# PAIRS OF NONTRIVIAL SOLUTIONS FOR RESONANT ROBIN PROBLEMS WITH INDEFINITE LINEAR PART.

1. INTRODUCTIONLet Q C RN be a bounded domain with a [C.sup.2]-boundary [partial derivative][OMEGA]. In this paper we study the following semilinear Robin problem:

(1.1) [mathematical expression not reproducible],

In this problem the potential function [xi] [member of] [L.sup.s]([OMEGA]) (with s > N) is in general sign-changing. So, the linear part of problem (1.1) is indefinite. The reaction term f is a Caratheodory function (that is, for all [zeta] [member of] R, z [??] f (z, [zeta]) is measurable and for a.a. z [member of] [OMEGA], [zeta] [??] f (z, [zeta]) is continuous). We assume that f (z, *) exhibits linear growth near and near zero and resonance is possible both at and at zero, but with respect to different eigenvalues of u [??] -[DELTA]u + [xi](z)u with Robin boundary condition. In the boundary condition, the coefficient [beta] [member of] [W.sup.1,[infinity]] ([partial derivative][OMEGA]) and [beta](z) [greater than or equal to] 0 for all z [member of] [partial derivative][OMEGA]. When [beta] [equivalent to] 0, we have the usual homogeneous Neumann problem. We prove a multiplicity theorem, producing a pair of nontrivial smooth solutions.

Recently semilinear Robin problems were studied by Shi-Li [17] (indefinite potential and superlinear reaction term), Qian-Li [16] (zero potential and superlinear reaction term), Zhang-Li-Xue [19] (positive potential, thus a coercive differential operator and an autonomous reaction term with zeros), Papageorgiou-Rhdulescu [15] (indefinite potential and a Caratheodory reaction term of arbitrary growth), D'AguiMarano-Papageorgiou [3] (indefinite potential and an asymmetric reaction term) and Filippakis-Papageorgiou [5] (indefinite potential and an odd reaction term of arbitrary growth). Also we mention the works of Papageorgiou-Papalini [11] (Dirichlet problems) and Papageorgiou-Radulescu [12, 14] (Neumann problems) on equations driven by the Laplacian plus an indefinite potential.

We prove a multiplicity theorem producing two nontrivial smooth solutions. Our approach is based on a variant of the reduction method of Amann [1] and Castro- Lazer [2] and on Morse theory (critical groups). However, note that in our case the reduction is done on an infinite dimensional component space and this makes the situation more complicated.

2. MATHEMATICAL BACKGROUND

Let X be a Banach space and [X.sup.*] its topological dual. By <*, *> we denote the duality brackets for the pair ([X.sup.*],X). Given a function [phi] [member of] [C.sup.1] (X; R), we say that [phi] satisfies the "Cerami condition", if the following is true:

"Every sequence [{[u.sub.n]}.sub.n[greater than or equal to]1] [subset or equal to] X such that [{[phi]([u.sub.n])}.sub.n[greater than or equal to]1] [subset or equal to] R is bounded and

(1 + [parallel][u.sub.n][parallel])([u.sub.n]) [right arrow] 0 in [X.sup.*] as n [right arrow] +[infinity],

admits a strongly convergent subsequence."

Our analysis of problem (1.1), will make use of the following three spaces:

[H.sup.1] ([OMEGA]), [C.sup.1] ([bar.[OMEGA]]), [L.sup.q]([partial derivative][OMEGA]) (1 [less than or equal to] q [less than or equal to] +[infinity]).

We know that [H.sup.1]([OMEGA]) is a Hilbert space with inner product given by

[mathematical expression not reproducible].

The corresponding norm is denoted by [parallel]*[parallel] and we have

[parallel]u[parallel] = [([[parallel]u[parallel].sup.2.sub.2] + [[parallel]Du[parallel].sup.2.sub.2]).sup.1/2] [for all]u [member of] [H.sup.1] ([OMEGA]).

On [partial derivative][OMEGA] we consider the (N - 1)-dimensional Hausdorff (surface) measure [sigma]. Then using this measure on [partial derivative][OMEGA], we can define in the usual way the Lebesgue spaces [L.sup.q]([partial derivative][OMEGA]) (1 [less than or equal to] q [less than or equal to] +[infinity]). From the theory of Sobolev spaces, we know that there exists a unique continuous linear map [[gamma].sub.0]: [H.sup.1]([OMEGA]) [right arrow] [L.sup.2]([partial derivative][OMEGA]) known as the "trace map" such that

[[phi].sub.0](u) = u[|.sub.[OMEGA]] [for all]u [member of] [H.sup.1]([OMEGA]) [intersection] C([bar.[OMEGA]]).

Hence the trace map assigns "boundary values" to every Sobolev function u [member of] [H.sup.1]([OMEGA]). We know that

* [[gamma].sub.0] is compact into [L.sup.q]([partial derivative][OMEGA]) for all q [member of] [1, 2(N - 1)/N - 2) if N [greater than or equal to] 3 and for all q [member of] [1, +[infinity]) if N = 1,2;

* R([[gamma].sub.0]) = [H.sup.1/2,2] ([partial derivative][OMEGA]) and ker [[gamma].sub.0] = [H.sup.1.sub.0]([OMEGA]).

