Implicit Equations Involving the p-Laplace Operator

In this work we study the existence of solutions u∈W01,p(Ω)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u \in W^{1,p}_0(\Omega )$$\end{document} to the implicit elliptic problem f(x,u,∇u,Δpu)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f(x, u, \nabla u, \Delta _p u)= 0$$\end{document} in Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Omega $$\end{document}, where Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Omega $$\end{document} is a bounded domain in RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathbb {R}}^N $$\end{document}, N≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ N \ge 2 $$\end{document}, with smooth boundary ∂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \partial \Omega $$\end{document}, 1<p<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ 1< p< \infty $$\end{document}, and f:Ω×R×RN×R→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f:\Omega \times {\mathbb {R}}\times {\mathbb {R}}^N \times {\mathbb {R}}\rightarrow {\mathbb {R}}$$\end{document}. We choose the particular case when the function f can be expressed in the form f(x,z,w,y)=φ(x,z,w)-ψ(y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ f(x, z, w, y)= \varphi (x, z, w)- \psi (y) $$\end{document}, where the function ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \psi $$\end{document} depends only on the p-Laplacian Δpu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \Delta _p u $$\end{document}. We also present some applications of our results.

We consider the special case f (x, z, w, y) = ϕ(x, z, w)−ψ(y), with ϕ : Ω×R× R N → R and ψ : Y → R. We require that ψ depends only on Δ p u. We further distinguish among the case when ϕ is a Carathéodory function depending on x, u, and ∇u, and the case when ϕ is allowed to be highly discontinuous in each variable. In this last case, the dependence on the gradient is no more allowed.
In both situations we first reduce problem (1.1) to an elliptic differential inclusion, but methods used are different and depend on the regularity of the function ϕ and on the structure of the problem. More precisely, in the first case we make use of a result in [19] to obtain the inclusion − Δ p u ∈ F (x, u, ∇u), (1.2) where F is a lower semicontinuous selection of the multifunction (x, z, w) → {y ∈ Y : ϕ(x, z, w) − ψ(y) = 0}.
We start with the general case Y = R and then we deduce, as a byproduct, the existence result when Y is a closed interval of R.
Existence of solutions to (1.2) is obtained by means of the following result, which is based on a selection theorem for decomposable-valued multifunctions, see [2,13]. Then, (1.2) has a solution u ∈ W 1,p 0 (Ω).
Here, λ 1,p is the first eigenvalue of the p-Laplacian in the space W 1,p 0 (Ω). The following is our main result, which extends [13,Theorem 3.2] to the case p = 2.
Then, there exists u ∈ W 1,p 0 (Ω) such that ψ(−Δ p u) = ϕ(x, u, ∇u) in Ω. When ϕ is discontinuous we essentially follow [16,Theorem 3.1] to construct an appropriate upper semicontinuous multifunction F related with ψ −1 and ϕ, and then we solve the elliptic differential inclusion −Δ p u ∈ F (x, u) using the following Theorem 1.3 (Theorem 2.2 of [14]). Let U be a nonempty set, let Φ: U → W 1,p 0 (Ω) and Ψ : U → L p (Ω) be two operators, and let F : has a closed graph for almost every x ∈ Ω; (i 4 ) There exists r > 0 such that the function Extending [16,Theorem 3.1] to the case p = 2, we obtain the following result. We denote by π 0 and π 1 the projections of Ω × R on Ω and R, respectively.

Structure of the Paper
In Sect. 2 we will introduce the functional analytic setting we will use throughout the work. In Sect. 3 we will suppose ϕ(x, · , ·) to be continuous. Here we will consider some cases, according to the growth conditions on ϕ or to the choice of the set Y . We will also give examples where these situations apply. In Sect. 4 we will consider the discontinuous framework.

