On the Schrödinger Equation in under the Effect of a General Nonlinear Term A. AZZOLLINI & A. POMPONIO ABSTRACT. In this paper we prove the existence of a positive solution to the equation u + V(x)u = g(u) in, assuming the general hypotheses on the nonlinearity introduced by Berestycki and Lions. Moreover we show that a minimizing problem, related to the existence of a ground state, has no solution. 1. INTRODUCTION This paper deals with the following equation: (1.1) u + V(x)u = g(u), u>0. x, N 3; An existence result of nontrivial solutions for this kind of problem has been obtained by Rabinowitz [8] assuming that g is superlinear and subcritical at infinity and satisfies the global growth Ambrosetti-Rabinowitz condition t (1.) µ>suchthat0<µ g(s)ds g(t)t, for all t R. 0 This condition is used to get the boundedness of Palais-Smale sequences. In [7], L. Jeanjean and K. Tanaka have been able to remove the hypothesis (1.) by means of an abstract tool, consisting in a suitable approximating method (see [4, Theorem 1.1]). However they preserved the condition on the superlinear growth at infinity. 1361 Indiana University Mathematics Journal cfl, Vol. 58, No. 3 (009)
136 A. AZZOLLINI & A. POMPONIO On the other hand, in the fundamental paper [], Berestycki and Lions proved the existence of a ground state, namely a solution which minimizes the action among all the solutions, for the problem u = g(u) in ; (1.3) u H 1 ( ), u 0, under the following assumptions on the nonlinearity g: (g1) g C(R,R), g odd; (g) < lim inf s 0 + g(s)/s lim sup s 0 + g(s)/s = m<0; (g3) < lim sup s + g(s)/s 1 0; (g4) there exists ζ>0 such that G(ζ) ζ g(s)ds >0; 0 here = N/(N ). So it seems that the natural assumptions on the nonlinearity do not require the superlinearity at infinity. In view of this result, the aim of this paper is to study the problem (1.1) preserving the same general assumptions of [] on g. Moreover we assume the following hypotheses on V : (V1) V C 1 (, R) and V(x) 0, for all x, and the inequality is strict somewhere; (V) ( V( ) ) + N/ < S; (V3) lim x V(x) = 0; (V4) V is radially symmetric; here ( V(x) x) + = max { ( V(x) x),0 } and S is the best Sobolev constant of the embedding D 1, ( ) L ( ), namely S = u inf u D 1, \{0} u. Our first main result is the following Theorem 1.1. Assume that (g1) (g4) and (V1) (V4) hold, then the problem (1.1) possesses at least a radially symmetric solution. Up to our knowledge, this is the first result on a problem as (1.1), where exactly the same general hypotheses of Berestycki and Lions [] are assumed on the nonlinearity g. As a consequence of Theorem 1.1, we can prove the following
On the Schrödinger Equation in under a General Nonlinear Term 1363 Theorem 1.. Assuming that (g1) (g4) and (V1) (V4) hold, then the problem (1.1) possesses a radial ground state solution, namely a solution minimizing the action among all the nontrivial radial solutions. Remark 1.3. We can generalize the hypotheses (V1)and(V3), requiring: (V1) V C 1 (, R) and V(x) V 0 > m,forallx, and the inequality is strict somewhere; (V3) lim x V(x) = V 0 ; and supposing that the function g(s) = g(s) V 0 s satisfies (g1) (g4). Remark 1.4. The geometrical