In the sequel, for the sake of notational simplicity, we drop the use of the trace map [[gamma].sub.0]. All restrictions of Sobolev functions on [partial derivative][OMEGA] are understood in the sense of traces.

We will use the spectrum of u [??] -[DELTA]u + [xi](z)u with Robin boundary condition.

So, we consider the following linear eigenvalue problem

(2.1) [mathematical expression not reproducible].

This problem was studied by D'Agui-Marano-Papageorgiou [3]. Suppose that [xi] [member of] [L.sup.s] ([OMEGA]) with s > N and let [gamma]: [H.sup.1]([OMEGA]) [right arrow] R be the [C.sup.2]-functional defined by

[mathematical expression not reproducible].

Problem (2.1) has a smallest eigenvalue [[??].sub.1] [member of] R given by

(2.2) [mathematical expression not reproducible].

Then we can find [mu] > 0 such that

(2.3) [gamma](u) + [mu][[parallel]u[parallel].sup.2.sub.2] [greater than or equal to] [c.sub.0][[parallel]u[parallel].sup.2.sub.2] [for all]u [member of] [H.sup.1]([OMEGA]),

for some [c.sub.0] > 0. Using (2.3) and the spectral theory for compact self- adjoint operators on a Hilbert space, we generate the spectrum of (2.1). This consists of a sequence [{[[??].sub.k]}.sub.k[greater than or equal to]1] of distinct eigenvalues such that [[??].sub.k] [right arrow] as k [right arrow] +[infinity]. Let E([[??].sub.k]) denote the eigenspace corresponding to the eigenvalue [[??].sub.k]. Using the regularity theory of Wang [18], we know that

E ([[??].sub.k]) [subset or equal to] [C.sup.1]([bar.[OMEGA]]) [for all]k [greater than or equal to] 1.

In addition, from de Figueiredo-Gossez [4], we have that each eigenspace E([[??].sub.k]), k [greater than or equal to] 1, exhibits the "unique continuation property", that is, if u [member of] E([[??].sub.k]) and u vanishes on a set of positive Lebesgue measure, then u [equivalent to] 0. We set

[mathematical expression not reproducible]

The space Hm is finite dimensional and we have the following orthogonal direct sum decomposition

[H.sup.1]([OMEGA]) = [[bar.H].sub.m] [direct sum] [[??].sub.m].

For the higher eigenvalues [{[[??].sub.m]}.sub.m[greater than or equal to]2] we have the following variational description:

(2.4) [mathematical expression not reproducible].

From (2.2) and (2.4) we see that we have variational characterizations for all the eigenvalues. In (2.2) the infimum is realized on E([[??].sub.1]), while in (2.4) both the infimum and supremum are realized on E([[??].sub.m]). We know that [[??].sub.1] is simple (that is, dim E([[??].sub.1]) = 1) and is the only eigenvalue with eigenfunctions of constant sign. For every m [greater than or equal to] 2, the elements of E(Am) are nodal (sign changing). By [[??].sub.1] we denote the L2-normalized (that is, [[parallel]u[parallel].sub.2] = 1) positive eigenfunction corresponding to [[??].sub.1]. The strong maximum principle implies that [[??].sub.1](z) > 0 for all z [member of] [OMEGA]. As a consequence of the variational characterizations of the eigenvalues (see (2.2) and (2.4)) and of the unique continuation principle, we have the following useful inequalities.

Proposition 2.1. (a) If [??] [member of] [L.sup.[infinity]]([OMEGA]), [partial derivative](z) [less than or equal to] Am for a.a. z [member of] [OMEGA], for some m [greater than or equal to] 1 and the inequality is strict on a set of positive measure, then we can find [c.sub.1] > 0 such that

[gamma](u) - [[integral].sub.[OMEGA]] [??](z)[u.sup.2]dz [greater than or equal to] [c.sub.1][[parallel]u[parallel].sup.2] [for all]u [member of] [[??].sub.m-1].

(b) If [??] [member of] [L.sup.[infinity]]([OMEGA]), [??](z) [greater than or equal to] [[??].sub.m] for a.a. z [member of] [OMEGA], for some m [greater than or equal to] 1 and the inequality is strict on a set of positive measure, then we can find [c.sub.2] > 0 such that

[gamma](u) - [[integral].sub.[OMEGA]] [??](z)[u.sup.2]dz [less than or equal to] - [c.sub.2][[parallel]u[parallel].sup.2] [for all]u [member of] [[bar.H].sub.m].

Next we recall some definitions and facts from Morse theory (critical groups) which will be used in the sequel.

Let X be a Banach space, [phi] [member of] [C.sup.1] (X; R) and c [member of] R. We introduce the following sets:

[[phi].sup.c] = {u [member of] X : [phi](u) [less than or equal to] c}, [K.sub.[phi]] = {u [member of] X : [phi]'(u) = 0}, [K.sup.c.sub.f] = {u [member of] [K.sub.[phi]] : [phi](u) = c}.

