Some recent results on the Dirichlet problem for (p,q)-Laplace equations

A short account of recent existence and multiplicity theorems on the Dirichlet problem for an elliptic equation with $(p,q)$-Laplacian in a bounded domain is performed. Both eigenvalue problems and different types of perturbation terms are briefly discussed. Special attention is paid to possibly coercive, resonant, subcritical, critical, or asymmetric right-hand sides.


Introduction
Let Ω be a bounded domain in R N with a C 2 -boundary ∂Ω, let 1 < q ≤ p < N, and let µ ∈ R + 0 . Consider the Dirichlet problem −∆ p u − µ∆ q u = g(x, u) in Ω, u = 0 on ∂Ω, (1.1) where ∆ r , r > 1, denotes the r-Laplacian, namely ∆ r u := div(|∇u| r−2 ∇u) ∀ u ∈ W 1,r 0 (Ω), p = q iff µ = 0, while g : Ω × R → R satisfies Carathéodory's conditions. The nonhomogeneous differential operator Au := ∆ p u + µ∆ q u that appears in (1.1) is usually called (p, q)-Laplacian. It stems from a wide range of important applications, including biophysics [14], plasma physics [37], reaction-diffusion equations [2,3], as well as models of elementary particles [4,5,6,13]. That's why the relevant literature looks daily increasing and numerous meaningful papers on this subject are by now available. This survey provides a short account of some recent existence and multiplicity results involving (1.1). To increase readability, we chose to report special but significant cases of more general theorems, and our statements are often straightforward extensions of known results. We refer to the original papers for the most up-to-date research on this topic. Section 2 contains basic properties of the operator u → −∆ p u − µ∆ q u, u ∈ W 1,p 0 (Ω). In particular, a scaling argument shows that if µ > 0 then there is no loss of generality in assuming µ := 1, which we will make throughout the work. Eigenvalue problems are treated in Section 3, where, accordingly, g(x, t) := α|t| p−2 t + β|t| q−2 t ∀ (x, t) ∈ Ω × R, with (α, β) ∈ R 2 . The results we present are chiefly taken from [9,12,23,35,36] and concern bounded domains; for the whole space see, e.g., [7,8,17]. Section 4 collects existence and multiplicity theorems, sometimes with a precise sign information. We shall suppose g purely autonomous as well as of the type g(x, t) := α|t| p−2 t + β|t| q−2 t + f (t), (x, t) ∈ Ω × R, where the perturbation f lies in C 1 (R) and exhibits a suitable growth rate at ±∞ and/or at zero. Results are diversified according to the asymptotic behavior at infinity, frequently under the simplified condition for some r > 1. In particular, Section 4.1 concerns the (p − 1)-sublinear case r ≤ p, possibly with coercivity and/or resonance additional assumptions. Section 4.2 treats the (p − 1)-superlinear case r > p and both subcritical, i.e., r < p * := N p N −p , or critical, namely r = p * , situations are examined. Finally, Section 4.3 deals with asymmetric nonlinearities, meaning that the asymptotic behavior at −∞ and +∞ is different. Due to limited space, we relied only on [10,12,16], [21]- [24], [26]- [29], [31,32], [38]- [40], where bounded domains are considered, and refer to [8,11,15,17] for the case Ω := R N .
We surely forgot to mention here significant works, something of which we apologize in advance.
2 Basic properties of the (p, q)-Laplacian Let 1 < q ≤ p < N and let µ ∈ R + 0 . We denote by p ′ the dual exponent of p, i.e., p ′ := p/(p − 1), while p * is the Sobolev dual in dimension N, namely If r ∈ [1, +∞) then Write A µ p,q for the differential of the C 1 , strictly convex functional Hence, where, as usual, ·, · indicates the duality coupling between W 1,p 0 (Ω) and its topological dual W −1,p ′ (Ω). Setting µ := 0 we obtain the p-Laplacian A p while µ = 1 produces the so-called (p, q)-Laplacian. Let us now discuss some features of A µ p,q .

-Homogeneity
The operator A µ p,q is not homogeneous whenever p = q. This prevents to apply, in a simple way, available tools from critical point theory. However, a reasoning analogous to the one exploited before allows to take advantage from such a trouble. Indeed, if u ∈ W 1,p (Ω) is a solution of the quasilinear equation A µ p,q (u) = g(·, u) then Setting µ := k q−p , the function v := µ 1/(q−p) u solves We thus achieve an equation for A p,q where the reaction exhibits a different asymptotic behavior both at zero and at infinity.
-Eigenvalues of A p Eigenvalues of A p play a key rôle in solving quasilinear equations where A p,q appears. So, some basic properties will be recalled. Denote by σ(A p , W 1,p 0 (Ω)) the spectrum of A p with (zero) Dirichlet boundary conditions. A whole sequence of variational eigenvalues can be constructed via the Ljusternik-Schnirelmann minimax scheme with Z 2 co-homological index [1, Section 4.2]. If p = 2 it reduces to the usual spectrum of (−∆, H 1 0 (Ω)), while for general p > 1 one has Moreover, as long as Ω is connected, the eigenspace corresponding to λ 1 (p) is onedimensional and spanned by a positive function u 1,p ∈ C 1,α (Ω), whereas eigenfunctions relative to higher eigenvalues must be nodal.