Preliminaries
Let X be a topological space and let V ⊂ X. We denote by int(V ) the interior of V and by V the closure of V . The symbol B(X) is used to denote the Borel σ-algebra of X.
If (X, d) is a metric space, for every x ∈ X, r ≥ 0 and every nonempty set V ⊂ X, we define Let X and Z be two nonempty sets. A multifunction Φ from X into Z (symbolically Φ : X → 2 Z ) is a function from X into the family of all subsets of Z. A function ϕ : Suppose that (X, A) is a measurable space and Z is a topological space. We say that the multifunction Φ is measurable if for every open set W ⊂ Z we have Φ − (W ) ∈ A. Suppose now that X and Z are two topological spaces. We say that Φ is lower semicontinuous (resp. upper semicontinuous) if for every open (resp. closed) set W ⊂ Z the set Φ − (W ) is open (resp. closed) in X. When (Z, δ) is a metric space, the multifunction Φ is lower semicontinuous if and only if, for every z ∈ Z, the real-valued function x → δ(z, Φ(x)), x ∈ X, is upper semicontinuous (see [20,Theorem 1.1]). If, moreover, X is first countable, then Φ is lower semicontinuous if and only if, for every x ∈ X, every sequence {x k } in X converging to x and every z ∈ Φ(x), there exists a sequence {z k } in Z converging to z and such that z k ∈ Φ(x k ), for all k ∈ N (see [10,Theorem 7 A general result on the lower semicontinuity of a multifunction is the following Then, the multifunction Q is lower semicontinuous, with nonempty closed values. From now on, Ω is a bounded domain in R N , N ≥ 2, with a smooth boundary ∂Ω. The symbol L(Ω) denotes the Lebesgue σ-algebra of Ω, while m(Ω) stands for the measure of Ω.
Let 1 ≤ r < ∞. We denote by L r (Ω), L r (Ω, R N ), and W 1,r (Ω) the usual Lebesgue and Sobolev spaces equipped with the norms · r and · 1,r given by For r = ∞ we recall that the norm of L ∞ (Ω) is given by It is well known that the Sobolev embedding theorem guarantees the existence of a linear, continuous map i : , with the critical exponent given by In particular, the embedding W 1,p 0 (Ω) → L r (Ω) is compact provided 1 ≤ r < p * .
If p = N , then to each r ∈ [1, p * ] there corresponds a constant c rp > 0 satisfying Given p ∈ (1, ∞), the symbol p denotes the conjugate exponent of p while W −1,p (Ω) stands for the dual space of W 1,p (Ω), with corresponding norm · −1,p . From [3, Theorem 6.4] we have the compact embedding L p (Ω) → W −1,p (Ω), and therefore there exists b > 0 such that and let λ 1,p be its first eigenvalue in W 1,p 0 (Ω). The following facts are well known (see, e.g., [18,Appendix A] or [11]): (p 1 ) A p is bijective and uniformly continuous on bounded sets;

The Case When ϕ is a Carathéodory Function
In this section we consider the following problem: find u ∈ W 1,p 0 (Ω) such that Δ p u ∈ L p (Ω) and We first suppose that Y = R and state the following assumptions Theorem 3.1. Let ϕ : Ω × R × R N → R be a Carathéodory function and let ψ : R → R be continuous. Suppose that (i)-(ii) hold true and, moreover, Then, there exists a solution u ∈ W 1,p 0 (Ω) to Eq. (3.1).
Proof. Fix x ∈ Ω. We want to apply Theorem 2.1. To this end, we choose Hypothesis (ii) directly yields (a). Moreover, in order to verify (b), we need to check that for all (z, w) ∈ R×R N the set U := {y ∈ R : ϕ(x, z, w)−ψ(y) = 0} is dense in R. Assumption (i) implies that R\U has empty interior, therefore U is dense in R, as desired. Let us next consider the set We want to show that A is dense in R×R. Suppose that there exist y , y ∈ R such that that is, ϕ(x, z, w) ∈ (ψ(y ), ψ(y )). Then the continuity of the function ϕ(x, · , ·) implies that the set is open. If it is not possible to find such y , y that realize (3.2), then the set B is empty. This implies that A = R × R, and then (c) follows. Thanks to Theorem 2.1, the multifunction F (x, · , ·) is lower semicontinuous, with nonempty closed values.
Moreover, thanks to [6, Lemma III.14], for all y , y ∈ R we have Arguing again as in [13,Theorem 3.2] we see that Then (3.3) implies that F − (A) ∈ L(Ω) ⊗ B(R × R N ) and therefore F is measurable. Finally, fix any y ∈ F (x, z, w). By hypothesis (iii) we have Therefore, all the hypotheses of Theorem 1.1 are satisfied, and there exists u ∈ W 1,p 0 (Ω) such that −Δ p u = F (x, u, ∇u). By definition of F we then have the result.
Since lim y→±∞ (y−λ sin y) = ±∞, the function y → ϕ(x, z, w)−ψ(y) changes sign, and then hypothesis (ii) follows. Moreover, ψ vanishes only at points of R and not in intervals, which implies that also hypothesis (i) is satisfied.
Since lim y→±∞ (y + λe y ) = ±∞, then hypotheses (i) and (ii) are fulfilled. In order to verify hypothesis (iii), we argue as in Corollary 3.3. Letỹ be a solution to ϕ(x, z, w) − ψ(y) = 0, then Young's inequality with exponents On the other hand we have In order to state our next theorem, we need some preliminary results. The following is an a priori estimate on ∇u L ∞ (Ω;R N ) , see [ Proposition 3.5 is used in the proof of the following Theorem 3.6. Let p ∈ (1, ∞), q > N, and let F : where δ Ω := diam(Ω) andĈ is given by Proposition 3.5.