Let ([Y.sub.1], [Y.sub.2]) be a pair of spaces such that [Y.sub.2] [subset or equal to] [Y.sub.1] [subset or equal to]

X. For every k [greater than or equal to] 0, by [H.sub.k]([Y.sub.1], [Y.sub.2]) we denote the k-th relative singular homology group for the pair ([Y.sub.1], [Y.sub.2]) with integer coefficients. Suppose that u [member of] [K.sup.c.sub.[phi]] is isolated. The critical groups of [phi] at u are defined by

[C.sub.k]([phi], u) = [H.sub.k]([[phi].sup.c] [intersection] U, [[phi].sup.c] [intersection] U\{u}) [for all]k [greater than or equal to] 0.

Here U is a neighbourhood of u such that [K.sub.[phi]] [intersection] [[phi].sup.c] [intersection] U = {u}. The excision property of singular homology, implies that the above definition of critical groups is independent of the choice of the isolating neighbourhood U.

Suppose that [phi] [member of] [C.sup.1](X; R) satisfies the Cerami condition at inf [phi]([K.sub.[phi]]) > - [infinity]. The critical groups of [phi] at infinity are defined by

[C.sub.k]([phi], [infinity]) = [H.sub.k](X, [[phi].sup.c]) [for all]k [greater than or equal to] 0,

with c < inf [phi]([K.sub.[phi]]). This definition is independent on the level c < inf [phi]([K.sub.[phi]]). Indeed, if c' < c < inf [phi]([K.sub.[phi]]), then [[phi].sup.c'] is strong deformation retract of (see for example Motreanu-Motreanu-Papageorgiou [10, Theorem 5.34, p. 110]). Therefore

[H.sub.k](X, [[phi].sup.c]) = [H.sub.k](X, [[phi].sup.c']) [for all]k [greater than or equal to] 0

(see Motreanu-Motreanu-Papageorgiou [10, Theorem 6.15, p. 145]).

Assume that Kv is finite. We introduce the following quantities:

[mathematical expression not reproducible].

Then the "Morse relation" says that

(2.5) [summation over (u[member of][K.sub.[phi]])] M(t, u) = P(t, [infinity]) + (1 + t)Q(t) [for all]t [member of] R,

where Q(t) = [summation over (k[greater than or equal to]0)] [[beta].sub.k][t.sup.k] is a formal series in t [member of] R with nonnegative integer coefficients [[beta].sub.k].

Suppose that X = H is a Hilbert space, [phi] [member of] [C.sup.2](H; R) and u [member of] [K.sub.[phi]]. We say that u is "nondegenerate", if [phi]"(u) [member of] L(H) is invertible. The "Morse index" m of u is defined to be the supremum of the dimensions of the vector subspaces of H on which [phi]"(u) is negative definite. If u [member of] [K.sub.[phi]] is isolated, nondegenerate, then

[C.sub.k]([phi], u) = [[delta].sub.k,m]Z [for all]k [greater than or equal to] 0,

where m is the Morse index of u and [[delta].sub.k,m] is the Kronecker symbol, that is,

[mathematical expression not reproducible].

3. PAIRS OF NONTRIVIAL SOLUTIONS

In this section we prove the existence of two nontrivial smooth solutions for problem (1.1) under conditions which permit resonance at [+ or -][infinity] and at zero.

More precisely the conditions on the reaction term f are the following:

[H(f).bar]: f: [OMEGA] x R [right arrow] R is a Caratheodory function such that f (z, 0) = 0 for a.a. z [member of] [OMEGA]

and

(i) : for every [??] > 0, there exists [a.sub.[??]] [member of] [L.sup.[infinity]] [([OMEGA]).sub.+] such that

[absolute value of (f(z, [zeta]))] [less than or equal to] [a.sub.[??]] (z) for a.a. z [member of] [OMEGA], all [absolute value of ([zeta])] [less than or equal to] [??];

(ii): there exist m [greater than or equal to] 1 and a function [??] [member of] [L.sup.[infinity]]([OMEGA]) such that

[mathematical expression not reproducible],

and

(f (z, [zeta]) - f(z, y)) ([zeta] - y) [greater than or equal to] [??](z) [([zeta] - y).sup.2] for a.a. z [member of] [OMEGA], all [zeta], y [member of] R;

(iii): [mathematical expression not reproducible] uniformly for a.a. z [member of] [OMEGA];

(iv): if F(z, [zeta]) = [[integral].sup.[zeta].sub.0] f (z, s) ds, then

[mathematical expression not reproducible];

(v) : there exist l [member of] N, l > m and [delta] > 0 such that

[[??].sub.l][[zeta].sup.2] [less than or equal to] f (z, [zeta])[zeta] [less than or equal to] [[??].sub.l+i][[zeta].sup.2] for a.a. z [member of] [OMEGA], all [absolute value of ([zeta])] [less than or equal to] [delta].

Remark 3.1. Hypothesis H(f)(iii) implies that at [+ or -][infinity] we can have resonance with respect to any nonprincipal eigenvalue. Similarly, hypothesis H(f)(v) says that at zero we can have double resonance at any spectral interval higher than the one corresponding to the asymptotic behaviour of and f(z, [zeta])/[zeta] and [zeta] [right arrow] [+ or -][infinity].

The hypotheses on the potential function [xi] and the boundary coefficient fl are the following:

[H([xi]).bar]: [xi] [member of] [L.sup.s]([OMEGA]) with s > N.