hypotheses on the potential V do not allow us to use concentration-compactness arguments as in [7]. As a consequence, we have to require a symmetry property on V to prevent any possible loss of mass at infinity. In the second part of the paper, we are interested in solving a minimization problem strictly related to the existence of a ground state solution for (1.1). Let I(u) = 1 u + V(x)u G(u), u H 1 ( ). When the nonlinearity g satisfies (1.), a standard method to look for the existence of a ground state solution for an equation as (1.1) is to study the minimizing problem (1.4) I(ū) = inf u N I(u), ū N, where N is the Nehari manifold related to I. In[8] the geometrical assumption on V (1.5) V(y) lim x V(x), for all y is used to solve such a minimizing problem. Moreover, in [7] it has been proved that, assuming (1.5), there exists a ground state solution for (1.1) also without the condition (1.). On the other hand, it is well known (see for example [1]) that if g satisfies (1.), and (1.5) holds with the reverse inequality, the minimizing problem (1.4) cannot be solved. In fact, a contradiction argument deriving from the comparison of the level η inf u N I(u) with η 0 inf u N0 I 0 (u) (here I 0 and N 0 are the functional and the Nehari manifold of the problem at infinity) is used and the Ambrosetti-Rabinowitz condition plays a fundamental role. Actually, when we do not assume such a type of growth condition on g, these arguments do not work any more. However, a similar study can be done by replacing the Nehari manifold with a more suitable one. Indeed, if we define by { (1.6) P 0 u H 1 ( ) \{0} N } u = N G(u),
1364 A. AZZOLLINI & A. POMPONIO the Pohozaev manifold related to (1.3), and we set S 0 {u H 1 ( ) \{0} uis a solution of (1.3)}, it is well known that S 0 P 0.Moreover, in [9] it has been proved that P 0 is a natural constraint for the functional related to (1.3) I 0 (u) = 1 u G(u), u H 1 ( ). In order to look for a ground state solution, a very natural question arises: is the infimum I 0 P0 achieved? Following two different ways, Jeanjean and Tanaka [6] and Shatah [9] have given a positive answer to this question, showing that (1.7) min u S 0 I 0 (u) = min u P 0 I 0 (u). Inspired by these papers and observed that each solution of (1.1) satisfies the following Pohozaev identity: (1.8) N u + N V(x)u + 1 ( V(x) x)u = N G(u), we indicate with P the Pohozaev manifold related to (1.1): (1.9) P= {u H 1 ( )\{0} usatisfies (1.8) } and we wonder if there exists a minimizer for I P. Proceeding in analogy with [1], we get the following result Theorem 1.5. If we assume (g1) (g4), (V1), (V3) and (V5) ( V(x) x) 0, for all x ; (V6) NV(x) + ( V(x) x) 0 for all x, and the inequality is strict somewhere, then b inf u P I(u) is not a critical level for the functional I. Requiring something more, instead of (V6), we have a further information on the level b: Corollary 1.6. Assuming the same hypotheses of Theorem 1.5,ifin(V6) we have the strict inequality almost everywhere, then the level b is not achieved as a minimum on the Pohozaev manifold P. Acknowledgment. The authors express their deep gratitude to Professor L. Jeanjean for stimulating discussions and useful suggestions, and to the anonymous referee for the careful reading of the paper and for all the fundamental hints.
