EXISTENCE RESULTS FOR AN ELLIPTIC DIRICHLET PROBLEM

LE MATEMATICHE Vol. LXVI (2) Fasc. I pp. 33 4 doi:.448/2.66..3 EXISTENCE RESULTS FOR AN ELLIPTIC DIRICHLET PROBLEM GIUSEPPINA D AGUÌ - GIOVANNI MOLICA BISCI The main purpose of this paper is to present recent existence results for an elliptic eigenvalue Dirichlet problem. Precisely our method ensures the existence of an exactly determined open interval (possibly unbounded) of positive parameters for which the problem admits infinitely many weak solutions.. Introduction This note concerns existence of infinitely many solutions to the Dirichlet problem { p u + q(x) u p 2 u = f (u) in Ω (D q f u Ω = ) where Ω is a bounded open subset of the Euclidean space (R N ) N with boundary Ω of class C and Lebesgue measure meas(ω). Moreover p u := div( u p 2 u) p > N is the usual Laplace operator f : R R is a continuous function q L (Ω) and is a positive real parameter. The existence of infinitely many solutions for problem (D f ) has been widely investigated. The most classical results in this topic are essentially based on the Ljusternik-Schnirelman theory. Entrato in redazione: 2 ottobre 2 AMS 2 Subject Classification: 35J2 35J65. Keywords: Critical points Variational methods Dirichlet problem Weak solutions.

34 GIUSEPPINA D AGUÌ - GIOVANNI MOLICA BISCI In them a key role is played by the oddness of the nonlinearity. Moreover conditions which do not allow an oscillating behaviour of the nonlinearity f are necessary in order to check the Palais-Smale condition. Multiplicity results with an oscillating behaviour of f are more rare. In this direction very recently in [2] under some hypotheses on the behavior of the potential of the nonlinear term at infinity the existence of an interval Λ such that for each Λ problem (D q f ) admits a sequence of pairwise distinct weak solutions is proved (see Theorem 3.). Moreover replacing the conditions at infinity of the potential by a similar at zero the same results hold and in addition the sequence of pairwise distinct solutions uniformly converges to zero (see Remark.4). In order to recall our result let us introduce some constants associated to the geometry of the set Ω. Define σ(n p) := inf µ ][ µ N µ N ( µ) p and consider µ ][ such that σ(n p) = µn µ N ( µ) p. Further let τ := sup x Ω dist(x Ω) and κ := where q := esssup x Ω q(x) and µ N meas(ω)m p (µ N σ(n p) + q g µ (pn)) () ( m := N p [Γ + N π 2 g µ (pn) := µ N + )] N ( ) p p meas(ω) N p p N ( µ) p NB (µ)(n p + ). Here B (µ) (N p + ) denotes the generalized incomplete beta function defined as follows B (µ) (N p + ) := t N ( t) (p+) dt µ and Γ is the Gamma function. Throughout the sequel ω τ denotes the measure of the N-dimensional ball of radius τ ξ := f (t)dt

for each ξ R and EXISTENCE RESULTS FOR AN ELLIPTIC PROBLEM 35 B := limsup ξ p. Our existence result obtained in [2] can be stated as follows Theorem.. Let f : R R be a continuous function. Assume that (i) for every ξ ; (ii) there exist two sequences {a n } and {b n } such that a n < ( µ N/p σ(n p) g µ (pn)) /p b n m /p + q ω µ N τ for every n N and lim n + b n = + such that where κ is given by () and Then for every A := lim n + A < meas(ω)κb (2) meas(ω)max t bn F(t) µ N ω τ F(a n ) b p n m p a p nω τ [ σ(n p) ] ( σ(n p) g µ (pn)) Λ qg := pb + q µ N g µ (pn)]. + q µ N µ N m p pa the problem (D q f ) admits a sequence of weak solutions which is unbounded in W p (Ω). The main tool in order to obtain Theorem. is a refinement of the Variational Principle of Ricceri (see the quoted paper []) recently obtained in []. Remark.2. Condition (ii) in Theorem. is technical and could be replaced by the more simple and sufficient assumption (ii max t ξ F(t) ) liminf ξ p < κ limsup ξ p. [

36 GIUSEPPINA D AGUÌ - GIOVANNI MOLICA BISCI In this setting if for every max t ξ F(t) A := liminf ξ p ] ( σ(n p) Λ g µ (pn)) qg := pb + q µ N m p pa the problem (D q f ) admits a sequence of weak solutions which is unbounded in W p (Ω). If in addition the nonlinearity f is non-negative hypothesis (ii ) can be written as follows (ii ) liminf ξ p < κ limsup ξ p. Remark.3. We observe that in the very interesting paper [8] the authors assume lim inf ξ p = and limsup ξ p = + which are conditions that imply our key assumptions. Moreover when q our theorems and the results in [7] and [8] are mutually independent (see Theorem. Example 4. and Remark 4. in [2]). Remark.4. Replacing the condition (ii) at infinity by the following one at zero (jj) there exist two sequences {a n } and {b n } such that a n < ( µ N/p σ(n p) g µ (pn)) /p b n m /p + q ω µ N τ for every n N and lim n + b n = such that A < κ meas(ω)limsup ξ + ξ p a sequence of pairwise distinct solutions uniformly converging to zero is obtained. Remark.5. We point out that the results contained in [4] are direct consequences of the main Theorem. On the other hand we do not require as in [6 Corollary 3.] that the function f is definitively non-positive. [

