Elliptic problems with mixed nonlinearities and potentials singular at the origin and at the boundary of the domain

We are interested in the following Dirichlet problem: -Δu+λu-μu|x|2-νudist(x,RN\Ω)2=f(x,u)inΩu=0on∂Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\textrm{dist}(x,\mathbb {R}^N \setminus \Omega )^2} = f(x,u) &{} \quad \text{ in } \Omega \\ u = 0 &{} \quad \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$\end{document}on a bounded domain Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \subset \mathbb {R}^N$$\end{document} with 0∈Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \in \Omega $$\end{document}. We assume that the nonlinear part is superlinear on some closed subset K⊂Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K \subset \Omega $$\end{document} and asymptotically linear on Ω\K\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega \setminus K$$\end{document}. We find a solution with the energy bounded by a certain min–max level, and infinitely, many solutions provided that f is odd in u. Moreover, we study also the multiplicity of solutions to the associated normalized problem.


Introduction
We are interested in the problem (1.1) where λ, µ ∈ R are real parameters, f : Ω × R → R, Ω ⊂ R N is a bounded domain in R N with 0 ∈ Ω, and K ⊂ Ω is a closed set with |int K| > 0.
Semilinear problems of general form appear when one looks for stationary states of time-dependent problems, including the heat equation ∂u ∂t − ∆u = h(x, u) or the wave equation ∂ 2 u ∂t 2 − ∆u = h(x, u).In nonlinear optics the nonlinear Schrödinger equation is studied and looking for standing waves Ψ (t, x) = e iλt u(x) leads then to a semilinear problem.The time-dependent equation (1.2) appears in physical models in the case of bounded domains Ω ( [11,12,23]), as well as in the case Ω = R N ( [8,18]).Two points of view of solutions to (1.1) are possible; either λ may be prescribed or may be considered as a part of the unknown.In the latter case a natural additional condition is the prescribed mass Ω u 2 dx.In the paper we will consider both cases, namely we will look for solutions for the unconstrained problem (1.1) as well as the constrained one, see (1.3)

below.
Date: December 16, 2022. 1 The equation (1.1) (and systems of such equations) on bounded domains has been studied in the presence of bounded potentials [4] and singular at the origin [13], see also [10,14,15] for the case of unbounded Ω.Its constrained counterpart without the potential has been studied e.g. in [19,20], where (1.3) was studied with f (x, u) = |u| p−2 u, ν = µ = 0 in the mass-subcritical, mass-critical and mass-supercritical cases.In this paper we are interested in the presence of a potential 2   which is singular in Ω as well as on the whole boundary ∂Ω.We mention here that Schrödinger operators were studied with potentials being singular at the point on the boundary [7], as well as with potentials being singular on the whole boundary [16,17].We assume that Ω is a domain satisfying the following condition This condition allows us to study the singular potential by means of Hardy-type inequalities (Section 2).As we will see in Section 2 (see Proposition 2.1), any convex domain Ω satisfies (C).
We impose the following condition on parameters appearing in the problem On the nonlinear part of (1.1) we propose the following assumptions.
In the last section we also study the normalized problem in Ω u = 0 on ∂Ω, Ω u 2 dx = ρ > 0, where ρ is fixed and (λ, u) ∈ R × H 1 0 (Ω) is an unknown.Then we obtain the following multiplicity result in the so-called mass-subcritical case.
In what follows, denotes the inequality up to a multiplicative constant.Moreover C denotes a generic constant which may vary from one line to another.

The domain Ω and the singular Schrödinger operator
We recall that if A ⊂ R N is a closed, nonempty set, we can define the distance function dist(•, A) : We collect the following properties of the distance function: We recall that we denote d(x) = dist(x, R N \ Ω).Observe that, due to Rademacher's theorem ([25, Theorem 2.2.1]) and (i), d is differentiable almost everywhere and, from (ii) |∇d| = 1, almost everywhere.We remind that the assumption (C) says that −∆d ≥ 0 in Ω holds in the sense of distributions.We note the following fact.
Proof.First note that d which completes the proof of concavity.Moreover, since d is concave on Ω, from [9, Theorem 6.8] there is a nonnegative Radon measure µ on Ω satisfying Clearly, for ϕ ≥ 0 we get Ω ∇d • ∇ϕ dx ≥ 0 and condition (C) holds.
To study singular terms in (1.1) we recall the following Hardy-type inequalities.If u ∈ H 1 0 (Ω), where Ω is a domain in R N with finite Lebesgue measure and 0 ∈ Ω, then (see [6]) Now let Ω ⊂ R N be a bounded domain satisfying (C).Then, for u ∈ H 1 0 (Ω), the following Hardy inequality involving the distance function holds (see [2])

