Nontrivial solutions for the Laplace equation with a nonlinear Goldstein-Wentzell boundary condition

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} \Delta u=0 \qquad&\text{in $\Omega$,}\\ u=0&\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u =|u|^{p-2}u\qquad&\text{on $\Gamma_1$,} \end{cases} $$ where $\Omega$ is a bounded open subset of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $\partial\Omega=\Gamma_0\cup\Gamma_1$, $\Gamma_0\cap\Gamma_1=\emptyset$, $\Gamma_1$ being nonempty and relatively open on $\Gamma$, $\mathcal{H}^{N-1}(\Gamma_0)>0$ and $p>2$ being subcritical with respect to Sobolev embedding on $\partial\Omega$. We prove that the problem admits nontrivial solutions at the potential--well depth energy level, which is the minimal energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.

Boundary conditions like the one in (1), but without the nonlinear source |u| p−2 u, are known in the literature as generalized Wentzell (sometimes spelled as Vencel) or Goldstein-Wentzell boundary condition, since they have been subject of several papers in the framework of linear evolutions problems.See for example [14,20,23,27,31] and [21], to which we refer for the physical motivations of this kind of problems.
On the other hand, to the author's knowledge, a Goldtsein-Wentzell boundary condition with a nonlinear source like |u| p−2 u in connection with the Laplace equation has never been considered in the literature.The motivation for studying it comes from a series of papers by the author concerning the wave equation with hyperbolic dynamical boundary conditions with boundary damping and source terms.The prototype of this kind of problems is the evolutionary boundary value problem (2) where u = u(t, x), t ≥ 0, x ∈ Ω, ∆ = ∆ x denotes the Laplacian operator respect to the space variable.Its associated initial-value problem was introduced in [33] and then studied, as a particular case, in [34,35] and [36].We refer to [13,34] for the physical derivation of the problem, describing the vibrations of a membrane with a part of the boundary carrying a linear density of kinetic energy.
In order to give clear-cut criteria on the initial data to discriminate between global existence and blow-up for solutions of (2) it is useful to know if it possesses nontrivial stationary solutions, which turns out to be solutions of (1), at some specific energy level.
In particular in the present paper we shall consider the case when the nonlinearity |u| p−2 u is sub-critical with respect to the Sobolev Embedding H 1 (Γ) ֒→ L p (Γ), that is we shall assume that (3) 2 < p < r, where r = Moreover, when dealing with problem (3), we shall also assume that the last assumption being related with well-posedness issues, see the papers quoted above. 1 We also remark that, although also the case p ≥ r (when N ≥ 4) was considered there, only the case p < r is of interest when dealing with the dichotomy between global existence and blow-up, see [36,Remark 1,p.6].
1 By the way assumption (4) may be skipped when dealing with stationary solutions, but we prefer to keep it to avoid re-discussing here problem (2) Moreover we shall denote by Tr the trace operator from H 1 (Ω) onto H 1/2 (Γ) and, for simplicity of notation, Tr u = u |Γ .
We introduce the Hilbert spaces H 0 = L 2 (Ω) × L 2 (Γ 1 ) and ( 6) with the topologies inherited from the products.For the sake of simplicity we shall identify, when useful, H 1 with its isomorphic counterpart )}, studied for example in [24], through the identification (u, u |Γ ) → u.So we shall write, without further mention, u ∈ H 1 for functions defined on Ω. Moreover we shall drop the notation u |Γ , when useful, so we shall write u L 2 (Γ) and so on, for u ∈ H 1 , referring to the restriction of the Hausdorff measure H N −1 to measurable subsets of Γ.We shall also drop the notation dH N −1 in boundary integrals, so By assumption (3) we can introduce in H 1 the nonlinear functional which represents the potential energy associated to problem (2).For this reason we shall call it the energy functional when dealing with (1).
We also introduce the potential-well depth d given by noticing that the identity between the two infima in ( 9) is essentially trivial and that we shall prove that d > 0.