-Regularity
The regularity theory for A p,q is well established after the works of Lieberman in the eighties (see, e.g., [20]) and parallels that concerning A p . In particular, weak solutions u ∈ W 1,p 0 (Ω) to the equation are C 1,α up to the boundary. Under natural conditions on g, a strong maximum principle as well as the Hopf boundary point lemma hold true. Precisely, define Its positive cone C + := {u ∈ C 1 0 (Ω) : u(x) ≥ 0 in Ω} has a nonempty interior given by where n(x) denotes the outward unit normal to ∂Ω at x. Because of [34, Theorems 5.5.1 and 5.3.1], if there exists δ > 0, C > 0 such that then, under (2.2), nonnegative nontrivial solutions to (2.1) actually lie in int(C + ). A similar statement holds true for nonpositive solutions.

Eigenvalue problems
In analogy with the Fučik spectrum, the eigenvalue problem for A p,q with homogeneous Dirichlet boundary conditions on a bounded domain Ω consists in finding all (α, β) ∈ R 2 such that the equation possesses a nontrivial weak solution u ∈ W 1,p 0 (Ω). The set of such (α, β) is called the (p, q)spectrum of A p,q and denoted by σ p,q . Additionally, σ + p,q indicates the set of (α, β) ∈ σ p,q α β λ 1 (p) for which there exists a positive solution to (3.1). This problem can evidently be recasted in the more general framework of weighted eigenvalues, namely to the equation where ξ and η are bounded functions, and most of the results presented here have suitable weighted (sometimes even sign-changing) variants. For the sake of simplicity, we will focus on the constant coefficient case. Nevertheless, a full description of σ p,q is out of reach, since it clearly presents additional difficulties with respect the comparatively simpler case of the spectrum of A p , a well-known open problem until today. Partial results on σ + p,q are scattered in the literature, since positive eigenfunctions turn out to be a useful tool for studying more general quasilinear problems. One always has Moreover, a quite complete picture of σ + p,q is available, according to whether the following conjecture holds true or not.
Let u 1,p and u 1,q be first eigenfunctions for A p and A q respectively. If p = q then u 1,p and u 1,q are linearly independent. (3.2) , Section 2). Suppose 1 < q < p < N. Then If, moreover, (3.2) is satisfied then there exists a continuous non-increasing function λ * : Therefore, the support of γ plays the rôle of a threshold for the existence of positive eigenfunctions, as described in Figure 1. More precisely, any line Some natural properties of λ * are yet to be understood. As an example, is it true that λ * (t) + t ≡ λ 1 (p) for t sufficiently negative? Besides, while some partial results are known on the borderline case σ + p,q ∩γ(R), the picture looks not complete until today. Let us finally point out that, still in [9], the case when Conjecture (3.2) fails is discussed, providing a simpler description of σ + p,q . This is particularly meaningful for weighted eigenvalue problems, where it may occur that (3.2) does not hold for some weights.
Let us mention that the existence results of [12,23] actually deal with more general equations, as we will see later. Eigenvalue problems on the whole space are investigated in [7,8,17].

Multiplicity results
In this section we will discuss existence and multiplicity of solutions to the general quasilinear Dirichlet problem where g : Ω × R → R satisfies Carathéodory's conditions. For the sake of simplicity, the case g smooth and purely autonomous, i.e. g(x, t) =ĝ(t) withĝ ∈ C 1 (R), is treated.
Most of the results can be recasted as perturbed eigenvalue problems, which meanŝ where the behavior of f at ±∞ and/or at zero is typically negligible when compared with the leading term α|t| p−2 t + β|t| q−2 t. To fix ideas, the sample equation is for various α, β, γ ∈ R and r > 1, the last term representing the perturbation. Let 1 < q < p < N and let f ∈ C 1 (R). We say that the problem is subcritical if f fulfills the growth rate with appropriate r ∈]1, p * [ and critical when this inequality holds true for r = p * but not for smaller r's. Moreover, (p − 1)-sublinear signifies r ≤ p, or, more generally, Otherwise (4.2) is called (p − 1)-superlinear. Finally, we say that Problem (4.2) turns out to be asymmetric if f (or F ) exhibits a different asymptotic behavior at ±∞. Due to the expository nature of the present work, we won't examine here possible formulations of the Ambrosetti-Rabinowitz (briefly, (AR)) condition on the nonlinearities involved. This assumption, especially fruitful to achieve the Palais-Smale condition for (p − 1)-superlinear problems, has been the object of many research papers in recent years. A deep discussion of various generalizations is made in [18] while [11,26,28] contain applications to (p, q)-Laplacian problems. We choose to somewhat oversimplify statements, substituting Condition (AR) with easier ones. So, the theorems given below are actually true under less restrictive hypotheses than those taken on here.