Variational setting and critical point theory
Suppose that (E, • ) is a Hilbert space and J : E → R is a nonlinear functional of the general form where I is of C 1 class and I(0) = 0. We introduce the so-called Nehari manifold Observe that To utilize the mountain pass approach, we consider the following space of paths and the following mountain pass level Moreover we set We propose an abstract theorem which is a combination of [3, Theorem 5.1] and [5, Theorem 2.1].
The proof is a straightforward modification of proofs of mentioned theorems, however we include it here for the reader's convenience.
Theorem 3.1.Suppose that (J1) there is r > 0 such that ) for all t > 0 and u ∈ N there holds Moreover there is a Cerami sequence for J on the level c, i.e. a sequence {u n } n ⊂ E such that Proof.Observe that there exists v ∈ Q \ {0} with v > r such that J (v) < 0. Indeed, fix u ∈ Q \ {0} and from (J2) there follows that → −∞ as t → +∞ and we may take v := tu for sufficiently large t > 0. In particular, the family of paths Γ Q is nonempty.Moreover, J (tu) → 0 as t → 0 + and for t = r u > 0 we get J (tu) > 0. Hence, taking (3.2) into account, (0, +∞) ∋ t → J (tu) ∈ R has a local maximum, which is a critical point of J (tu) and tu ∈ N .Hence N ∩ Q = ∅.Suppose that u ∈ N ∩ Q.Then, from (J3), and therefore u is a maximizer (not necessarily unique) of J on R + u := {su : s > 0}.Hence, for any u ∈ N ∩Q there are 0 Since inf there follows, under (J1), that The existence of a Cerami sequence follows from the mountain pass theorem.
To study the multiplicity of solutions we will recall the symmetric mountain pass theorem.We consider the following condition (J4) there exists a sequence of subspaces Then, the following theorem holds.
We work in the usual Sobolev space H 1 0 (Ω) being the completion of C ∞ 0 (Ω) with respect to the norm

Define the bilinear form
Lemma 3.3.B defines an inner product on H 1 0 (Ω).Moreover, the associated norm is equivalent with the usual one.Proof.To check that B is positive-definite we utilize (2.1), (2.2), and (N) to get and the statement follows from the Poincaré inequality.Moreover, from there follows that B generates a norm on H 1 0 (Ω) equivalent to the standard one.Let • denote the norm generated by B, namely u := B(u, u), u ∈ H 1 0 (Ω).Then we can define the energy functional J : where G(x, u) := u 0 g(x, s) ds and F is given in (F3).It is well-known that under (F1), (F2) the functional is of C 1 class and Hence, its critical points are weak solutions to (1.1).

Cerami sequences and proofs of main theorems
Lemma 5.1.Any Cerami sequence for J is bounded.
Proof.Suppose that u n → +∞ up to a subsequence.We define v n = un un .Then v n = 1 and and almost everywhere.
We consider three cases.
• Suppose that v 0 = 0. Condition (J3) implies that Taking t → t un we obtain that for any t > 0 -a contradiction.• Now we suppose that v 0 = 0 and |supp v 0 ∩ K| > 0. Then Observe that for a.e.x ∈ supp ϕ ∩ supp v we get that pointwise, a.e. on supp ϕ ∩ supp v. Combining (F4) and (F5) we also get that Thus, from Lebesgue dominated convergence theorem and the Hölder inequality we get In particular, 0 is an eigenvalue of the operator with Dirichlet boundary conditions on Ω \ K, which contradicts (A).
Proof of Theorem 1.1.Since Cerami sequence u n is bounded we have following convergences (up to a subsequence): , and in L p (Ω), u n → u 0 a.e. on Ω.
Hence, for any ϕ ∈ C ∞ 0 (Ω), because obviously weak convergence of u n implies that and we will use the Vitali convergence theorem to prove that Hence we need to check the uniform integrability of the family {(f (x, u n ) − f (x, u 0 )) ϕ} n .Using (F1) and Lemma 5.1 we obtain that for any measurable set Then, for any ε > 0, we can choose δ > 0 small enough that Proof of Theorem 1.2.The statement follows directly from Theorem 3.2 and Lemma 4.1.

Multiple solutions to the mass-subcritical normalized problem
In what follows we are interested in the normalized problem (1.3), where λ is not prescribed anymore and is the part of the unknown (λ, u) ∈ R × H 1 0 (Ω).Then, solutions are critical point of the energy functional and λ arises as a Lagrange multiplier.We recall the well-known Gagliardo-Nirenberg inequality (6.1) , u ∈ H 1 0 (Ω), where δ p := N 1 2 − 1 p and C p,N > 0 is the optimal constant.Lemma 6.1.J 0 is coercive and bounded from below on S.
Proof.Let (u n ) ⊂ S be a Palais-Smale sequence for J 0 | S .Then Lemma 6.1 implies that (u n ) is bounded in H 1 0 (Ω).Hence we may assume that (up to a subsequence) Observe that, from (F1) Therefore (λ n ) ⊂ R is bounded, and (up to a subsequence) λ n → λ 0 .Therefore, up to a subsequence, It is clear that u n , u → u 2 and that Moreover, from (F1) Hence u n → u and therefore u n → u in H 1 0 (Ω).
Proof of Theorem 1.3.From [22, Theorem II.5.7] we obtain that J 0 has at least γ(S) critical points, where γ(S) := sup{γ(K) : K ⊂ S -symmetric and compact} and γ denotes the Krasnoselskii's genus for symmetric and compact sets.We will show that γ(S) = +∞.Indeed, fix k ∈ N. It is sufficient to construct a symmetric and compact set K ⊂ S with γ(K) = k.Choose functions w 1 , w 2 , . . ., w k ∈ C ∞ 0 (Ω) ∩ S with pairwise disjoint supports, namely w i w j = 0 for i = j.Now we set It is clear that K ⊂ S is symmetric and compact.We will show that γ(K) = k.In what follows S m−1 denotes the (m − 1)-dimensional sphere in R m of radius 1 centered at the origin.Note that h : K → S k−1 given by is a homeomorphism, which is odd.Hence γ(K) ≤ k.Suppose by contradiction that γ(K) < k.