[H([beta]):.bar] [beta] [member of] [W.sup.1,[infinity]]([partial derivative][OMEGA]) with [beta](z) [greater than or equal to] 0 for all z [member of] [partial derivative][OMEGA].

Let <p: H 1(Q) --? R be the energy (Euler) functional for problem (1.1) defined

by

[phi](u) = 1/2 [phi](u) - [[integral].sub.[OMEGA]] F(z, u) dz [for all]u [member of] [H.sup.1]([OMEGA])

Evidently [phi] [member of] [C.sup.1]([H.sup.1]([OMEGA])). We set

[mathematical expression not reproducible]

We have the following orthogonal direct sum decomposition:

[H.sup.1]([OMEGA]) = Y [direct sum] V.

Proposition 3.2. If hypotheses H(f), H([xi]) and H([beta]) hold, then there exists a continuous map [tau]: V [right arrow] Y such that

[mathematical expression not reproducible].

Proof. Fix v [member of] V and consider the C 1-functional [[phi].sub.v]: [H.sup.1](Q) [right arrow] R defined by

[[phi].sub.v](u) = [phi](v + u) [for all]u [member of] [H.sup.1]([OMEGA]).

Let [i.sub.Y]: Y [right arrow] [H.sup.1]([OMEGA]) be the inclusion map. We set

[[??].sub.v] = [[phi].sub.v] [omicron] i.

Using the chain rule, we have

(3-1) [[??]'.sub.v] = [p.sub.Y*] [omicron] [[phi]'.sub.v],

with [p.sub.Y*] being the orthogonal projection of [H.sup.1][([OMEGA]).sup.*] onto [Y.sup.*]. In the sequel by <*, *> we denote the duality brackets for the pair ([Y.sup.*], Y). For [y.sub.1], [y.sub.2] [member of] Y we have

(3.2) [mathematical expression not reproducible]

(see (3.1), hypothesis H(f)(ii) and Proposition 2.1(b)).

Therefore -[[??]'.sub.v] is strongly monotone and -[[??].sub.v] is strictly convex. Note that

(3.3) [mathematical expression not reproducible],

for some [c.sub.3] > 0 (see (3.2)), so

(3.4) -[[??]'.sub.v] is coercive.

Since -[[??]'.sub.v] is continuous, monotone, it follows that

(3.5) -[[??]'.sub.v] is maximal monotone.

From (3.4) and (3.5) we infer that -[[??]'.sub.v] is surjective (see Gasmski- Papageorgiou [6, Corollary 3.2.31, p. 319]). Therefore we can find [y.sub.0] [member of] Y such that

(3.6) [[??]'.sub.v] ([y.sub.0]) = 0.

Since -[[??]'.sub.v] is strongly monotone, the solution [y.sub.0] [member of] Y of (3.6) is unique and is the unique minimizer of the strictly convex functional -[[??].sub.v] = - [[phi].sub.v][|.sub.Y]. Now, let [tau]: V [right arrow] Y be the map which to each v [member of] V assigns the unique solution [y.sub.0] [member of] Y of (3.6). From (3.1) and (3.6), we have

(3.7) [mathematical expression not reproducible].

We examine the continuity of the map [tau] : V [right arrow] Y. So, let [v.sub.n] [right arrow] v in V. We have

[mathematical expression not reproducible]

(see (3.6) and (3.3)), so the sequence [{[tau]([v.sub.n])}.sub.n[greater than or equal to]1] [subset or equal to] Y is bounded. Since Y is finite dimensional, by passing to a subsequence if necessary, we may assume that

(3.8) [tau]([v.sub.n]) [right arrow] [??] [member of] Y in Y.

From the continuity of [phi], we have

(3.9) [mathematical expression not reproducible]

(see (3.8)). From (3.7), we have

[phi]([v.sub.n] + y) [less than or equal to] [phi]([v.sub.n] + [tau](vn)) [for all]y [member of] Y, n [greater than or equal to] 1,

so

[phi](v + y) [less than or equal to] [phi](v + [??]) [for all]y [member of] Y

(see (3.9)), thus [??] = [tau](v) (see (3.7)).

By the Urysohn criterion for the convergence of subsequences (see, for example, Gasinski-Papageorgiou [7, Problem 1.3, p. 33]), for the original sequence we have

[tau]([v.sub.n]) [right arrow] [tau](v) in Y,

so [tau]: V [right arrow] Y is continuous.

We set

[psi](v) = [phi](v + [tau](v)) [for all]v [member of] V.

From Proposition 3.2 it follows that [psi] [member of] C(V; R). In fact we can say more, namely that f is continuously differentiable on V.

Proposition 3.3. If hypotheses H(f), H([xi]) and H([beta]) hold, then [psi] [member of] [C.sup.1](V; R) and

[psi]'(v) = [p.sub.V*][phi](v + [tau](v)) [for all]v [member of] V,

here [p.sub.V*] is the orthogonal projection of [H.sup.1][([OMEGA]).sup.*] onto [V.sup.*].

