Existence and regularity results for a degenerate elliptic nonlinear equation on weighted Sobolev spaces

Abstract This paper deals with the existence of many solutions a degenerate weighted elliptic equation which its the number of solutions depend on degeneracy term a i.e., the number of subdomains of Ω\a−1(0) whose boundary is made by submanifolds with 1-codimension. Moreover, by the Ljusternik-Schnirelman principle, we would study the corresponding eigenvalue problem and obtained the nonnegative eigenvalues sequence of main problem. Finally, we will discuss of the regularity of the solutions on C1,α (Ω).


Introduction.
In this article, we will investigate, under suitable conditions, the existence and regularity results for a weighted degenerate nonlinear elliptic system with a potential term as following −div(a(x)∇u) + V (x)u = f (x, u), where the degeneracy term a : Ω → [0, +∞) is a positive continuous function on the whole of domain Ω except for a 1-codimensional submanifolds contained in Ω where it vanishes.Moreover, V (x) is a potential function of the system which 158 M. Alimohammady, C. Cattani and M.K. Kalleji satisfies on a suitable assumption and f (x, u) is a nonlinear term of the system.These systems are called degenerate due to unboundedness a −1 .
It is well-known that many physical phenomena are described by degenerate evolution equations.For example, degenerate equations are appropriate to describe fluid diffusion in nonhomogeneous porous media taking into account saturation and porosity of the medium.For more applications and problems involving degenerate operators one can see [5,10].In mathematical view, the degenerate systems contain equations with singularities in the coefficients.Therefore, the researchers usually study and investigate the solutions of these type systems in weighted Sobolev spaces.
In this moment, we recall some definitions and preliminary results about the weighted Sobolev spaces which is used in the next sections.For more details and applications of weighted Sobolev spaces see [1,12].First, let us assume that Ω ⊂ R N be a smooth and bounded domain.Similar to [9], a Muckenhoupt weight function in class A p for p > 1 is a function w : Ω → [0, +∞) which satisfies sup where the supremum is taken over all balls B ⊂ Ω. Suppose that p ≥ 1, the weighted Lebesgue space L p (Ω, w) is a Banach space of measurable functions u : Ω → R such that It is well-known that if the weight function w(x) is in Muckenhoupt A p class, then L p (Ω, w) ⊂ L 1 loc (Ω).Therefore, it makes sense to speak about weak derivatives and Sobolev spaces.By the similar way in [1,12], the weighted Sobolev space H 1 (Ω, w) is the set of functions u ∈ L 2 (Ω, w) such that the weak derivative of first order are all in L 2 (Ω, w).The weighted Sobolev space H 1 (Ω, w) is a Hilbert space with the norm Moreover, one can consider H 1 0 (Ω, w) as the closure of C ∞ c (Ω) with respect to the following norm The following results are important and fundamental in this text.