On the Schrödinger Equation in under a General Nonlinear Term 1365. NOTATION For any 1 s<+, L s (R 3 ) is the usual Lebesgue space endowed with the norm u s s u s ; R 3 H 1 ( ) is the usual Sobolev space endowed with the norm u u + u ; R 3 D 1, ( )is completion of C0 (R3 ) with respect to the norm u D 1, ( ) R 3 u ; for any r>0,x R 3 and A R 3 B r (x) {y R 3 y x r}, B r {y R 3 y r}, A c R 3 \A. 3. THE EXISTENCE RESULT The aim of this section is to prove Theorems 1.1 and 1.. Set H 1 r ( ) {u H 1 ( ) uis radial } and, following [], define s 0 min{s [ζ, + [ g(s) = 0} (s 0 = + if g(s) 0foranys ζ)andset g:r Rthe function such that g(s) on [0,s 0 ]; (3.1) g(s) = 0 on R + \[0,s 0 ]; g( s) on R. By the strong maximum principle, a solution of (1.1) with gin the place of g is a solution of (1.1). So we can suppose that g is defined as in (3.1), so that (g1), (g), (g4) and then the following limit (3.) lim s g(s) s 1 = 0
1366 A. AZZOLLINI & A. POMPONIO hold. Moreover, we set for any s 0, and we extend them as odd functions. Since g 1 (s) (g(s) + ms) +, g (s) g 1 (s) g(s), (3.3) (3.4) g lim 1 (s) = 0, s 0 s lim s g 1 (s) s 1 = 0, and (3.5) g (s) ms, s 0, by some computations, we have that for any ε>0there exists C ε > 0 such that (3.6) g 1 (s) C ε s 1 + εg (s), s 0. If we set G i (t) then, by (3.5)and(3.6), we have t g i (s) ds, i = 1,, 0 (3.7) G (s) m s, s R and for any ε>0 there exists C ε > 0 such that G 1 (s) C ε (3.8) s + εg (s), s R. Using an idea from [4], we look for bounded Palais-Smale sequences of the following perturbed functionals I λ (u) = 1 u + V(x)u + G (u) λ G 1 (u), for almost all λ near 1. Then we will deduce the existence of a non-trivial critical point v λ of the functional I λ at the mountain pass level. Afterward, we study the convergence of the sequence (v λ ) λ,asλgoes to 1 (observe that I 1 = I). We will apply the following slight modified version of [4, Theorem 1.1] (see [5]).
On the Schrödinger Equation in under a General Nonlinear Term 1367 Theorem 3.1. Let ( X, ) be a Banach space and J R + an interval. Consider the family of C 1 functionals on X I λ (u) = A(u) λb(u), λ J, with B nonnegative and either A(u) + or B(u) + as u and such that I λ (0) = 0. For any λ J we set { (3.9) Γ λ γ C([0,1], X) γ(0) = 0,I λ (γ(1)) < 0 }. If for every λ J the set Γ λ is nonempty and (3.10) c λ inf max γ Γ λ t [0,1] I λ (γ(t)) > 0, then for almost every λ J there is a sequence (v n ) n X such that (i) (v n ) n is bounded; (ii) I λ (v n ) c λ ; (iii) (I λ ) (v n ) 0 in the dual X 1 of X. In our case, X = H 1 r (RN ) and A(u) 1 u + V(x)u + G (u), B(u) G 1 (u). In order to apply Theorem 3.1, we have just to define a suitable interval J such that Γ λ,foranyλ J,and(3.10)holds. Observe that, according to [], there exists a function z H 1 r ( ) such that G 1 (z) G (z) = G(z) > 0. Then there exists 0 < δ <1 such that (3.11) δ G 1 (z) G (z) > 0. We define J as the interval [ δ, 1]. Lemma 3.. Γ λ, for any λ J.