Our first main result shows that problem (1) admits nontrivial weak solutions, see Definition 3 below, coinciding with critical points of the functional I, at the positive energy level d.We shall also recognize that they are stationary weak solutions of (3) provided this class of solutions is well-defined, see Definition 2 below.
Theorem 1 will be proved by applying a variant, maybe less well-known than other ones, of the Mountain Pass Theorem, explicitly given in § 2.
To show the relevance of the potential-well depth d, beside its interest in evolution problems, we give some relevant properties of weak solutions of (1) having energy 2 here ∇ Γ denotes the Riemannian gradient on Γ and | • | Γ , the norm associated to the Riemannian scalar product on the tangent bundle of Γ. See Section 2.
d.At first we introduce the norm B of the bounded linear trace operator from H 1 to L p (Γ 1 ), that is noticing that we shall prove that B < ∞.We can then state our second main result.
Theorem 2. Let (3) hold and set 2) , and Then we have Moreover, if u is a weak solution of (1) with I(u) = d we also have Finally weak solutions at the energy level d are least energy non-trivial solutions of (1), that is for any non-trivial weak solutions u of (1) one has I(u) ≥ d.
The proof of the minimality of the energy of solutions at level d, stated in Theorem 2, is of elementary nature.Hence it is easier than most proofs in the literature for internal sources, see for example [17,18,29].By the way the homogeneity of the source |u| p−2 u allows this simple approach.
Finally, to show that the minimality asserted in Theorem 2 is of some use, since there are solutions at an higher level, we give our last main result, which can be also of independent interest.
Theorem 3. When (3) holds there is a sequence (u n ) n of nontrivial weak solutions of (1) The proof of Theorem 3 relies on applying the Z 2 -version of the Mountain Pass Theorem in a different variational setting, which turns out to be equivalent to the one illustrated in this Section.See § 4 for details.
We would like to mention that, although in the paper we give Theorems 1-3 for the prototype nonlinearity f (x, u) = |u| p−2 u, they can be easily extended to the problem under suitable assumptions on f which have been widely used in the literature.
Here, for the sake of simplicity, we preferred to concentrate on the prototype problem.
Moreover clearly Theorems 1-3 also show the existence of stationary solutions for other evolution problems not considered in the present paper, which can be of future interest.
The paper is organized as follows: in Section 2 we shall give all preliminaries needed in the paper.Section 3 will be devoted to prove Theorems 1 and 2, while in Section 4 we shall prove Theorem 3.

Preliminaries
2.1.Notation.We shall adopt the standard notation for (real) Lebesgue and Sobolev spaces in Ω, referring to [1].For simplicity we shall denote by Given a Banach space X we shall denote by X ′ its dual and by •, • X the duality product between them.Moreover we shall use the standard notation for X-valued Lebesgue and Sobolev spaces in a real interval.When another Banach space Y is given we shall denote by L(X, Y ) the space of bounded linear operators between X and Y and by • L(X,Y ) the standard norm on it.
2.2.Function spaces and Riemannian operators on Γ. Lebesgue spaces on Γ and Γ 1 will be intended with respect to (the restriction to measurable subset of them of) the Hausdorff measure H N −1 , and for simplicity we shall denote, for Sobolev spaces on Γ and on its relatively open subsets are classical objects, and we shall use the standard notation for them.We refer to [15] for their definition in the present case in which Γ is merely C 1 .
Since Γ is C 1 it inherits from R N the structure of a Riemannian C 1 manifold, see [28], so in the sequel we shall use some notation of geometric nature, which is quite common when Γ is smooth, see [8,16,19,30], and which can be easily extended to the C 1 case, see for example [22].Moreover, since Γ 1 is relatively open on Γ, this notation will apply (by restriction) to it, without further mention.