Existence main results.
Consider the following degenerate weighted elliptic nonlinear equation According to [11], one can define u * ∈ D ′ (Ω) as a solution when u * ∈ C 1 (Ω) and satisfies the following identity in the distribution sense: On the other hand, it follows from 2.1 that In particular case, if V (x) = a(x) and x ∈ a −1 (0) then it follows from 2.1 that f (x, u * ) = 0. Thus u * (x) = 0 is a solution of our problem.Motivated by this fact, we investigate in this section the existence of weak solutions for a degenerate weighted elliptic operator in a bounded domain with Dirichlet boundary condition and with the additional condition that our solutions are zero on the set where the degeneracy term a is zero.
In order to do this, we consider the following assumptions : Let Ω ⊂ R N , N ≥ 2 be a smooth and bounded domain, a ∈ C(Ω), a(x) ≥ 0 and f (x, .)∈ C(R) are functions satisfying: union of a finite number k of compact, connected without boundary and 1-codimensional smooth submanifolds Γ l of R N , a2) a ∈ A 2 , the standard Mouckenhoupt class and 1 a ∈ L t (Ω) for some t > N 2 , a3) a(x) V (x), i.e., there exists a positive constant C such that a(x) ≤ CV (x), f 1) with respect to the second variable, f (x, .)has a local minimum with f (x, 0) = 0 and there is s * > 0 such that f (x, s * ) = 0 and f (x, s) > 0 for any s ∈ (0, s * ) and x ∈ Ω, and M j := max x∈Dj (a(x)+V (x)) < γ λ1(Dj ) , where λ 1 (D j ) is the first eigenvalue of the Dirichlet Laplacian in D j and D j is any connected component of Ω \ a −1 (0).
Upon the above assumptions we consider the following problem: Similar to that one [1], a weak solution of problem 2.2 is a function We note that, since a ∈ C(Ω) and f (x, .)∈ C(R), the above equality makes sense.The choice of the space W 1,1 0 (Ω \ a −1 (0)) in place of the usual space H 1 0 (Ω \ a −1 (0)) is due to the fact that we do not know if the gradient of the solution u * is in In order to obtain the main result of this section, first we investigate the existence of a weak solution for the problem 2.2.
Thus, we consider the following degenerate weighted elliptic problem (2.4) It can be seen that b(x) may be zero somewhere on the boundary ∂D.Moreover, note that the space ), using the Hölder inequality and since 2t 1+t > 1, so Theorem 2.1.Suppose that the assumptions a1), a2), a3), f 1) and f 2) hold.The problem 2.3 has at least one non-negative weak solution.
Proof.Let us consider the energy functional I : H 1 0 (D, b) → R corresponding to the degenerate problem 2.3 as follows where F is the primitive of for some α * > 0 such that f (x, s) > 0 for all s ∈ [−α * , 0).Since, functional I is well-defined and f * is bounded then there exists positive constant C such that Furthermore, from the Hölder inequality and the assumption b ∈ A 2 one obtains , from the inequalities 2.6 and 2.7 and the assumption b(x) V (x) where From the compactness of embedding theorem one can obtain that a.e. in D. From the assumption a2) . Hence, by using the Lebesgue dominated convergence theorem, Therefore, the energy functional I is weakly lower semicontinuous in This means u * is a weak solution of problem 2.3.To complete our proof, we show that the weak solution u * is non-trivial.It is enough to realize that I takes negative values.Indeed, let e 1 be a positive eigenfunction associated to the first eigenvalue λ 1 (D) of Laplacian operator in D with Dirichlet boundary condition, then for any s > 0. Using the assumption f 2), L ′ Hospital rule, Lebesgue dominated convergence theorem and passing to the limit s → 0 + , For s > 0 small enough, one gets I(u * ) ≤ I(se 1 ) < 0, show that u * is nontrivial.According to the definition of f * and assumption From the positivity of b(x) and V (x) one can obtain that ∇u * − = 0 a.e. in D. 4 and following, similar.Hence, f * (x, u * ) = f (x, u * ) and the proof is complete. 2 Now, we consider our main problem as follows Because of the degeneracy of a we prove the existence of multiple solutions of problem 2.9 in two steps.First one can consider a class of problem for i = 1, ..., m and j = 1, ..., l i as follows with weighted operator containing coefficients degenerating on the boundary of the domain.It follows from a1) that each set Ω i j is bounded domain of R n with smooth boundary, on which function a can be zero.In the second step, one can use the solutions obtained in the problems 2.10 to construct solutions of problem 2.9 which have different numbers of positive bumps.Now, let us consider Γ k+1 = ∂Ω and suppose that π 0 ) is the usual quotient space of Ω \ a −1 (0) under the equivalence relation which identifies points that can be joint with a continuous arc.
Hence one can write where j i denotes the number of subdomains of Ω \ a −1 (0) whose boundary is made exactly by i connected 1-codimensional submanifolds of R N which is denoted by Theorem 2.2.Assume that the conditions (a1), (a2), a3), (f 1) and (f 2) hold.Then problem 2.9 has at least 2 χ − 1 non-negative weak solutions.More precise, the number of positive solutions with n bumps, for n ∈ {1, ..., χ}, is given by the binomial coefficient Proof.In the first step, we prove the existence of χ one-bump weak solution to the problems 2.10 for i = 1, ..., m and j = 1, ..., l i .In fact, from the assumption (a1) one gets that each of domain Ω i j is a bounded domain in R n with a smooth boundary for which weighted function a can be zero on the boundary.By using of Theorem 2.1 with together the conditions (a1), (f 1) and (f 2) the existence of χ one-bump weak solutions is straightforward.Now, we apply the solutions obtained in the first step which denote them by u i j to construct the solutions of problem 2.9.To do this, we extend the solutions u i j from Ω i j to the whole of domain Ω as follows From the assumption a ∈ C(Ω) and since ).According to Theorem 2.1, ) is a weak solution of problem 2.10 so where the summation runs over all the possible combinations of indexes i, j so that to include all the connected components of Ω \ a −1 (0).According to the above identity and the assumptions it follows that U i j is a non-negative nontrivial weak solution of problem 2.9 for i = 1, ..., m and j = 1, ..., l i .By assumption (a1), the sum of n of the previous weak solutions U i j for n ∈ [2, χ], is still a solution so that the result holds.On the other hand, ∫ ) ( a(x) Hence, the solutions obtained here for problem 2.9 are in H 1 0 (Ω, a). 2 3. Eigenvalue problem for weighted Laplacian operator.We recall a version of the Ljusternik-Schnirelman (L-S) principle which was discussed by Browder [2].We shall apply the L-S principle to establish the existence of a sequence of eigenvalues for eigenvalue problems in closed subspaces of H 1 (Ω, a).Let X be a real reflexive Banach space and J, K be two functionals on X.Consider the following eigenvalue problem: where Γ J is the level Γ J = {u ∈ X ; J(u) = 1}.Moreover, we assume that: H1) J, K : X → R are even functionals and that