Proof. Let v, w [member of] V and t > 0. We have

1/t ([psi](v + tw) - [psi](v)) [greater than or equal to] 1/t ([phi](v + tw + [tau](v)) - [phi](v + [tau](v)))

(see (3.7)), so

(3.10) [mathematical expression not reproducible]

Also, we have

1/t ([psi] (v + tw) - [psi](v)) [less than or equal to] 1/t ([phi]{v + tw + t(v + tw)) - [phi](v + [tau](v + w)))

(see (3.7)), so

(3.11) [mathematical expression not reproducible]

(since T is continuous by Proposition 3.2 and [phi] [member of] [C.sup.1]([H.sup.1]([OMEGA]); R)). Then from (3.10) and (3.11) it follows that

(3.12) [mathematical expression not reproducible].

In a similar fashion we show that

(3.13) [mathematical expression not reproducible].

By [<*, *>.sub.V] we denote the duality brackets for the pair ([V.sup.*], V). From (3.12) and (3.13), we conclude that

[<[psi]'(v),w>.sub.V] = <[phi]'(v + [tau](v)),w> [for all]v, w [member of] V,

so

[psi]'(v) = [p.sub.V*][phi]'(v + [tau](v)),

thus [psi] [member of] (V;R) (recall that [tau] is continuous; see Proposition 3.2).

Note that in contrast to the usual reduction method (see Amann [1] and Castro- Lazer [2]), in our case the reduction is done on infinite dimensional space (the space V). This is a consequence of hypothesis H(f)(ii) (in the spectral interval [[[??].sub.m], [[??].sub.m+1]] at [+ or -][infinity] we can have resonance with Am+1, but only nonuniform nonresonance with respect to [[??].sub.m]). In addition, here the reaction term f is only a Caratheodory function (no differentiability condition on f (z, *) is required) and so the energy functional is only [C.sup.1] and not [C.sup.2] on [H.sup.1]([OMEGA]). These special features, lead to some technical difficulties. Nevertheless, with no extra conditions, we are able to overcome these difficulties and prove the following result.

Proposition 3.4. If hypotheses H(f), H([xi]) and H([beta]) hold, then the functional f is coercive.

Proof. Let [??] = [phi][|.sub.V]. Clearly [??} [member of] [C.sup.1](V; R) and so as before via the chain rule, we have

(3.14) [mathematical expression not reproducible] (recall that [p.sub.V*] denotes the orthogonal projection of [H.sup.1][([OMEGA]).sup.*] onto [V.sup.*]).

Claim 1. The functional [??] satisfies the Cerami condition.

We consider a sequence {[v.sub.n]} [subset or equal to] V such that

(3.15) [absolute value of ([??]([v.sub.n]))] [less than or equal to] [M.sub.1] [for all]n [member of] N (for some [M.sub.1] > 0),

(3.16) (1 + [parallel][v.sub.n][parallel][parallel])[??]'([v.sub.n]) [right arrow] 0 in [V.sup.*] as n [right arrow] +[infinity].

From (3.16), we have

[absolute value of ([<[??]([v.sub.n]), h>.sub.V])] [less than or equal to] [[epsilon].sub.n][parallel]h[parallel]/1 + [parallel][v.sub.n][parallel] [for all]n [greater than or equal to] 1, h [less than or equal to] V,

with [[epsilon].sub.n] [right arrow] [0.sup.+], so

(3.17) <[phi]([v.sub.n]),h> [less than or equal to] [epsilon]n[parallel]h[parallel]/1+[parallel][v.sub.n][parallel], [for all]n [greater than or equal to] 1, h [member of] V

(see (3.14)).

In (3.17) we choose h = [v.sub.n] [member of] V. Then

(3.18) [gamma]([v.sub.n]) - [[integral].sub.[OMEGA]]f(z, [v.sub.n])[v.sub.n]dz [less than or equal to] [[epsilon].sub.n] [for all]n [greater than or equal to] 1.

We will show that the sequence {vn} C V is bounded. Arguing by contradiction, suppose that at least for a subsequence, we have

(3.19) [parallel][v.sub.n][parallel] [right arrow] +[infinity].

Let [[??].sub.n] = [v.sub.n]/[parallel][v.sub.n][parallel], n [greater than or equal to] 1. Then [parallel][[??].sub.n][parallel] = 1 for all n [less than or equal to] 1 and so passing to a next subsequence if necessary, we may assume that

(3.20) [mathematical expression not reproducible].

Hypotheses H(f) imply that

(3.21) [absolute value of (f (z, [zeta]))] [less than or equal to] [c.sub.4] [absolute value of ([zeta])] for a.a. z [member of] [OMEGA], all [zeta] [member of] R,

for some [c.sub.4] > 0. We return to (3.18) and use (3.21). We obtain

[gamma]([v.sub.n]) - [c.sub.4][[parallel][v.sub.n][parallel].sup.2.sub.2] [[epsilon].sub.n] [[for all].sub.n] [greater than or equal to] 1,

so

[gamma]([v.sub.n]) + [mu][[parallel][v.sub.n][parallel].sup.2.sub.2] [member of] ([c.sub.4] + [mu])[[parallel][v.sub.n][parallel].sub.2] [member of] [[epsilon].sub.n],

with [mu] > 0 as in (2.3). Using also (2.3), we get

[c.sub.0] [less than or equal to] ([c.sub.4] + [mu]) [[parallel][??][parallel].sup.2.sub.2]

so

[c.sub.0] - ([c.sup.4] + [mu]) [[parallel][[??].sub.n][parallel].sup.2.sub.2] [[epsilon].sub.n]/[[parallel][v.sub.n][parallel].sup.2] [for all]n[greater than or equal to]1,

thus by (3.19) and (3.20), we get

[c.sub.0] [less than or equal to] ([c.sub.4] + [mu])[[absolute value of (v)].sub.2.sub.2],

hence

(3.22) [??] [not equal to] 0.