We shall denote by T (Γ) and T * (Γ) the tangent and cotangent bundles, and by (•, •) Γ the Riemannian metric inherited from R N , given in local coordinates by (u, v) Γ = g ij u i v j for all u, v ∈ T (Γ) (here and in the sequel the summation convention being in use).The metric induces the fiber-wise defined musical isomorphisms , where •, • T (Γ) denotes the fiber-wise defined duality pairing.The induced bundle metric on T * (Γ), still denoted by (•, •) Γ , is then defined by the formula (α, •) Γ we shall denote the associated bundle norms on T (Γ) and T * (Γ).Denoting by d Γ the standard differential on Γ, the Riemannian gradient operator ∇ Γ is defined by setting, for u ∈ C 1 (Γ) and thus by density for u ∈ H , so in the sequel the use of vectors or forms is optional.
It is well known, see for example [22,Chapter 3], that H 1 (Γ) can be equipped with the equivalent norm • H 1 (Γ) given by In the sequel we shall also deal with the closed subspace of H 1 (Γ) , which is then an Hilbert space.Since for all u ∈ H 1 Γ0 (Γ) one has ∇ Γ u = 0 a.e. on Γ \ Γ 1 and H N −1 (Γ 0 ∩ Γ 1 ) = 0, we have ( 16) Remark 1.Although the definition of the space H 1 Γ0 (Γ) given above is adequate for our purpose, we would like to point out two characterizations of it in two different geometrical situations: i) when Γ 0 ∩ Γ 1 = ∅, so both Γ 0 and Γ 1 are relatively open, by identifying the elements of H 1 (Γ i ), i = 0, 1, with their trivial extensions to Γ, one easily gets the splitting , and consequently a characterization is false, since one easily sees that the characteristic function χ Γ1 of Γ 1 does not belong to H 1 Γ0 (Γ), while its restriction to Γ 1 trivially belongs to H 1 (Γ 1 ).Indeed in this case the elements of H 1 Γ0 (Γ) "vanish" at the relative boundary ∂Γ 1 = Γ 0 ∩ Γ 1 of Γ 1 on Γ, although such a notion can be made more precise only when ∂Γ 1 is regular enough.For example, when Γ is smooth and Γ 1 is a manifold with boundary ∂Γ 1 , see [30, §5.1], H 1 Γ0 (Γ) is isometrically isomorphic to the space The Laplace-Beltrami operator ∆ Γ can be defined in a geometrically elegant way by using ∇ Γ and the Riemannian divergence operator, as in [22, § 2.3], at least when Γ is C 2 .To avoid the need of introducting Sobolev spaces of tensor fields we shall adopt here a less elegant approach.Indeed we set, when Γ is in local coordinates.Since g, g ij are continous and Γ is compact, formula (17) extends by density to u ∈ H 2 (Γ), so defining an operator Since Γ is compact, by (17), integrating by parts and using a C 2 partition of the unity one gets that Formula (18) motivates the definition of the operator −∆ Γ ∈ L(H 1 (Γ); H −1 (Γ)), also when Γ is merely C 1 , given by By density, when Γ is C 2 , the so defined operator is the unique extension of −∆ Γ ∈ L(H 2 (Γ); L 2 (Γ)).
In § 4 we shall deal with the realization of −∆ Γ between the space H 1 Γ0 (Γ) and its dual.The different nature of the space H 1 Γ0 (Γ) in the two cases i) and ii) has been pointed out in Remark 1.To explain the definition of the realization we shall give we recall that, when Γ 0 ∩ Γ 1 = ∅, so Γ 1 is compact, formula (18) holds, when Γ is C 2 , also when replacing Γ with Γ 1 , so making natural to set fails to hold on Γ 1 , since a boundary integral on ∂Γ 1 appears.On the other hand, taking into account the homogeneous Dirichlet boundary condition in the space Hence, taking into account the characterizations given in Remark 1, to simultaneously deal with the two cases i) and ii), in the sequel we shall deal with the operator noticing that, by (19),

2.3.
