svλ we can find u ∈ Svλ such that u ≤ u₁, u ≤ u₂). Then Lemma 3.10, p.178 of Hu-Papageorgiou [28] implies that we can find a decreasing sequence {u_n}_n≥1 ⊆ Svλ such that infn≥1un=infSvλ. We have

If we test (3.31) with h = u_n ∈ W01,p (Ω) and use (3.32) and hypothesis H₁(i), we infer that {u_n}_n≥1 ⊆ W01,p (Ω) is bounded. So, we may assume that

In (3.31), we choose h=un−u~vλ∈W01,p(Ω), pass to the limit as n → ∞ and use (3.33). Reasoning as in the last part of the proof of Proposition 3.1, we obtain

Pass to the limit as n → ∞ in (3.31) and using (3.34), we obtain

From (3.35) and (3.36), it follows that

□

We can define the minimal solution map ξ_λ : C01 (Ω̄) → C01 (Ω̄) by

Clearly a fixed point of this map will be a positive smooth solution of (E_λ) λ ∈ (0,λ^*). To produce a fixed point of ξ_λ(⋅), we will use Theorem 2.3 (Leray-Schauder Alternative Principle). This is done in the next section.

Proposition 4.1

If hypotheses H₀, H₁ hold, v_n → v in C01 (Ω̄), λ ∈ (0, λ^*) and u ∈ Svλ , then for m ≥ 2 big we can find u_n ∈ Svnλ , n ∈ ℕ such that u_n → u in C01 (Ω̄).

Proof

From Proposition 3.3, we know that u ∈ [ū_λ) ⋂ intC₊. We consider the following Dirichlet problem

We have g^vnλ(⋅,u(⋅))≠0,g^vnλ(⋅,u(⋅))≥0 (see hypothesis H₁(i) and recall ū_λ ∈ int C₊, u¯λ−η ∈ L^s(Ω)). In fact we have