Let [[OMEGA].sup.*] = {z [member of] [OMEGA] : [??](z) = 0}. We have [[absolute value of ([[OMEGA].sup.*])].sub.N] > 0, with [absolute value of (*)]N being the Lebesgue measure on [R.sup.N] and

[absolute value of ([v.sub.n](z))] [right arrow] +[infinity] for a.a. z [member of] [[OMEGA].sup.*], so

f (z, [v.sub.n](z))Vn(z) - 2F(z,Vn(z)) [right arrow] +[infinity] for a.a. z [less than or equal to] [[OMEGA].sup.*]

(see hypothesis H(f)(iv)), thus

(3.23) [[integral].sub.[OMEGA]*] (f (z,[v.sub.n])[v.sub.n] - 2F(z,[v.sub.n])) dz [right arrow] +[infinity]

(see hypothesis H(f)(iv) and use Fatou's lemma).

Hypothesis H(f)(iv) implies that we can find [M.sub.2] > 0 such that

(3.24) f(z, [zeta])[zeta] - 2F(z, [zeta]) [greater than or equal to] 0 for a.a. z [member of] [OMEGA], all [absolute value of [zeta]] [greater than or equal to] [M.sub.2].

Then we have

[mathematical expression not reproducible],

for some [c.sub.5] > 0 (see (3.24) and hypothesis H(f)(i)), so

(3.25) [[integral].sub.[OMEGA]] (f(z,[v.sub.n])[v.sub.n] - 2F(z,[v.sub.n])) dz [right arrow] as n [right arrow] +[infinity]

(see (3.23)).

From (3.15), we have

(3.26) [gamma]([v.sub.n]) + [integral].sub. 2] F(z, [V.sub.n]) dz [less than or equal to] 2[M.sub.1] [greater than or equal to] 1.

Also from (3.17) with h = [v.sub.n] [member of] V, we have

(3.27) -[gamma]([v.sub.n])+ [[integral].sub.[OMEGA]] f(z, [v.sub.n])[v.sub.n] dz [less than or equal to] [[epsilon].sub.n] [for all]n [greater than or equal to] 1.

We add (3.26) and (3.27) and obtain

(3.28) [[integral].sub.[OMEGA]] (f(z, [v.sub.n])[v.sub.n] - 2F (z, [v.sub.n])) dz [less than or equal to] [M.sub.3] [for all] [greater than or equal to] 1,

for some [M.sub.3] > 0. Comparing (3.25) and (3.28) we get a contradiction. This proves that the sequence [{[v.sub.n]}.sub.n[greater than or equal to]1] [subset or equal to] V is bounded.

Therefore, passing to a subsequence if necessary, we may assume that

(3.29) [v.sub.n] [??] v in [H.sup.1]([OMEGA]) and [v.sub.n] [right arrow] v in [L.sup.2](Q) and in [L.sup.2]([partial derivative][OMEGA]).

From (3.17), we have

(3.30) [mathematical expression not reproducible],

with [[epsilon].sub.n] [right arrow] [0.sup.+]. Choosing h = [v.sub.n] - v [member of] [H.sup.1]([OMEGA]) in (3.30), passing to the limit as n [right arrow] +[infinity] and using (3.29), we obtain

[mathematical expression not reproducible],

so

[[parallel]D[v.sub.n][parallel].sub.2] [right arrow] [[parallel][D.sub.v][parallel].sub.2],

thus, by the Kadec-Klee property for Hilbert spaces (see (3.29)), we get

hence [??] satisfies the Cerami condition. This proves Claim 1.

Claim 2. [[??].sub.m+1][[xi].sup.2] - 2F(z, [xi]) [right arrow] uniformly for a.a. z [member of] [OMEGA] as [xi] [right arrow] +[infinity].

Hypothesis H(f)(iv) implies that given n > 0, we can find [M.sub.4] = [M.sub.4]([eta]) > 0 such that

(3.31) f (z, [zeta])[zeta] - 2F(z, Z) [greater than or equal to] [eta] for a.a. z [member of] [OMEGA], all [absolute value of (Z)] [greater than or equal to] [M.sub.4].

We have

[mathematical expression not reproducible]

(see (3.31)), so

(3.32) [mathematical expression not reproducible].

Hypotheses H(f)(ii) and (iii) imply that

[mathematical expression not reproducible],

so

(3.33) [mathematical expression not reproducible].

In (3.32) we let [absolute value of (y)] [right arrow] +[infinity]. Using (3.33) we obtain

[[??].sub.m+1][[absolute value of (u)].sup.2] - 2F(z,u) [greater than or equal to] [eta] for a.a. z [member of] [OMEGA], all [absolute value of (u)] [greater than or equal to] [M.sub.4].