(p − 1)-sublinear problems
The present section collects some multiplicity results devoted to the sublinear case. Accordingly, the perturbation f will fulfill

Coercive setting
This means that the energy functional stemming from (4.2), namely turns out to be coercive. As long as the behavior of f (t) at infinity is negligible with respect to |t| p−1 , we are thus studying (4.2) for α < λ 1 (p). Very recently, a multiplicity result has been proved in [31] under coercivity conditions. The associated sample equation is (4.1) with q = 2, r ∈ ]2, p[, γ > 0. The more general case 1 < q < p is treated in [23]. It corresponds to r ∈ [p, p * [ and γ < 0 for the model equation (4.1).
A concave perturbation is added in [24], obtaining the next multiplicity result.
Under additional hypotheses on F , a fifth solution is found in the same paper.
• Resonance from the right. On the other hand, the condition gives at least one nontrivial solution.

Problems neither coercive nor resonant
Let us now come to the case where f is (p − 1)-sublinear but the energy functional associated with (4.2) fails to be coercive and is indefinite. This occurs when, e.g., The following result has already been mentioned in Section 3 for f ≡ 0.
where m(β) is as in Theorem 3.2, then (4.2) possesses at least one nontrivial solution.
The sample equation for the next result is (4.1) with q := 2, p < r, γ < 0. The setting is not purely resonant (meaning that α = λ 1 (p) is not allowed in (4.2)), but falls inside the so-called near resonance problems.
for appropriate 2 < r < p then (4.2) has a nontrivial solution.
Analogous results on the whole space, which causes further difficulties, are established in [8,17]. Finally, we refer to the survey paper [25] for non-variational problems involving the (p, q)-Laplace operator.

Subcritical setting
Through the Nehari manifold approach, a ground state solution (namely a positive solution, which minimizes the associated energy functional) has been obtained in [16].
Then (4.2) possesses a ground state solution.
A more sophisticated result is proved in [28].
For additional results on the whole space, see [8,11].

Critical setting
In this framework, the most relevant term behaves as |u| p * −2 u. Thus, we shall be concerned with equations of the type where the perturbation h is strictly subcritical. All the results we will present involve odd nonlinearities, so that any positive solution directly gives rise to a negative one. Under the hypothesis 1 < r < q < p < N, a first result on the sample problem u ∈ W 1,p 0 (Ω), A p,q (u) = |u| p * −2 u + γ|u| r−2 u (4.3) has been obtained in [19] and then generalized as follows.
Let us note that Theorem 4.11 actually extends to more general equations of the form (4.4), still for r ∈ ]p, p * [, cf. [39,.
Then paper [10] contains further existence results concerning the situation 1 < q < p < q * < N.
It should be noted that, even for the sample problem (4.3), the intermediate case q < r < p is, as far as we know, still open. Finally, the existence of ground states for critical equations on the whole space is studied in [15].

Asymmetric nonlinearities
We now discuss quasilinear Dirichlet problems with reactions having different asymptotic behaviors at −∞ and +∞. The sample equation stems from the Fučik spectrum theory, and is of the type where t ± := max{0, ±t} while f exhibits suitable (possibly asymmetric as well) growth rates at zero and ±∞. For the sake of simplicity, we singled out the Fučik structure only on the higher order term of the reaction with respect to which resonance can occur, but, as before, most of theorems can be recasted in a more general setting. A first result in this framework is obtained provided 1 < q < p := 2, both for subcritical and critical nonlinearities.  1 and 1.2). Suppose 1 < q < s < p := 2 < r ≤ 2 * , α ∈ ]λ 1 (2), +∞[ \σ 2 , and λ > 0. Then the problem u ∈ W 1,2 0 (Ω), A 2,q (u) = αu − λ|u| s−2 u + µu r−1 + admits at least three nontrivial solutions if µ > 0 is sufficiently small and either r < 2 * , N ≥ 3 or r = 2 * , N ≥ 4.
Let us finally point out that infinitely many solutions are also obtained in [22] under a symmetry condition near zero. For further results concerning asymmetric nonlinearities, see [30,33].