2. Main Results EXISTENCE RESULTS FOR AN ELLIPTIC PROBLEM 37 Taking Remark.2 into account a particular case of Theorem. reads as follows. Theorem 2.. Let f : R R be a continuous non-negative function and assume that lim inf ξ p < m p meas(ω)σ(n p) limsup ξ p. Then for each the prob- lem σ(n p) p limsup ξ p pm p meas(ω)liminf ξ p { p u = f (u) in Ω u Ω = (D f ) admits a sequence of pairwise distinct positive weak solutions in W p (Ω). The next result is a simpler but less general form of Theorem. in the ordinary case (see [5 Theorem.]). Theorem 2.2. Let f : R R be a continuous non-negative function. Assume that the following condition holds f 2 ) liminf ξ 2 < 4 limsup ξ 2. Then for each Λ := 8 lim sup ξ 2 2 lim inf ξ 2 for every non-negative continuous function g : R R such that g ) G := lim ξ g(t) dt < + ξ 2

38 GIUSEPPINA D AGUÌ - GIOVANNI MOLICA BISCI and for every µ [ µ g [ where µ g := ( G 2 liminf ξ 2 ) the following problem { u = f (u) + µg(u) in ][ u() = u() = admits a sequence of pairwise distinct positive classical solutions. (P f g ) Example 2.3. For instance for each (µ) Λ [+ [ the following Dirichlet problem { u = f (u) + µ u in ][ where u() = u() = ucos 2 (ln(u)) if u > f (u) := if u. possesses a sequence of pairwise distinct positive classical solutions. Here we present a simple and direct consequence of Theorem. that improves Proposition. in [4] (see also Remark 2.5). Proposition 2.4. Let {a n } {b n } be two sequences in ]+ [ a n < b n < a n+ n n for some n N lim b n = + and lim = +. Moreover let n + n + a n ϕ C ([]) be a non-negative and non-zero function such that ϕ() = ϕ() = ϕ () = ϕ () = and let g : R R be the function defined by ( t bn ) ϕ if t [b n a n+ ] g(t) := a n+ b n n n (3) otherwise. Then for every > ϕ := b n ( σ(n p) g µ (pn)) p max ϕ(s) + q µ N s [] the problem { p u + q(x) u p 2 u = h(u) in Ω where u Ω = h(u) := u p (pg(u) + ug (u)) admits an unbounded sequence of positive weak solutions in W p (Ω). (D qh )

EXISTENCE RESULTS FOR AN ELLIPTIC PROBLEM 39 Proof. Let {a n }{b n } be two positive sequence satisfying our assumptions. We claim that all the hypotheses of Theorem. are verified. In fact taking into account the definition of ϕ one has that g is a C (R)-function and ξ = h(t)dt = ξ p g(ξ ) ξ R +. Hence hypothesis (i) is verified. Moreover < a n < ( µ N/p σ(n p) g µ (pn)) /p b n m /p + q ω µ N τ for every n sufficiently large. Then condition (2) holds. Indeed direct computations ensure that lim sup ξ p = limsupg(ξ ) = max ϕ(s) s [] and A =. Hence for every > ϕ Theorem. guarantees the existence of an unbounded sequence in W p (Ω) of weak solutions of problem (D qh ). Finally arguing as in [2 Remark 3.3] the Strong Maximum Principle (see [9 Theorem.]) ensures that the obtained solutions are positive. Remark 2.5. We point out that if {a n } {b n } are two sequences in ]+ [ such that b n+ < a n < b n n n for some n N lim b n = lim n + and g : R R is defined by ( t bn+ ) ϕ if t [b n+ a n ] g(t) := a n b n+ n n otherwise n + b n a n = + then for every > ϕ the problem (D qh ) admits a sequence of pairwise distinct positive weak solutions which strongly converges to zero in W p (Ω). For instance let a n := and b n := n!n n! for every n 2 and ϕ C ([]) given by ( ) ϕ(s) := exp s [] s(s ) and zero otherwise.

4 GIUSEPPINA D AGUÌ - GIOVANNI MOLICA BISCI Hence let g be the C (R)-function given by g(t) := exp ( n(n + )! t )( n (n + )! ( t for every t [ (n + )! ] and zero otherwise. n!n n 2 Consider the problem { p u + e u p 2 u = h(u) in Ω where Then for every u Ω = h(t) := u p (pg(u) + ug (u)). > e4 p ( σ(n p) + e g µ(pn) µ N ) ) (n + )! (n + )! ) (D eh ) problem (D eh ) possesses a sequence of pairwise distinct positive weak solutions which strongly converges to zero in W p (Ω). Finally we just observe that by using the same variational approach in [3] the existence of infinitely many weak solutions for quasilinear elliptic systems has been widely investigated. A special case reads as follows; see [3 Theorem.2]. Theorem 2.6. Let Ω R N be a non-empty bounded open set with boundary of class C. Let f g : R 2 R be two positive C (R 2 )-functions such that the differential -form ω := f (ξ η)dξ + g(ξ η)dη is integrable and let F be a primitive of ω such that F() =. Fix pq > N with p q and assume that Then the problem F(y y) lim inf y + y p = and limsup y + F(y y) y q p u = f (uv) in Ω q v = g(uv) in Ω u Ω = v Ω = = +. admits an unbounded sequence {(u n v n )} W p (Ω) W q (Ω) of positive weak solutions.

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