Since [eta] > 0 is arbitrary, it follows that

[[??].sub.m+1] [[absolute value of (u)].sup.2] - 2F(z,u) [right arrow] + [infinity] uniformly for a.a. z [member of] [OMEGA], as u [right arrow] [+ or -][infinity].

This proves Claim 2.

For every v [member of] V, we have

[mathematical expression not reproducible],

for some [c.sub.5] > 0 (recall that [??] = [phi][|.sub.V], see (2.4), Claim 2 and hypothesis H(f)(i)), so

(3.34) [??] is bounded below.

Then (3.34), Claim 1 and Proposition 5.22 of Motreanu-Motreanu-Papageorgiou [10, p. 103] imply that (is coercive. From Proposition 2.1 we have

[mathematical expression not reproducible],

so ft is coercive (since (p is coercive).

Corollary 3.5. If hypotheses H(f), H([xi]) and H([beta]) hold, then [psi] is bounded below, satisfies the Cerami condition and

[C.sub.k]([phi], [infinity]) = [[delta].sub.k,0]Z [for all]k [greater than or equal to] 0.

We assume that [K.sub.[phi]] is finite. Otherwise we already have an infinity of solutions and so we are done.

Proposition 3.6. If hypotheses H(f), H([xi]) and H([beta]) hold, then

[mathematical expression not reproducible],

with [mathematical expression not reproducible].

Proof. Let

[mathematical expression not reproducible].

We have the following orthogonal direct sum decomposition

[mathematical expression not reproducible].

Then every u G H 1(Q) admits a unique sum decomposition of the form

(3.35) [mathematical expression not reproducible].

Let [lambda] [member of] ([[??].sub.l], [[??].sub.l+1]) and consider the [C.sup.2]-functional (0: [H.sup.1]([OMEGA]) [right arrow] R defined by

[[phi].sub.0](u) = 1/2 [gamma](u) - [lambda]/2[[parallel]u[parallel].sup.2.sub.2] [for all]u [member of] [H.sup.1]([OMEGA]).

We consider the homotopy h(t, u) defined by

h(t, u) = (1 - t)<[phi](u) + t[[phi].sub.0](u) [for all](t, u) [member of] [0,1] x [H.sup.1]([OMEGA]).

Suppose that we can find two sequences [{[t.sub.n]}.sub.n[greater than or equal to]1] [subset or equal to] [0,1] and [{[u.sub.n]}.sub.n[greater than or equal to]1] [subset or equal to] [H.sup.1]([OMEGA])\{0} such that

(3.36) [mathematical expression not reproducible].

Since [K.sub.[phi]] is finite, we may assume that [t.sub.n] [not equal to] 0 for all n [greater than or equal to] 1. We have

(3.37) (1 - [t.sub.n])<[phi]'([u.sub.n]),h> + [t.sub.n]<[[phi]'.sub.0]([u.sub.n]), h> = 0 [for all]n [greater than or equal to] 1, h [member of] [H.sup.1]([OMEGA]), so

(3.38) [mathematical expression not reproducible],

so

[mathematical expression not reproducible]

(see Papageorgiou-Radulescu [13]). The regularity theory of Wang [18] implies that there exists [alpha] [member of] (0,1) and [M.sub.5] > 0 such that

(3.39) [mathematical expression not reproducible].

Recall that [C.sup.1,[alpha]]([OMEGA]) is embedded compactly into [C.sup.1]([bar.[OMEGA]]). So, from (3.36) and (3.39), we have

(3.40) [u.sub.n] [right arrow] 0 in [C.sup.1]([bar.[OMEGA]]) as n [right arrow] +[infinity].

From (3.40) it follows that we can find [n.sub.0] [greater than or equal to] 1 such that

(3.41) [u.sub.n](z) [member of] [-[delta],[delta]] [for all]z [member of] [bar.[OMEGA]], all n [greater than or equal to] [n.sub.0].

In (3.38) we choose [mathematical expression not reproducible] (see (3.35)). Exploiting the orthogonality of the component spaces, we have

(3.42) [mathematical expression not reproducible].

Note that when [u.sub.n](z) [not equal to] 0, we have

(3.43) [mathematical expression not reproducible]

(see (3.35) and hypothesis H(f)(v)).

When [u.sub.n](z) = 0, then f (z, [u.sub.n](z)) = 0 and [[??].sub.n](z) = - [[??].sub.n](z) (see (3.35)). Hence

(3.43) remains valid.

We return to (3.42) and use (3.43). Recalling that [t.sub.n] [not equal to] 0 for all n [greater than or equal to] 1 and that [lambda] [member of] ([[??].sub.1], [[??].sub.l+1]), we have

[mathematical expression not reproducible]

so

[mathematical expression not reproducible]

(see Proposition 2.1), thus

[mathematical expression not reproducible],

hence

[u.sub.n] = 0 [for all]n [greater than or equal to] [n.sub.0],

a contradiction. This shows that (3.36) cannot occur. Then the homotopy invariance of critical groups (see Gasmski-Papageorgiou [8, Theorem 5.125, p. 836]) implies that

(3.44) [C.sub.k]([phi], 0) = [C.sub.k] ([[phi].sub.0], 0) [for all]k [greater than or equal to] 0.