The space H 1 .We recall, see [31, Lemma 1, p. 2147] which trivially extends to Γ of class C 1 , that the space with the topology inherited from the product, can be identified with the space {u ∈ H 1 (Ω) : u |Γ ∈ H 1 (Γ)} and equivalently equipped with the norm given by The identification made in § 1 between the spaces H 1 and H 1 Γ0 (Ω, Γ), respectively defined by ( 6) and (7), is a simple consequence of the identification above, and, by ( 16), H 1 can be equivalently equipped with the norm |||•||| H 1 given by ( 23) On the other hand, to get advantage of the assumption H N −1 (Γ 0 ) > 0, made in the present paper, we point out the following well-known result, the proof of which is given only for the reader's convenience. • By the Trace Theorem there is a positive constant where , by combining ( 23), ( 25) and ( 26) we get for all u ∈ H 1 , where c 3 = 1 + c 2 (1 + c 1 ), from which the statement trivially follows.
2.4.Some results from Critical Point Theory.We now recall some wellknown notions of Critical Point Theory for a functional I ∈ C 1 (X) = C 1 (X; R) on any Banach space X with norm • X .By I ′ ∈ C(X; X ′ ) we shall denote the Fréchet differential of I and (PS) will stands, in short, for Palais-Smale.See [26].
We also say that I ∈ C 1 (X) satisfies the (PS) condition if any (PS) sequence has a (strongly) convergent subsequence.
The following result is nothing but a well-known version of the celebrated Mountain Pass Theorem, see [26, Chapter 1, p. 4].
Theorem 4. Let I ∈ C 1 (X) satisfies the (PS) condition and i) I(0) = 0; ii) there are ρ, α > 0 such that I(u) ≥ α for all u ∈ X such that u X = ρ; iii) there is l ∈ X such that l X > ρ and I(l) ≤ 0.
Then I possesses a critical value c l ≥ α given by It is rarely pointed out in textbooks that the critical level c l above may depend on l, as the following trivial example shows.
Example 1.Let X = R and Trivially I ∈ C 1 (R) satifies the (PS) condition as well as assumptions i)-iii).Moreover its critical points are exactly x = 0, −1/2, √ 2/2 from which one easily sees that Since in the present paper we are interested in characterizing our critical level as the potential-well depth of the functional I, we now state a less known variant of the Mountain Pass Theorem under slighty more restrictive assumptions on the functional which look similar (although not identical) to the assumptions of [4, Theorem 2.1, p. 354], this one being the first version of this celebrated result.
Then I possesses a critical value c ≥ α given by Proof.One can repeat the proof of [4, Theorem 2.1, p. 354] verbatim, since by assumption iv) for any σ ∈ Σ one has σ(1) X > ρ.Alternatively one can also deduce the statement from Theorem 4 by the following simple argument.By assumption iv) one has Σ = l∈A Σ l , where A = {l ∈ X : l X > ρ and I(l) < 0}.Hence, by ( 27) and ( 28) we have c = inf l∈A c l .Hence, since by assumption ii) one has α ≤ c < ∞, there is a sequence (l n ) n in A such that c ln → c.By Theorem 4 there ia a corresponding sequence (u n ) n in X such that I(u n ) = c ln and I ′ (u n ) = 0, which is then a (PS) sequence and consequently, up to a subsequence, u n → u.
In the sequel we shall also use the following well-known Z 2 -version of the Mountain Pass Theorem, see [26, Chapter 9, Theorem 9.12, p. 55 and Proposition 9.33, p. 58].
Theorem 6.Let X be infinite dimensional and I ∈ C 1 (X) be even, satisfying the (PS) condition, assumptions i)-ii) of Theorem 4 and Then I possesses a sequence (u n ) n of critical points such that I(u n ) → ∞.

3.
Existence of solutions of (1) at level d and their minimality.
The aim of this section is to prove Theorems 1 and 2. We start by recalling what we mean by a weak solution of (2), referring to [35, §2.2 and Definition 3.1, p. 4896].We shall also make precise the use of the term "stationary" when referring to them.