1368 A. AZZOLLINI & A. POMPONIO Proof. Let λ J. Set θ>0sufficiently large and z = z( / θ). Define γ : [0, 1] H 1 r (RN ) in the following way 0, if t = 0, γ(t) = z t = z( /t), if 0 <t 1. It is easy to see that γ is a continuous path from 0 to z. Moreover, we have that I λ (γ(1)) θ N z + θ N V ( ) θx z ) + (R θ N G (z) δ G 1 (z) N and then, by (3.11), (V3) and the Lebesgue theorem, for a suitable choice of θ, certainly γ Γ λ. Lemma 3.3. There exists a constant c >0such that c λ c >0for all λ J. Proof. Observe that for any u Hr 1(RN ) and λ J,using(3.7)and(3.8)for ε<1, we have I λ (u) 1 u + V(x)u + G (u) G 1 (u) 1 u + (1 ε) m u C ε u and then, by Sobolev embeddings, we conclude that there exists ρ>0 such that for any λ J and u H 1 r (RN ) with u 0and u ρ,it results I λ (u) > 0. In particular, for any u =ρ, we have I λ (u) c >0.Now fix λ J and γ Γ λ. Since γ(0) = 0andI λ (γ(1)) < 0, certainly γ(1) >ρ.by continuity, we deduce that there exists t γ ]0, 1[ such that γ(t γ ) =ρ. Therefore, for any λ J, (3.1) c λ inf γ Γ λ I λ (γ(t γ )) c >0. We present a variant of the Strauss compactness lemma [10] (see also [, Theorem A.1]), whose proof is similar to that contained in []. It will be a fundamental tool in our arguments: Lemma 3.4. Let P and Q : R R be two continuous functions satisfying P(s) lim s Q(s) = 0,
On the Schrödinger Equation in under a General Nonlinear Term 1369 (v n ) n,vand z be measurable functions from to R,withzbounded, such that sup Q(v n (x))z dx <+, n P(v n (x)) v(x) a.e. in. Then (P(v n ) v)z L 1 (B) 0, for any bounded Borel set B. Moreover, if we have also lim s 0 P(s) Q(s) = 0, lim x sup v n (x) =0, n then (P(v n ) v)z L 1 ( ) 0. In analogy with the well-known compactness result in [3], we state the following result: Lemma 3.5. For any λ J, each bounded Palais-Smale sequence for the functional I λ admits a convergent subsequence. Proof. Let λ J and (u n ) n be a bounded (PS) sequence for I λ, namely, (I λ (u n )) n is bounded, (3.13) lim n (I λ ) (u n ) = 0 in (H 1 r ( )). Up to a subsequence, we can suppose that there exists u H 1 r ( ) such that (3.14) (3.15) u n u weakly in H 1 r (RN ) u n (x) u(x) a.e. in. By weak lower semicontinuity we have: (3.16) u lim inf u n ; n (3.17) V(x)u lim inf V(x)u n n. If we apply Lemma 3.4 for P(s) = g i (s), i = 1,, Q(s)= s 1,(v n ) n = (u n ) n,v =g i (u), i = 1, andz C0 (RN ), by (3.), (3.4) and(3.15) we deduce that g i (u n )z g i (u)z, i = 1,.
1370 A. AZZOLLINI & A. POMPONIO As a consequence, by (3.13)and(3.14)wededuce(I λ ) (u) = 0 and hence (3.18) u + V(x)u = λg 1 (u)u g (u)u. If we apply Lemma 3.4 for P(s) = g 1 (s)s, Q(s) = s + s,(v n ) n =(u n ) n, v = g 1 (u)u, and z = 1, by (3.), (3.4), (3.15) and the well-known Strauss radial lemma (see [10]), we deduce that (3.19) g 1 (u n )u n g 1 (u)u. Moreover, by (3.15) and Fatou s lemma (3.0) g (u)u lim inf g (u n )u n. n By (3.18), (3.19)and(3.0), and since (I λ ) (u n ), u n 0 (3.1) lim sup u n + V(x)u n n [ ] = lim sup λ g 1 (u n )u n g (u n )u n n λ g 1 (u)u g (u)u R N = u + V(x)u. By (3.16), (3.17)and(3.1), we get (3.) lim u n n R = u N R N lim V(x)u n R N n = V(x)u, hence (3.3) lim g (u n )u n = g (u)u. n Since g (s)s = ms +q(s),withqapositive and continuous function, by Fatou s Lemma we have q(u) lim inf q(u n ); n R N u lim inf u n n.