Recall that [lambda] [member of] ([[??].sub.l], [[??].sub.l+1]). Hence [mathematical expression not reproducible] = {0} and u = 0 is a nondegenerate critical point of [[phi].sub.0] with Morse index [mathematical expression not reproducible]. Since [[phi].sub.0] [member of] [C.sup.2]([H.sup.1]([OMEGA])), we can apply Theorem 6.51 of Motreanu-Motreanu-Papageorgiou [10, p. 155] and have that

[mathematical expression not reproducible],

so

[mathematical expression not reproducible]

(see (3.44))

Using Proposition 3.6 and Theorem 1.2 of Li-Liu [9], we obtain the following result.

Corollary 3.7. If hypotheses H(f), H([]xi) and H([beta]) hold, then

[mathematical expression not reproducible].

The next result is an easy observation which relates the critical sets [K.sub.[phi]] and [K.sub.[psi]].

Proposition 3.8. If hypotheses H(f), H([xi]) and H([beta]) hold, then v [member of] [K.sub.[psi]] if and only if v + T(v) [member of] [K.sub.[psi]].

Proof. "[??]": We have

(3.45) 0 = [psi]'(v) = [p.sub.V*][phi]'(v + [tau](v))

(see Proposition 3.3). We know that [H.sup.1][([OMEGA]).sup.*] = [Y.sup.*] [direct sum] [V.sup.*]. So, from (3.45), it follows that [phi]'(v + [tau](v)) [member of] [Y.sup.*]. But from (3.7), we have [p.sub.Y]*[phi]'(v + [tau](v)) = 0, so p'(v + [tau](v)) = 0, thus v + [tau](v) [member of] [K.sub.[phi]].

"[??]": Follows from Proposition 3.3.

Now we are ready for the multiplicity theorem for problem (1.1). We produce two nontrivial smooth solution.

Theorem 3.9. If hypotheses H(f), H([xi]) and H([xi]) hold, then problem (1.1) admits at least two nontrivial solutions

[u.sub.0], [??] [member of] [C.sup.1] ([bar.[OMEGA]]), [u.sub.0] [not equal to] [??].

Proof. From Proposition 3.4 we know that f is coercive. Also, the Sobolev embedding theorem, the compactness of the trace operator and the continuity of the map T (see Proposition 3.2) imply that f is sequentially weakly lower semicontinuous. So, by the Weierstrass-Tonelli theorem, we can find [v.sub.0] [member of] V such that

[mathematical expression not reproducible],

so

(3.46) [mathematical expression not reproducible].

From Corollary 3.7, we have

(3.47) [mathematical expression not reproducible].

According to hypothesis H(f)(v), l > m. Hence [d.sub.l] > [d.sub.m]. So, comparing (3.46) and

(3.47), we infer that [v.sub.0] [not equal to] 0, therefore [u.sub.0] = [v.sub.0] + t([v.sub.0]) [member of] [K.sub.[phi]]\{0} (see Proposition 3.8).

Hypotheses H(f) imply that

(3.48) [absolute value of (f (z, [zeta]))] [less than or equal to] [c.sub.6][absolute value of ([zeta])] for a.a. [zeta] [member of] H, all [zeta] [member of] R,

for some [c.sub.6] > 0. We have

(3.49) [mathematical expression not reproducible],

We define

[mathematical expression not reproducible].

Evidently [eta] [member of] [L.sup.[infinity]](H) (see (3.48)). From (3.49), we have

[mathematical expression not reproducible],

with [??] = [eta] - [xi] [member of] [L.sup.s]([OMEGA]), s > N (see hypothesis H([xi])). From Lemma 5.1 of Wang [18], we have that [u.sub.0] [member of] [L.sup.[infinity]](Q). Then using the Calderon-Zygmund estimates (see Lemma 5.2 of Wang [18]), we obtain

[u.sub.0] [member of] [W.sup.2,s]([OMEGA])

so

[u.sub.0] [member of] [C.sup.1,[alpha]] ([bar.[OMEGA]]),

with [alpha] = 1 - N/s > 0 (by the Sobolev embedding theorem).

Suppose that K[psi] = {0, [v.sup.0]}. Then from (3.46), (3.47), Corollary 3.5 and the Morse relation with t = -1 (see (2.5)), we have

[mathematical expression not reproducible],

a contradiction. So, there exists [??] [member of] [K.sub.[psi]]\{0,[v.sub.0]}. Then

[mathematical expression not reproducible]

and as before using the regularity theory of Wang [18], we have [??] [member of] [C.sup.1]([bar.[OMEGA]]).

Received February 3, 2017

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LESZEK GASINSKI AND NIKOLAOS S. PAPAGEORGIOU

Faculty of Mathematics and Computer Science, Jagiellonian University, ul. Lojasiewicza 6, 30-348 Krakow, Poland Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece.

The research was supported by the National Science Center of Poland under Projects No. 2015/19/B/ST1/01169 and 2012/06/A/ST1/00262.

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Author: | Gasinski, Leszek; Papageorgiou, Nikolaos S. |
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Publication: | Dynamic Systems and Applications |

Article Type: | Report |

Date: | Mar 1, 2017 |

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