Definition 2. Let (3) and ( 4) hold.A weak solution of (2) is We also make precise what we mean by weak solutions of (1).Definition 3. Let 2 < p < r.A weak solution of ( 1) is u ∈ H 1 such that ( 31) Actually weak solutions of (1) and stationary weak solutions of (2) coincide when they are both defined, as the following result shows.
Lemma 2. Let (3), ( 4) hold and u 0 ∈ H 1 .Then u ≡ u 0 is a stationary weak solution of (2) if and only if u 0 is a weak solution of (1).
Proof.If u 0 is a weak solution of ( 1) by ( 31) one immediately gets that u ≡ u 0 satisfies (30).To prove the converse we recall that, by [35,Lemma 3.3,p. 4896], any weak solution u of (2) satisfies the alternative form of the distribution identity (30) ( 32) Hence, when u ≡ u 0 ∈ H 1 is a stationary weak solution of (2), taking in (32) test functions ψ ≡ φ ∈ H 1 for an arbitrary T > 0 we get (31), concluding the proof.Equation ( 31) has a variational nature, as the next result highlights.Lemma 3. Il (3) holds the functional I defined in (8) belongs to C 1 (H 1 ) and its critical points coincide with the weak solutions of (1).
We now establish some geometrical properties of the functional I. Proof.By (8) trivially I(0) = 0, proving i).To prove ii) and iv) we remark that, by Sobolev Embedding Theorem there is c 4 = c 4 (p, Ω) > 0 such that Consequently, by Lemma 1, there is Hence, by (8), for all u ∈ H 1 we have from which ii) and iv) follow, by taking for example , and α = 1 4 To prove iii) we remark that, for any u ∈ H 1 such that u |Γ1 = 0 and s > 0 we have The last relevant property of I is given by the following result.Proof.Let (u n ) n be a (PS) sequence in H 1 .Then there are c 6 , c 7 ≥ 0, depending on (u n ) n , such that (37) Since, by ( 8) and (35), from which one immediately yields that (u n ) n is bounded in H 1 .Consequently, up to a subsequence, u n ⇀ u in H 1 .To prove that the convergence is actually strong we remark that, by (35), for all φ ∈ H 1 we have The first two terms in the right hand side of (38) converges to 0 since I ′ (u n ) → 0 in (H We can then give the proof of our first main result. Proof of Theorem 1.By simply combining Lemmas 2-5, Theorem 5, with I = I, and the fact that I is even we get the statement, but for the critical level, since by Theorem 5 we have that I(u) = I(−u) = c, where c is given by (28).To complete the proof we then have to recognize that c = d 1 = d 2 , where Since, for any λ > 0, (39) This fact is well-known for similar problems, see for example [2,Chapter 8,p. 117] and [6,11,17,18,32].Here we shall essentially conveniently adapt the argument in [32].By (39), for any u ∈ H 1 with u |Γ ≡ 0, the function λ → I(λu) has a unique critical point λ u > 0, max  We now turn to prove Theorem 2. We remark at first that the number B defined in (10) is finite because of the estimate (36).We now introduce the auxiliary functional K ∈ C 1 (H 1 ) given by ( 40) The key point in the proof of Theorem 2 is the following result, of possible independent interest.Lemma 6.Let (3) hold and λ 1 , λ 2 be given by (11).Then (12) holds.Moreover, for any u ∈ H 1 such that u |Γ ≡ 0 and I(u) ≤ d, the following implications hold: Proof.To prove (12) we remark that, for any u ∈ H 1 with u |Γ ≡ 0 an easy calculations shows that max λ>0 .
By using (10) and (11) in the last formula we get (12).Now let u ∈ H 1 such that u |Γ ≡ 0 and I(u) ≤ d.To prove (41) we shall first prove the implications If K(u) ≥ 0, supposing by contradiction that u H 1 > λ 1 , by (34) we get To complete the proof of (41) we are then going to prove the further implications If K(u) ≤ 0, by (43), we have a contradiction, so K(u) ≤ 0 and the proof is complete.