H3) J
′ is continuous, bounded and satisfies condition Γ 0 , i.e., as n → ∞ Existence and regularity results 165 H4) The level set Γ J is bounded and u ̸ = 0 implies It is known that (u, µ) solves problem 3.1 if and only if u is critical point of functional J with respect to Γ J see [16] (Proposition 43.21).For any positive integer n, denote by U the class of all compact, symmetric subsets E of Γ J such that J(u) > 0 and γ(E) ≥ n where γ(E) denotes the genus of E i.e., We define We now state the L − S principle.
where β, ν ∈ L ∞ (∂Ω, a) such that β, ν ≥ 0 [8].It easy to see that the functionals J and K are C 1 functional.Set A(u) := 1 2 J ′ (u) and B(u Then problem 3.1 means Bu = µAu, where According to assumption f 1) and the above definition, one can get that all eigenvalues λ are non-negative by choosing u = v.Let (u, λ) ∈ X × R + be an eigenpair, then, it is a weak solution of problem 2.9.Proposition 3.3.Let K defined in 3.5 with ν(s) ≡ 0. Then K ′ satisfies the assumption H2.
Proof.Upon the functional K ′ , it is enough to prove that B is strongly continuous.
Let u n ⇀ u in X, we have to show that Bu n → Bu in X * .According to the inequality (2.6) Moreover, ⟨Au − Av, u − v⟩ = 0 if and only if u = v in Ω.
Proof.As A = 1 2 J ′ , it suffices to show this for A. Using Sobolevs embedding theorem, Hölders inequality and following the arguments used in the proof of Proposition 3.3 one can easily see that A is continuous and bounded.It remains to show that A satisfies condition Γ 0 .That means if {u n } is a sequence in X such that u n ⇀ u Au n ⇀ v and ⟨Au n , u n ⟩ → ⟨v, u⟩ for some v ∈ X * and u ∈ X, then it follows that u n → u in X.By Sobolev's compact embedding theorem we have u n → u [14].Since X is a Hilbert space, one can find an equivalent norm such that X with this norm is locally uniformly convex.In such a space weak convergence and ∥u n ∥ → ∥u∥ implies strongly convergence.Thus to show that u n → u in X, we only need to show that ∥u∥ → ∥u∥.To this end, we first observe that On the other hand, it follows from Lemma 3.4 that Hence, ∥u n ∥ → ∥u∥ as n → ∞.Therefore, A satisfies condition Γ 0 . 2 Theorem 3.6.(Existence of L − S sequences) Let X be a closed subspace of H 1 (Ω, a) such that H 1 0 (Ω, a) ⊆ X and let K, J be two functionals defined in 3.4 and 3.5.Then there exists a non-increasing sequence of non-negative eigenvalues µ n obtained from L − S principle such that µ n → 0 as n → ∞ where and each µ n is eigenvalue of K Proof.The existence of such a sequence {µ n } follows from Theorem 3.1.To verify 3.8, from 3.4 , 3.5.3.6 and 3.7 Combining this with 3.2 implies 3.8. 2 Theorem 3.7.Let K, J be defined in 3.4 and 3.5 with β(s) = ν(s) ≡ 0 and let X = H 1 0 (Ω, a).Then there exists a nondecreasing sequence of nonnegative eigenvalues {λ n } of problem 2.9 obtained by the L − S principle such that λ n = 1 µn → ∞ as n → ∞, where each µ n is an eigenvalue of the corresponding equation Comparing the last equation to problem 2.9 and applying Theorem 3.1 we obtain the result.2 4. Regularity results.In this section we shall prove boundedness of eigenfunctions and use this fact to obtain C 1,α (Ω) and C 1,α ( Ω) smoothness of (weak) eigenfunctions of the nonlinear eigenvalue problem (2.9).

Proof.
Let u be a non-negative eigenfunction.Then from Theorem 4.2 u ∈ C 1,α (Ω, a) ∩ L ∞ (Ω, a) and u ̸ ≡ 0. Suppose that u(x 0 ) = 0 for some x 0 ∈ Ω.From the Haranack inequality (see Theorem 1.1 page 724 and Corollary 1.1 page 725 in [14]) u is is identically zero on any cube in Ω containing x 0 and connectedness of Ω we obtain u ≡ 0 in Ω which is a contradiction.Therefore, u is strictly positive in Ω. 2 ) where D ⊂ R N is smooth, open and bounded, b ∈ C(D), b(x) > 0 for x ∈ D, b ∈ A 2 and 1 b ∈ L t (D), t > N 2 and f satisfies in f 1) and f 2) for Ω replaced by D and a by the function b.As pervious, a weak solution for problem 2.3 is a function u