On the Schrödinger Equation in under a General Nonlinear Term 1371 These last two inequalities and (3.3) imply that, up to a subsequence, u = lim n u n, which, together with (3.), shows that u n u strongly in H 1 r ( ). Lemma 3.6. For almost every λ J, there exists u λ Hr 1(RN ), u λ 0, such that (I λ ) (u λ ) = 0 and I λ (u λ ) = c λ. Proof. By Theorem 3.1, for almost every λ J, there exists a bounded sequence (u λ n) n Hr 1 ( ) such that (3.4) (3.5) I λ (u λ n ) c λ; (I λ ) (u λ n) 0in(H 1 r ( )). Up to a subsequence, by Lemma 3.5, we can suppose that there exists u λ Hr 1 ( ) such that u λ n u λ in Hr 1 ( ). By Lemma 3.3, (3.4) and(3.5) we conclude. Now we are able to provide the proof of our main result: Proof of Theorem 1.1. By Lemma 3.6, we are allowed to consider a suitable λ n 1 such that for any n 1 there exists v n Hr 1 ( ) \{0}satisfying (3.6) (3.7) I λn (v n ) = c λn, (I λn ) (v n ) = 0in(H 1 r ( )). We want to prove that (v n ) n is a bounded Palais-Smale sequence for I at the level c c 1. By standard argument, since H 1 r ( ) is a natural constraint, we have that v n is a weak solution of the problem and it satisfies the Pohozaev equality w + V(x)w +g (w) λ n g 1 (w) = 0 (3.8) v n + N V(x)v N n + 1 N + N N ( V(x) x)v n G (v n ) λ n G 1 (v n ) = 0. Therefore, by (3.6), (3.7)and(3.8) we have that the following system holds:
137 A. AZZOLLINI & A. POMPONIO (3.9) 1 (α n + β n ) + γ,n λ n γ 1,n = c λn, α n + β n + δ,n λ n δ 1,n = 0, α n + N N β n + 1 N η n + N N γ,n N N λ nγ 1,n = 0, where α n = v n, β n = V(x)v n, η n = ( V(x) x)v n, γ i,n = G i (v n ), δ i,n = g i (v n )v n, i =1,. By the first and the third equations of the system we get 1 N α n 1 N η n = c λn and then, using Hölder s inequality, by (V) and the boundedness of (c λn ) n (indeed the map λ c λ is non-increasing), we have (3.30) α n C for all n 1. By the second equation of the system we have δ,n λ n δ 1,n = α n β n 0 and then by (3.6), there exists 0 <ε<1andc ε >0 such that δ,n δ 1,n C ε v n +εδ,n. Therefore, by the Sobolev embedding D 1, ( ) L ( ) (1 ε)δ,n C ε v n Cα / n andthen,by(3.30), δ,n is bounded. By (3.5) and(3.30) we deduce that (v n ) n is bounded in H 1 r (RN ). Up to a subsequence, there exists v H 1 r ( ) such that (3.31) v n vweakly in H 1 r ( ). By (3.7), we have that I (v n ) = (I λn ) (v n ) + (λ n 1)g 1 (v n ) = (λ n 1)g 1 (v n )
On the Schrödinger Equation in under a General Nonlinear Term 1373 so (3.3) I (v n ) 0 in (H 1 r (RN )), if ( g 1 (v n ) ) n is bounded in ( Hr 1 ( ) ). But this is true by the Banach-Steinhaus theorem, since Lemma 3.4 implies that for any z Hr 1 ( ) g 1 (v n )z g 1 (v)z. Moreover, from (3.6) and the boundedness of (v n ) n, we deduce that (3.33) I(v n ) = I λn (v n ) + (λ n 1) G 1 (v n ) c. Therefore, by (3.3) and(3.33), (v n ) n is a Palais-Smale sequence for the functional I and so, by Lemma 3.5, v is a nontrivial mountain pass type solution for (1.1). To conclude, observe that, by standard arguments, we can use the strong maximum principle to get v>0. In order to prove Theorem 1.,weset S r { u H 1 r (RN )\{0} I (u) = 0 }, σ r inf u S r I(u). Lemma 3.7. We have σ r > 0. Proof. By (3.6) and Sobolev embedding we have that for any u S r u R u + V(x)u +(1 ε) g (u)u N C ε u C u where 0 <ε<1andc ε,c >0.So we deduce that inf u S r u > 0. Now, since S r P(see (1.9)), for any u S r by (V)wehave I(u) = 1 N u 1 N ( V(x) x)u C>0. Finally we provide the following:
1374 A. AZZOLLINI & A. POMPONIO Proof of Theorem 1.. Let (u n ) n S r such that I(u n ) σ r. Arguing as in the proof of Theorem 1.1 we have that the sequence is bounded. By Lemma 3.5, there exists u H 1 r (RN ) such that u n u in H 1 r (RN ) and then the conclusion follows by Lemma 3.7. 4. THE NONEXISTENCE RESULT In this section we give the proof of Theorem 1.5 and we will assume that V satisfies hypotheses (V1), (V3), (V5) (V6). Lemma 4.1. We have inf u P u > 0. Proof. The conclusion follows immediately by (1.8), together with (3.7), (3.8) and (V6). Let us show that the functional I is bounded below on the manifold P: Lemma 4.. We have (4.1) b = inf I(u) > 0. u P Proof. It is easy to see that for any u P, (4.) I(u) = 1 N u 1 N ( V(x) x)u. Now the conclusion follows by (V5) and Lemma 4.1. Lemma 4.3. Let w H 1 ( ) be such that G(w) > 0. Then there exists RN θ >0such that w θ = w( / θ) P. In particular this result is true for any w P 0 (see (1.6)). Proof. For any θ>0, we set f(θ) I(w θ ) = θn w + θn V(θx)w θ N G(w). By the Lebesgue theorem and (V3), we get lim V(θx)w = 0, θ + and so lim f(θ) =. θ + We argue that there exists θ >0 such that f ( θ) = 0: hence w θ P.
On the Schrödinger Equation in under a General Nonlinear Term 1375 Let w P 0.Foranyy,wesetw y w( y) P 0.Setθ y >0such that w y = w y ( /θ y ) P. Lemma 4.4. We have lim y θ y = 1. Proof of Lemma 4.4. Step 1: lim sup y θ y < +. Suppose, by contradiction, that θ yn +,for y n.foranyy, we have (4.3) I( w y ) = θn y w + θn y ) V(θ y x)w (x yθy Let us show that ) (4.4) lim V(θ y x)w (x yθy = 0. y Indeed we have (4.5) V(θ y x)w (x yθy ) ) = V(θ y x)w (x yθy B r sup V(x) x B r ( y/θ y ) θ N y G(w). ( + V(θ y x)w x y ) Br c θ y w + sup V(θ y x) w. x Br c By the absolute continuity of the Lebesgue integral, for any ε >0 there exists r >0 such that, for any r< rand for any y,weget (4.6) sup V(x) w ε. x B r ( y/θ y ) Therefore, since we are supposing that θ yn +,as y n,by(4.5), (4.6) and (V), we get (4.4). As a consequence, by (4.3) and(4.4), we infer that I( w yn ),as y n, and we get a contradiction with Lemma 4.. Step : lim y θ y = 1. Since w P 0 and w y P,weget (4.7) N(θ y 1) G(w) = θ y [ NV(θy x + y) + ( V(θ y x +y) (θ y x + y)) ] w.
1376 A. AZZOLLINI & A. POMPONIO By (V3), (V5) and(v6), using the dominated convergence and the conclusion of the Step 1, the right hand side in (4.7) goes to zero as y, and so the lemma is proved. We set (see (1.7)) b 0 min u S0 I 0 (u) = min u P0 I 0 (u). Lemma 4.5. b b 0. Proof. Let w H 1 ( ) be a ground state solution of (1.3). Then w P 0 and I 0 (w) = b 0.Foranyy,wesetw y =w( y). By the invariance by translations of (1.3), we have that w y P 0 and I 0 (w y ) = b 0. By Lemma 4.3,for any y there exists θ y > 0 such that w y = w y ( /θ y ) P.Weget I( w y ) b 0 = I( w y ) I 0 (w y ) θn y 1 w + θn y V(θ y x +y)w + θ N y 1 G(w). Therefore, by Lemma 4.4, we infer that lim y I( w y ) = b 0, hence b b 0. Lemma 4.6. Let z H 1 ( ) be such that G(z) > 0. Then there exists θ >0such that z θ = z( / θ) P 0. In particular, by (V6) this result is true for any z Pwith θ 1. Proof. The first part of the statement follows from the fact that, for any z H 1 ( ) such that G(z) > 0, certainly there exists θ >0 such that (4.8) N z = N θ G(z). Consider now the case of z P.Since (4.9) N z + N V(x)z + 1 ( V(x) x)z = N G(z), by (V6)wehave G(z) > 0. Let θ >0suchthat(4.8) holds. Combining (4.8) and (4.9) weget (4.10) 1 [ ] NV(x) + ( V(x) x) z = N(1 θ ) G(z). By (V6), we get the conclusion.