We can now prove our second main result.
To prove that also assumption v) in Theorem 6 holds true let Y be any finite dimensional subspace of H 1 Γ0 (Γ).Since on Y all norms are equivalent there are c Using it in (55) we then get from which, since p > 2, one gets that J(v) ≤ 0 for v p,Γ1 sufficiently large.By (58) then assumption v) in Theorem 6 holds true.
To complete checking of the assumptions of Theorem 6 we give the following result.
Proof.We shall prove the statement by using Lemma 5.With this aim we make some preliminary remarks concerning the space H 1 and the functional I.At first H 1 admits the orthogonal (with respect to (•, •) H 1 given in (24)) splitting (59) H 1 = A 1 ⊕ H 1 0 (Ω), the respective orthogonal projectors Π A 1 ∈ L(H 1 , A 1 ) and Π H 1 0 (Ω) ∈ L(H 1 , H 1 0 (Ω)) being given by Π A 1 u = Du |Γ and Π H 1 0 (Ω) u = u − Du |Γ for all u ∈ H 1 .Using this splitting we can rewrite (35), for any u ∈ A 1 , as (60) A 1 (u), φ A 1 for all φ ∈ A 1 and ψ ∈ H 1 0 (Ω), where I A 1 = I |A 1 , that is I A 1 is the restriction of I to A 1 . 3o prove the statement let now (v n ) n be a (PS) sequence for J. Since J = I • D = I A 1 • D, by (50) we get that (Dv n ) n is a (PS) sequence for the functional I A 1 in A 1 .Hence (I(Dv n )) n is bounded in H 1 and, using (60), I ′ (Dv n ) → 0 in [H 1 ] ′ , so (Dv n ) n is a (PS) sequence for I.
By Lemma 5 then, up to a subsequence, (D(v n )) n converges in H 1 which, using (50) again, concludes the proof.
We can finally give the proof of our last main result.
Proof of Theorem 3. By Lemmas 8-10 the functional J satisfies the assumptions of Theorem 6, so it possesses a sequence (v n ) n of critical points such that J(v n ) → ∞.By Lemma 7 the sequence (u n ) n given by u n = Dv n for all n ∈ N is thus a sequence of critical points of I with I(u n ) = J(v n ) → ∞.By Lemma 4 the proof is thus concluded.

Theorem 1 .
When (3) holds problem (1) has at least a couple (u, −u) of antipodal weak solutions in H 1 such that I(u) = I(−u) = d > 0. When (4) holds they are also stationary weak solutions of problem (2).Moreover d coincides with the Mountain Pass level of the functional I, that is

Lemma 4 .
Il (3)  holds the functional I satisfies the assumptions i)-iii) of Theorem 4 and iv) of Theorem 5.

I
(σ u (t)) ≥ c.Hence d ≥ c.On the other hand, if u is a critical point of I with I(u) = c, already found above, by (35) we have u 2 H 1 = u p p,Γ1 .Then, since u = 0, we also get that u |Γ ≡ 0. Consequently, since d dλ I(λu) = I ′ (u), u H 1 , we have λ u = 1 and consequently c = I(u) = max λ>0 I(λu) ≥ d, completing the proof.
1) ′ and u n − u ⇀ 0, hence being norm bounded.As to the third term in it, since the embedding H 1 (Γ) ֒→ L p (Γ) is compact and the operator u → u |Γ from H 1 to H 1 (Γ) is bounded, up to subsequence we have u n|Γ → u |Γ strongly in L p (Γ).By standard properties of the Nemitskii operators, see [3, Chapter 1, Theorem 2.2, p. 16], we also have|u n|Γ | p−2 u n|Γ → |u |Γ | p−2 u |Γ strongly in L p ′ (Γ).Consequently the third term in the right hand side of (38) converges to 0 goes to 0 by Hölder inequality.