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

Abstract:We consider a nonlinear parametric Dirichlet problem driven by the (p,q)-Laplacian (double phase problem)with a reaction exhibiting the competing e ects of three di erent terms.Aparametric one consisting of the sum of a singular term and of a drift term (convection) and of a nonparametric perturbation which is resonant. Using the frozen variable method and eventually a xed point argument based on an iterative asymptotic process, we show that the problem has a positive smooth solution.

In problem (E λ ) we have the sum of two such operators with di erent exponents. Hence the di erential operator of the problem is not homogeneous. In the reaction (right hand side of (E λ )), we have the combined e ects of three terms, each of di erent nature. There is a parametric contribution which is the sum of a singular term and of a gradient dependent term (a drift term). Both are multiplied with the parameter λ > . So, we have dependence on the gradient of u (convection). The third term is the perturbation f (z, x) which is a Carathéodory function (that is, for all x ∈ R z → f (z, x) is measurable and for a.a z ∈ Ω x → f (z, x) is continuous). We assume that f (z, ·) exhibits (p-1)-linear growth as x → +∞ and it is resonant with respect to a nonprincipal variational eigenvalue of (∆p , W ,p (Ω)). We are looking for positive solutions of the problem. The presence of the drift term makes the problem nonvariational and so ultimately our method of proof will be topological. Our approach uses the so called "frozen variable method". More speci cally, we x (freeze) the gradient (drift) term and this way we have a variational problem. However, the presence of the singular term is a source of di culties, since the energy (Euler) functional of this variational problem is not C and so the results and methods of critical point theory can not be used directly on this function. We need to nd a way to bypass the singularity and work with a C -functional. This is done by introducing an auxiliary problem which we solve and then use its solution and suitable truncation techniques to neutralize the singularity. We are able to show that for all small parameter values, the "frozen problem" has a positive solution. Next, we need to nd a canonical way to choose a solution of the "frozen problem". To this end, we show that each such problem has a minimal positive solution (a smallest positive solution). We choose this solution and we have the minimal positive solution map. We show that this map has a xed point using the Leray-Schauder Alternative Principle (see Theorem 4). To show that the minimal solution map is compact (requirement in the Leray-Schauder theorem), we employ an iterative asymptotic process.

Mathematical Background, Hypotheses
The main spaces in the study of problem (E λ ) are the Sobolev space W ,p (Ω) and the Banach space C (Ω) = {u ∈ C (Ω) : u| ∂Ω = }. By · , we denote the norm of the Sobolev space W ,p (Ω). From the Poincaré inequality, we have u = Du p for all u ∈ W ,p (Ω).
The Banach space C (Ω) is ordered with positive (order) cone C+ = {u ∈ C (Ω) : u(z) ≥ for all z ∈ Ω}. This cone has a nonempty interior given by with n(·) being the outward unit normal on ∂Ω.
We will also use a weighted version of the eigenvalue problem (2.1). So, let m ∈ L ∞ (Ω), m(z) ≥ for a.a. z ∈ Ω, m ≠ . We consider the following nonlinear eigenvalue problem (2. 3) The same results hold for the eigenvaluesλ(p, m) of problem (2.3). In this case the Rayleigh quotient in the variational characterization ofλ (p, m) > , is given by for all u ∈ W ,p (Ω).
Then using this fact, we can easily prove the following monotonicity property of the map m →λ (p, m).
Now, we introduce the hypotheses on the data of (E λ ).
Remark 2.2. Since our goal is to nd positive solutions and the above hypotheses concern the positive semiaxis R+ = [ , +∞), without any loss of generality, we may assume that Hypothesis H (ii) implies that the problem is resonant with respect to a nonprincipal variational eigenvalueλm(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: 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 ). Consider the Banach space C (Ω) = {u ∈ C(Ω) : u| ∂Ω = }. This is an ordered Banach space with positive (order) cone K+ = {u ∈ C (Ω) : u(z) ≥ for all z ∈ Ω}. This cone has a nonempty interior given by intK+ = {u ∈ K+ : cud ≤ u for some cu > }, withd(z) = d(z, ∂Ω) for all z ∈Ω. According to Lemma 14.16, p. 355, of Gilbarg-Trudinger [23], there exists δ > such thatd ∈ C (Ω δ ), where Ω δ = {z ∈Ω : d(z, ∂Ω) < δ }. It follows thatd ∈ intC+ and so from Proposition 4.1.22, p.274, of Papageorgiou-Rădulescu-Repovš [22], we can nd < c < c such that Let s > N and considerû (p) s ∈ K+. Then using Proposition 4.1.22, p. 274, of Papageorgiou-Rădulescu-Repovš [22], we know that we can nd c > such that In the next section, this fact will help us bypass the singular term and use variational tools on the "frozen" problem.

