On some variational problems in Riemannian and Fractal Geometry - PowerPoint PPT Presentation

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References A problem on Riemannian manifolds A remarkable case of problem ( P λ ) is σ ∈ S d , w ∈ H 2 1 ( S d ) , ( S λ ) − ∆ h w + s (1 − s − d ) w = λK ( σ ) f ( w ) , where S d is the unit sphere in R d +1 , h is the standard metric induced by the embedding S d ֒ → R d +1 , s is a constant such that 1 − d < s < 0, and ∆ h denotes the Laplace-Beltrami operator on ( S d , h ). Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References A problem on Riemannian manifolds Existence results for problem ( S λ ) yield, by using an appropriate change of coordinates, the existence of solutions to the following parameterized Emden-Fowler equation x ∈ R d +1 \ { 0 } . − ∆ u = λ | x | s − 2 K ( x/ | x | ) f ( | x | − s u ) , ( F λ ) Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References A problem on Riemannian manifolds Moreover, we observe that the existence of a smooth positive solution 4 d − 2 t, for problem ( S λ ), when s = − d/ 2 or s = − d/ 2 + 1 , and f ( t ) = | t | can be viewed as an affirmative answer to the famous Yamabe problem on S d . For these topics we refer to Aubin, Cotsiolis and Iliopoulos, Hebey, Kazdan and Warner, V´ azquez and V´ eron, and to the excellent survey by Lee and Parker. In these cases the right hand-side of problem ( S λ ) involves the critical Sobolev exponent. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References A problem on Riemannian manifolds Cotsiolis and Iliopoulos as well as V´ azquez and V´ eron studied problem ( F λ ) in A. Cotsiolis and D. Iliopoulos ´ Equations elliptiques non lin´ eaires ` a croissance de Sobolev sur-critique, Bull. Sci. Math. 119 (1995), 419–431. J.L. V´ azquez and L. V´ eron Solutions positives d’´ equations elliptiques semi-lin´ eaires sur des vari´ et´ es riemanniennes compactes , C. R. Acad. Sci. Paris, S´ er. I Math. 312 (1991), 811–815. by applying either minimization or minimax methods, provided that f ( t ) = | t | p − 1 t , with p > 1. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Successively, in A. Krist´ aly, V. R˘ adulescu Sublinear eigenvalue problems on compact Riemannian manifolds with applications in Emden-Fowler equations , Studia Math. 191 (2009), 237–246. the authors are interested on the existence of multiple solutions of problem ( P λ ) in order to obtain solutions for parameterized Emden-Fowler equation ( F λ ) considering nonlinear terms of sublinear type at infinity. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References A problem on Riemannian manifolds In particular, for λ sufficiently large, the existence of two nontrivial solutions for problem ( P λ ) has been successfully obtained through a careful analysis of the standard mountain pass geometry. Theorem 9.2 p. 220 in A. Krist´ aly, V. R˘ adulescu and Cs. Varga Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems , Cambridge University press, 2010. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References A problem on Riemannian manifolds Further, Krist´ aly, R˘ adulescu and Varga proved the existence of an open interval of positive parameters for which problem ( P λ ) admits two distinct nontrivial solutions by using an abstract three critical points theorem due to Bonanno. G. Bonanno Some remarks on a three critical points theorem , Nonlinear Anal. TMA 54 (2003), 651–665. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Some basic facts We start this section with a short list of notions in Riemmanian geometry. We refer to T. Aubin Nonlinear Analysis on Manifolds. Monge–Ampe` ere Equations , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252. Springer-Verlag, New York, 1982. and E. Hebey Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities , Courant Lecture Notes in Mathematics, New York, 1999. for detailed derivations of the geometric quantities, their motivation and further applications. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Some basic facts Let ( M , g ) be a smooth compact d -dimensional ( d ≥ 3) Riemannian manifold without boundary and let g ij be the components of the metric g . As usual, we denote by C ∞ ( M ) the space of smooth functions defined on M . Let α ∈ C ∞ ( M ) be a positive function and put � α � ∞ := max σ ∈M α ( σ ). Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Some basic facts For every w ∈ C ∞ ( M ) , set � � � w � 2 |∇ w ( σ ) | 2 dσ g + α ( σ ) | w ( σ ) | 2 dσ g , α := H 2 M M where ∇ w is the covariant derivative of w, and dσ g is the Riemannian measure. In local coordinates ( x 1 , . . . , x d ), the components of ∇ w are given by ∂ 2 w ∂w ( ∇ 2 w ) ij = ∂x i ∂x j − Γ k ∂x k , ij where ij := 1 � ∂g lj ∂x i + ∂g li ∂x j − ∂g ij � Γ k g lk , ∂x k 2 are the usual Christoffel symbols and g lk are the elements of the inverse matrix of g . Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Some basic facts Here, and in the sequel, the Einstein’s summation convention is adopted. Moreover, the measure element dσ g assume the form dσ g = √ det g dx , where dx stands for the Lebesgue’s volume element of R d . Hence, let � Vol g ( M ) := dσ g . M In particular, if ( M , g ) = ( S d , h ), where S d is the unit sphere in R d +1 and h is the standard metric induced by the embedding S d ֒ → R d +1 , we set � ω d := Vol h ( S d ) := S d dσ h . Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Some basic facts The Sobolev space H 2 α ( M ) is defined as the completion of C ∞ ( M ) α . Then H 2 with respect to the norm � · � H 2 α ( M ) is a Hilbert space endowed with the inner product � � � w 1 , w 2 � H 2 α = �∇ w 1 , ∇ w 2 � g dσ g + α ( σ ) � w 1 , w 2 � g dσ g , M M where �· , ·� g is the inner product on covariant tensor fields associated to g. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Some basic facts Since α is positive, the norm � · � H 2 α is equivalent with the standard norm � 1 / 2 �� |∇ w ( σ ) | 2 dσ g + | w ( σ ) | 2 dσ g � w � H 2 1 := . M M Moreover, if w ∈ H 2 α ( M ), the following inequalities hold σ ∈M α ( σ ) 1 / 2 }� w � H 2 α ≤ max { 1 , � α � 1 / 2 min { 1 , min 1 ≤ � w � H 2 ∞ }� w � H 2 1 . (1) Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Some basic facts From the Rellich-Kondrachov theorem for compact manifolds without boundary one has H 2 → L q ( M ) , α ( M ) ֒ for every q ∈ [1 , 2 d/ ( d − 2)] . In particular, the embedding is compact whenever q ∈ [1 , 2 d/ ( d − 2)). Hence, there exists a positive constant S q such that for all w ∈ H 2 � w � q ≤ S q � w � H 2 α , α ( M ) . (2) Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Some basic facts From now on, we assume that the nonlinearity f satisfies the following structural condition: f : I R → I R is a locally H¨ older continuous function sublinear at infinity, that is, f ( t ) (h ∞ ) lim = 0 . t | t |→∞ Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Weak solutions of problem ( P λ ) A function w ∈ H 2 1 ( M ) is said a weak solution of ( P λ ) if � � � �∇ w, ∇ v � g dσ g + α ( σ ) � w, v � g dσ g − λ K ( σ ) f ( w ( σ )) v ( σ ) dσ g = 0 , M M M for every v ∈ H 2 1 ( M ) . Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Weak solutions of problem ( P λ ) From a variational stand point the weak solutions of ( P λ ) in H 2 1 ( M ), are the critical points of the C 1 -functional given by � w � 2 � H 2 J λ ( u ) := − λ K ( σ ) F ( w ( σ )) dσ g , α 2 M for every u ∈ H 2 1 ( M ). Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Theorem (G. Bonanno and S.A. Marano, Appl. Anal. 2010) Let X be a reflexive real Banach space, Φ : X → R be a coercive, continuously G ˆ a teaux differentiable and sequentially weakly lower semicontinuous functional whose G ˆ a teaux derivative admits a continuous inverse on X ∗ , Ψ : X → R be a continuously G ˆ a teaux differentiable functional whose G ˆ a teaux derivative is compact such that Φ(0) = Ψ(0) = 0 . Assume that there exist r > 0 and ¯ x ∈ X , with r < Φ(¯ x ) , such that: sup Ψ( x ) < Ψ(¯ x ) Φ( x ) ≤ r ( a 1 ) x ); r Φ(¯ � Φ(¯ x ) r � ( a 2 ) for each λ ∈ Λ r := x ) , the functional Ψ(¯ sup Φ( x ) ≤ r Ψ( x ) Φ − λ Ψ is coercive. Then, for each λ ∈ Λ r , the functional Φ − λ Ψ has at least three distinct critical points in X . Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Main results on the existence of at least three solutions Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Notations We set 2 � 1 / 2 � κ α := , � α � L 1 ( M ) and S q K 1 := S 1 q √ 2 � α � L 1 ( M ) , K 2 := � α � L 1 ( M ) . 2 − q 2 q 2 Further, let � ξ F ( ξ ) := f ( t ) dt, 0 for every ξ ∈ I R. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Existence of three solutions Theorem (G. Bonanno,—–, V. R˘ adulescu; Nonlinear Anal. (2011)) Let f : R → R be a function such that (h ∞ ) holds and assume that ( h 1 ) There exist two nonnegative constants a 1 , a 2 such that | f ( t ) | ≤ a 1 + a 2 | t | q − 1 , for all t ∈ R , where q ∈ ]1 , 2 d/ ( d − 2)[; ( h 2 ) There exist two positive constants γ and δ , with δ > γκ α , such that F ( δ ) � K � ∞ � a 1 K 1 � γ + a 2 K 2 γ q − 2 > . δ 2 � K � L 1 ( M ) Then, for each parameter λ ∈ Λ ( γ,δ ) the problem ( P λ ) , possesses at least three solutions in H 2 1 ( M ) . Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Existence of three solutions Where   δ 2 � α � L 1 ( M ) � α � L 1 ( M )   Λ ( γ,δ ) := ,  .   K 1 2 F ( δ ) � K � L 1 ( M ) � γ + a 2 K 2 γ q − 2 �  2 � K � ∞ a 1 Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Existence of three solutions for the problem σ ∈ S d , w ∈ H 2 1 ( S d ) − ∆ h w + α ( σ ) w = λK ( σ ) f ( ω ) , Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Three solutions on the sphere Let α, K ∈ C ∞ ( S d ) be positive and set κ 1 � α � L 1 ( S d ) K ⋆ 1 := � . (3) √ � σ ∈ S d α ( σ ) 1 / 2 2 min 1 , min Further, for q ∈ ]1 , 2 d/ ( d − 2)[, we will denote κ q q � α � L 1 ( S d ) K ⋆ 2 := � . (4) � 2 − q 2 q min σ ∈ S d α ( σ ) q/ 2 2 1 , min Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Three solutions on the sphere Where 2 − q  2 q ω if q ∈ [1 , 2[ ,  d        1 / 2 κ q :=      q − 2 1 � 2 d �    max , if q ∈ 2 , .    q − 2 q − 2 d − 2   q 2 q dω ω      d d Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Three solutions on the sphere Corollary Let f : R → R be a function such that (h ∞ ) and (h 1 ) hold. Further, assume that there exist two positive constants γ and δ , with δ > γκ α , and K ⋆ � � 2 ) F ( δ ) � K � ∞ ( h ⋆ 1 + a 2 K ⋆ 2 γ q − 2 > a 1 , δ 2 � K � L 1 ( S d ) γ where K ⋆ 1 and K ⋆ 2 are given respectively by (3) and (4). Then, for each parameter λ belonging to Λ ⋆ ( γ,δ ) the problem ( S α λ ) possesses at least three distinct solutions. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Three solutions on the sphere Where   δ 2 � α � L 1 ( S d ) � α � L 1 ( S d ) Λ ⋆   ( γ,δ ) := ,  .  K ⋆  2 F ( δ ) � K � L 1 ( S d ) � 2 γ q − 2 � 1  + a 2 K ⋆ 2 � K � ∞ a 1 γ Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Existence of three solutions for the Emden-Fowler problem x ∈ R d +1 \ { 0 } − ∆ u = λ | x | s − 2 K ( x/ | x | ) f ( | x | − s u ) , Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Emden-Fowler problems Next, we consider the following parameterized Emden-Fowler problem that arises in astrophysics, conformal Riemannian geometry, and in the theories of thermionic emission, isothermal stationary gas sphere, and gas combustion: x ∈ R d +1 \ { 0 } . − ∆ u = λ | x | s − 2 K ( x/ | x | ) f ( | x | − s u ) , ( F λ ) The equation ( F λ ) has been studied when f has the form f ( t ) = | t | p − 1 t, p > 1 , see Cotsiolis-Iliopoulos, V´ azquez-V´ eron. In these papers, the authors obtained existence and multiplicity results for ( F λ ), applying either minimization or minimax methods. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Emden-Fowler problems The solutions of ( F λ ) are being sought in the particular form u ( x ) = r s w ( σ ) , (5) where, ( r, σ ) := ( | x | , x/ | x | ) ∈ (0 , ∞ ) × S d are the spherical coordinates in R d +1 \ { 0 } and w be a smooth function defined on S d . This type of transformation is also used by Bidaut-V´ eron and V´ eron, where the asymptotic of a special form of ( F λ ) has been studied. Throughout (5), taking into account that � � ∆ u = r − d ∂ r d ∂u + r − 2 ∆ h u, ∂r ∂r the equation ( F λ ) reduces to σ ∈ S d , w ∈ H 2 1 ( S d ) , − ∆ h w + s (1 − s − d ) w = λK ( σ ) f ( w ) , see also Krist´ aly and R˘ adulescu. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Three solutions for Emden-Fowler problems Corollary Assume that d and s are two constants such that 1 − d < s < 0 . Further, let K ∈ C ∞ ( S d ) be a positive function and f : R → R as in the previous Corollary. Then, for each parameter λ belonging to   s (1 − s − d ) ω d δ 2 s (1 − s − d ) ω d Λ s,d   ( γ,δ ) := ,  ,  K ⋆  2 F ( δ ) � K � L 1 ( S d ) � 2 γ q − 2 � 1 + a 2 K ⋆  2 � K � ∞ a 1 γ the following problem x ∈ R d +1 \ { 0 } , − ∆ u = λ | x | s − 2 K ( x/ | x | ) f ( | x | − s u ) , ( F λ ) admits at least three distinct solutions. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Example and Application Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Example Let ( M , g ) be a compact d -dimensional ( d ≥ 3) Riemannian manifold without boundary, fix q ∈ ]2 , 2 d/ ( d − 2)[ and let K ∈ C ∞ ( M ) be a positive function. Moreover, let h : R → R be the function defined by  1 + | t | q − 1 if | t | ≤ r    h ( t ) := (1 + r 2 )(1 + r q − 1 )  if | t | > r,   1 + t 2 where r is a fixed constant such that 1 �� 2 � 1 / 2 � K � ∞ q − 2 � 1 � � r > max , q ( K 1 + K 2 ) . (6) q − 2 Vol g ( M ) � K � L 1 ( M ) Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Example From our Theorem, for each parameter qr 2 Vol g ( M ) � � Vol g ( M ) λ ∈ , , 2( qr + r q ) � K � L 1 ( M ) 2 � K � ∞ ( K 1 + K 2 ) the following problem σ ∈ M , w ∈ H 2 − ∆ g w + w = λK ( σ ) h ( w ) , 1 ( M ) , possesses at least three nontrivial solutions. Giovanni Molica Bisci On some variational problems in...

Index Preliminaries Abstract Abstract result A problem on Riemannian manifolds Main Results Infinitely many solutions Three solutions on the sphere Infinitely many solutions for the Sierpi´ nski fractal An Example References Three solutions for Elliptic problems We just mention that similar results for elliptic problems on bounded domains of the Euclidean space are contained in G. Bonanno, —– Three weak solutions for elliptic Dirichlet problems , J. Math. Anal. Appl. , in press. and G. D’Agu` ı, —– Three non-zero solutions for elliptic Neumann problems , Analysis and Applications, 2010, 1-9. Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References Infinitely many weak solutions for Elliptic problems Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References Let X be a reflexive real Banach space, let Φ , Ψ : X → R be two Gˆ ateaux differentiable functionals such that Φ is strongly continuous, sequentially weakly lower semicontinuous and coercive and Ψ is sequentially weakly upper semicontinuous. For every r > inf X Φ, put � � sup v ∈ Φ − 1 (] −∞ ,r [) Ψ( v ) − Ψ( u ) ϕ ( r ) := inf r − Φ( u ) u ∈ Φ − 1 (] −∞ ,r [) and γ := lim inf r → + ∞ ϕ ( r ) , δ := r → (inf X Φ) + ϕ ( r ) . lim inf Then, one has Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References Theorem (G. Bonanno,—–; Bound. Value Probl. 2009) � � 1 (a) For every r > inf X Φ and every λ ∈ 0 , , the restriction of the ϕ ( r ) functional I λ := Φ − λ Ψ to Φ − 1 (] − ∞ , r [) admits a global minimum, which is a critical point (local minimum) of I λ in X . � � 0 , 1 (b) If γ < + ∞ then, for each λ ∈ , the following alternative holds: γ either ( b 1 ) I λ possesses a global minimum, or ( b 2 ) there is a sequence { u n } of critical points (local minima) of I λ such that lim n → + ∞ Φ( u n ) = + ∞ . 