Then, there exists at least one solution
Finally, we state our result.
Proof. We aim to apply Theorem 3.6. As before, fix x ∈ Ω and for all (z, w) Reasoning as in Theorem 3.1 ensures that F has nonempty closed values, is lower semicontinuous w.r.t. (z, w), and L(Ω) ⊗ B(R × R N )-measurable. Fix now y ∈ F (x, z, w), that is y ∈ ψ −1 (ϕ(x, z, w)). Then hypothesis (iii) implies that Taking into account (iv), we see that all the hypotheses of Theorem 3.6 are fulfilled. Therefore, there exists u ∈ W 1,p 0 (Ω) such that −Δ p u ∈ F (x, u, ∇u). According to the definition of F , it turns out that u is a solution to Eq. (3.1).
The following result is an application of the previous theorem and has been inspired by [9,Corollary 1]. Observe that, unlike [9], here we consider a function ϕ which is not necessarily continuous w.r.t. the variable x, but only lies in a suitable L q (Ω). Moreover, here we deal with partial differential equations. Proof. Fix k ∈ R and for all (x, z, w) ∈ Ω × R × R N and all y ∈ R define ϕ(x, z, w) := h(x) + z 3 + |w| 2 as well as ψ(y) := y − k sin y.
In order to verify hypothesis (iv), we have to check the existence of R > 0 such that If 0 < R << 1, then choosingh in such a way that h L q (Ω) < R 2 gives immediately (3.10), since the terms containing R 2 and R 3 are negligible with respect to R. Therefore, all the hypotheses of Theorem 3.7 are fulfilled, and we have the thesis. The next result provides solutions to Eq. (3.1) when the function ψ is of the form y → y − h(y), with h continuous and bounded. Note that here a specific growth condition on ϕ is required. Theorem 3.9. Let ϕ : Ω × R × R N → R be a Carathéodory function and let h ∈ L ∞ (R) be continuous. Suppose that (i)-(ii) hold true and, moreover, Then, there exists a solution u ∈ W 1,p 0 (Ω) to the equation ϕ(x, u, ∇u). (3.11) Proof. We fix x ∈ Ω and for all (z, w) ∈ R × R N define Reasoning as in the above proofs ensures that F is lower semicontinuous w.r.t. (z, w), L(Ω) ⊗ B(R × R N )-measurable, and has nonempty, closed values.
We now consider two applications of the previous result, which differ by the behavior of the function ψ. In both cases, the boundedness of ϕ will play a central role. Proof. Assumption (1) is clearly satisfied. Moreover, for every x ∈ Ω, we have which gives hypothesis (2). Thanks to Theorem 3.10, there exists at least a solution u ∈ W 1,p 0 (Ω) to Eq. (3.12). Note that the interval [α, β] could be unbounded, as the following example shows. Proof. Define ψ(y) := λe −y − y for every y ∈ [0, +∞). Observe that hypothesis (1) is immediately satisfied. Moreover, thanks to (3.13), for every that is hypothesis (2), and hence the conclusion follows from Theorem 3.10.