On the Schrödinger Equation in under a General Nonlinear Term 1377 Remark 4.7. Let z Pand let θ (0, 1] be such that z θ P 0.Wehave I(z) = 1 z 1 ( V(x) x)z N N θn z N = I 0 (z θ ) b 0, namely,b b 0. By Lemma 4.5,weinferthatb=b 0. Now we can prove Theorem 1.5: Proof of Theorem 1.5. Suppose by contradiction that there exists z H 1 ( ) critical point of the functional I at level b: in particular, z Pand I(z) = b. Let θ (0, 1] be such that z θ P 0. Let us show that θ<1. By standard arguments and using the strong maximum principle, we infer that z does not change sign and so we can assume that z>0. Therefore, by (V6) and (4.10), we get that θ<1. By (4.) and(v5), we infer that b = I(z) = 1 z 1 ( V(x) x)z N N > θn z N = I 0 (z θ ) b 0, and we get a contradiction with Lemma 4.5. Proof of Corollary 1.6. If the strict inequality in (V6) is satisfied almost everywhere, then for any z Pthere exists θ (0, 1) such that z θ P 0. Arguing as in the proof of Theorem 1.5 we conclude. Remark 4.8. In view oftheorem1.5, the proofofcorollary 1.6 wouldfollow immediately if P was a natural constraint for the functional I. REFERENCES [1] V. BENCI, C.R. GRISANTI,andA.M. MICHELETTI,Existence and non existence of the ground state solution for the nonlinear Schrödinger equations with V( ) = 0, Topol. Methods Nonlinear Anal. 6 (005), 03 19. [] H. BERESTYCKI and P.L. L IONS, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 8 (1983), 313 345. [3], Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 8 (1983), 347 375.
1378 A. AZZOLLINI & A. POMPONIO [4] L. JEANJEAN, On the existence of bounded Palais-Smale sequences and application to a Landesman- Lazer-type problem set on, Proc. R. Soc. Edinb., Sect. A, Math. 19 (1999), 787 809. [5], Local condition insuring bifurcation from the continuous spectrum, Math. Z. 3 (1999), 651 664. [6] L. JEANJEAN and K. TANAKA, A remark on least energy solutions in, Proc. Am. Math. Soc. 131 (003), 399 408. [7], A positive solution for a nonlinear Schrödinger equation on, Indiana Univ. Math. J. 54 (005), 443 464. [8] P.H. R ABINOWITZ, On a class of nonlinear Schrödinger equations, Z.Angew.Math.Phys.43 (199), 70 91. [9] J. SHATAH, Unstable ground state of nonlinear Klein-Gordon equations, Trans. Amer. Math. Soc. 90 (1985), 701 710. [10] W.A. STRAUSS, Existence of solitary waves in higher dimensions, Comm. Math. Phys. 55 (1977), 149 16. A. AZZOLLINI: Dipartimento di Matematica Università degli Studi di Bari ViaE.Orabona4 I-7015 Bari, Italy E-MAIL: azzollini@dm.uniba.it A. POMPONIO: Dipartimento di Matematica Politecnico di Bari ViaE.Orabona4 I-7015 Bari, Italy E-MAIL: a.pomponio@poliba.it KEY WORDS AND PHRASES: nonlinear Schrödinger equation; general nonlinearity; Pohozaev identity. 000 MATHEMATICS SUBJECT CLASSIFICATION: 35J60; 58E05. Received: February 13th, 008; revised: April 1st, 008. Article electronically published on May 14th, 009.