The "Frozen" Problem
In this section, we develop the "frozen variable method" for problem (E λ ). So, we x v ∈ C (Ω) and consider the Carathéodory function As we already mentioned in the Introduction, due to the singular term λx −η , the function g λ v (z, x) leads to an energy function which is not C . For this reason, we use (2.5) and consider the following truncation of Then we consider the following parametric double phase Dirichlet problem ("the frozen problem") This problem is variational and its energy (Euler) function is C . So, we can use the results of critical point On account of (2.5) (see also Papageorgiou-Smyrlis [25], Proposition 3), we have that ψ λ v ∈ C (W ,p (Ω)).
From (2.5),(3.6) and hypothesis H (i), we see that In (3.13) we choose h = yn − y ∈ W ,p (Ω), pass to the limit as n → ∞ and use (3.11) (recall q < p), We may assume that This proves that the functionalψ λ v (·) satis es the C-condition.
The next proposition will help us satisfy the mountain pass geometry for the functionalψ λ v (·).
Proof. From Proposition 19 of Papageorgiou-Rădulescu-Repovš [4], we know that the solution set S λ v is downward directed (that is given u , u ∈ S λ v we can nd u ∈ S λ v such that u ≤ u , u ≤ u ). Then Lemma 3.10, p.178 of Hu-Papageorgiou [28] implies that we can nd a decreasing sequence {un} n≥ ⊆ S λ v such that inf n≥ un = inf S λ v . We have Pass to the limit as n → ∞ in (3.31) and using (3.34), we obtain

Ap(un), h + Aq(un), h =
We can de ne the minimal solution map ξ λ : C (Ω) → C (Ω) by Clearly a xed point of this map will be a positive smooth solution of (E λ ) λ ∈ ( , λ * ). To produce a xed point of ξ λ (·), we will use Theorem 2.3 (Leray-Schauder Alternative Principle). This is done in the next section.

Positive Solution
According to Theorem 2.3, to produce a xed point for the minimal solution map, we need to show that ξ λ (·) is compact and that D(ξ λ ) is bounded (λ ∈ ( , λ * )). First, we show that ξ λ (·) is compact. To this end the following proposition will be helpful.
Reasoning as above, we see that this problem has a unique solution w n ∈ W ,s (Ω), w n ≥ , w n ≠ and w n → u in C (Ω) as n → ∞.
We continue this way and generate a sequence {w k n } n,k∈N such that For each n ∈ N, we consider the sequence {w k n } k∈N . We claim that this sequence is bounded in W ,p (Ω). Arguing by contradiction, we may assume that w k n → ∞ as k → ∞ and { w k n } k∈N is increasing. On account of hypothesis H (i) and of (4.38), (4.39), we see that we act on (4.43) with y k − y ∈ W ,p (Ω) and then pass to the limit as k → ∞ and use ( If m ≥ is big so that η > (λ (p)

λm(p)
) p− , then it follows that y = or y is nodal, both possibilities leading to a contradiction (see (4.45)). This proves that for every n ∈ N, {w k n } k∈N ⊆ W ,p (Ω) is bounded. Then Theorem 7.1, p.286, of Ladyzhenskaya-Uraltseva [29] implies that {w k n } k∈N ⊆ L ∞ (Ω) is bounded. The nonlinear regularity theory of Lieberman [27] implies that for each n ∈ N, we can nd α ∈ ( , ) and c > such that w k n ∈ C ,α (Ω), w k n C ,α (Ω) ≤ c for all k ∈ N.
As before, the compact embedding of C ,α (Ω) into C (Ω), implies that at least for a subsequence, we have Passing to the limit as k → ∞ in (4.39), we obtain in Ω, un| ∂Ω = . (4.48) Moreover, as in the proof of Proposition 3.3, using (3.6) and the fact that f ≥ (see hypothesis H (i)), we havē u λ ≤ w k n for all n, k ∈ N, ⇒ū λ ≤ un for all n ∈ N, ⇒ un ∈ S λ vn ⊆ int C+ for all n ∈ N.
As above, using (4.48) and a contradiction argument, we show that {un} n≥ ⊆ W ,p (Ω) is bounded, hence relatively compact in C (Ω) (nonlinear regularity). Then the double limit lemma (see , Problem 1.175, p.61), implies that un → u in C (Ω) as n → ∞, un ∈ S λ vn for all n ∈ N.
Using this proposition, we can show that the minimal solution map ξ λ (·) is compact. Proof. First, we show that ξ λ (·) maps bounded sets in C (Ω) to relatively compact sets in C (Ω). So, let D ⊆ C (Ω) be bounded. Then as in the proof of Proposition 4.1, with a contradiction argument and using the fact that m ≥ is big, we show that ξ λ (D) ⊆ W ,p (Ω) is bounded and from this by the nonlinear regularity theory (see [27,29]), we obtain that ξ λ (D) ⊆ C (Ω) relatively compact. Next, we show that ξ λ (·) is continuous. So, let vn → v in C (Ω) and letũ λ v = ξ λ (v). According to Proposition 4.1, we can nd un ∈ S λ vn , n ∈ N such that un →ũ λ v . (4.49) We know that ξ λ (vn) ≤ un for all n ∈ N, (4.50) and from the rst part if the proof, we have that ξ λ (vn) n∈N ⊆ C (Ω) is relatively compact.
Therefore, we have proved that the minimal map ξ λ (·) is compact.
On account of Propositions 4.2 and 4.3, we can apply Theorem 2.3 and produce a xed point for ξ λ (·), λ ∈ ( , λ * ). So, we can state the following existence theorem for problem (E λ ).