0 , 1 � � (c) If δ < + ∞ then, for each λ ∈ , the following alternative holds: δ either ( c 1 ) there is a global minimum of Φ which is a local minimum of I λ , or ( c 2 ) there is a sequence { u n } of pairwise distinct critical points (local minima) of I λ which weakly converges to a global minimum of Φ , with lim n → + ∞ Φ( u n ) = inf X Φ . . Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References We point out that this result is a refinement of Theorem 2.5 in B. Ricceri (2000) A general variational principle and some of its applications , J. Comput. Appl. Math. 113 , 401-410 Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References Theorem (G. Bonanno,—–; Proc. Roy. Soc. Edinburgh, A 2009) Let f : R → R be a continuous non-negative function and p > N . Put 1 − µ N σ ( N, p ) := inf µ N (1 − µ ) p , τ := sup dist( x, ∂ Ω) , µ ∈ ]0 , 1[ x ∈ Ω N � p − 1 �� 1 � 1 − 1 m := N − 1 � � 1 + N p p N − 1 1 p , √ π Γ | Ω | 2 p − N τ p and κ := m p | Ω | σ ( N, p ) . Assume that F ( ξ ) F ( ξ ) lim inf < κ lim sup ξ p . ( g ) ξ p ξ → + ∞ ξ → + ∞ σ ( N, p ) 1 � � Then, for each λ ∈ , , the problem F ( ξ ) F ( ξ ) pτ p lim sup m p p | Ω | lim inf ξ p ξ p ξ → + ∞ ξ → + ∞ ( D f λ ) admits a sequence of positive weak solutions which is unbounded in W 1 ,p (Ω) . 0 Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References Assumption (g) could be replaced by ( g ′ ) There exist two sequences { a n } and { b n } such that 1 0 ≤ a n < b n mµ N/p σ 1 /p ( N, p ) ω 1 /p τ τ for every n ∈ N and n → + ∞ b n = + ∞ such that lim | Ω | F ( b n ) − µ N ω τ F ( a n ) F ( ξ ) lim < κ | Ω | lim sup ξ p , σ ( N, p ) n → + ∞ b p ξ → + ∞ n − m p a p µ N n ω τ τ p where π N/ 2 ω τ := τ N � . 1 + N � Γ 2 Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References We point out that the results contained in F. Cammaroto, A. Chinn` ı and B. Di Bella (2005) Infinitely many solutions for the Dirichlet problem involving the p -Laplacian , Nonlinear Anal. 61 (2005) 41-49 are direct consequences of main Theorem by using condition (g ′ ). Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References Example Assume that p ∈ N and 1 ≤ N < p . Put a n := 2 n !( n + 2)! − 1 b n := 2 n !( n + 2)! + 1 , , 4( n + 1)! 4( n + 1)! for every n ∈ N . Let { g n } be a sequence of non-negative functions such that: g 1 ) g n ∈ C 0 ([ a n , b n ]) such that g n ( a n ) = g n ( b n ) = 0 for every n ∈ N ; � b n g 2 ) g n ( t ) dt � = 0 for every n ∈ N . a n For instance, we can choose the sequence { g n } as follows � 1 ξ − n !( n + 2) � 2 � g n ( ξ ) := 16( n + 1)! 2 − , ∀ n ∈ N . 2 Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References Example Define the function f : R → R as follows ∞  g n ( ξ ) [( n + 1)! p − n ! p ] � if ξ ∈ [ a n , b n ]  � b n   f ( ξ ) := g n ( t ) dt n =1 a n    0 otherwise . From our result, for each λ > σ ( N, p ) the problem p 2 p τ p � − ∆ p u = λf ( u ) in Ω ( D f λ ) u | ∂ Ω = 0 , possesses a sequence of weak solutions which is unbounded in W 1 ,p (Ω). 0 Giovanni Molica Bisci On some variational problems in...

Index Abstract Abstract result A problem on Riemannian manifolds Infinitely many positive weak solutions for Elliptic problems Infinitely many solutions Elliptic problems and Orlicz-Sobolev spaces Infinitely many solutions for the Sierpi´ nski fractal References We observe that in the very interesting paper P. Omari and F. Zanolin (1996) Infinitely many solutions of a quasilinear elliptic problem with an oscillatory potential , Commun. Partial Differential Equations 21 (5-6) the authors, assuming that f (0) ≥ 0 and that F ( ξ ) F ( ξ ) lim inf = 0 and lim sup = + ∞ , ξ p ξ p ξ → + ∞ ξ → + ∞ proved problem � − ∆ p u = f ( u ) in Ω , u | ∂ Ω = 0 . admits a sequence of non-negative solutions which is unbounded in C 0 (Ω). Giovanni Molica Bisci On some variational problems in...

On some variational problems in Riemannian and Fractal Geometry - PowerPoint PPT Presentation

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