%%7-4-95 Last version to appear in Differential Integral Equations 1995% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \input amstex \baselineskip=16 pt plus 1 pt \abovedisplayskip=16 pt plus 1 pt minus 7 pt \belowdisplayskip=16 pt plus 1 pt minus 7 pt \pageno=1 \TagsAsMath \TagsOnRight \documentstyle{amsppt} %\magnification=\magstep1 \NoBlackBoxes \pageno=1 %%%%%%%%%%%%%%%%%%%%% %definici\'on de letras griegas %%%%%%%%%%%%%%%%%%%% \define \a {\alpha } \redefine \b {\beta } \redefine \d {\delta } \redefine \D {\Delta } \define \e {\epsilon } \define \g {\gamma} \define \G {\Gamma } \redefine \l {\lambda } \redefine \L {\Lambda } \define \m {\mu } \define \n {\nabla } \redefine \var {\varphi } \define \r {\rho } \define \s {\sigma } \redefine \Sig {\Sigma } \redefine \t {\tau } \define \th {\theta } \redefine \o {\omega } \redefine \O {\Omega } \define \z {\zeta } %%%%%%%%%%%%%%%%%%%% %definiciones matematicas %%%%%%%%%%%%%%%%%%%%%%% \define \iy {\infty} \define \p {\partial} \define \re {\bold R} \define \ren {{\bold R}^N} \define \qu {\bold Q} \define \ene {\bold N} \define \zet {\bold Z} \define \x {\times} \define \irn {\int\limits_{\re^N}} \define \iot {\int_0^T} \define \sign {\text{sign\,}} \define \Div {\text{div\,}} \define \meas {\text{meas\,}} \define \lp {\Delta_p} \define \bil {\Delta^2} \define \sob {W_0^{1,p}(\O)} \define \esc {W^{2,2}(\O)\cap W_0^{1,2}(\O)} \define \esd {W_0^{2,2}(\O)} \define \ci {\Cal C^{\iy}\,} \define \cu {\Cal C^{1}\,} \define \cdos {\Cal C^{2}\,} \define \cio {\Cal C_0^{\iy}\,} \define \sf {\Cal C_0^{\iy}(\ren)} \define \Do {\Cal D (B)} \define \io {\int\limits_{\O}} \define \db {\rightharpoonup} \define \dbs {\overset * \to{\db}} \define \ps {p^*} \define \pss {2^{*}} \define \un {u^n} \define \uun {u^1} \define \unn {u^{n+1}} \define \apl {\longrightarrow} \define \implica {\Longrightarrow} %%%%%%%%%%%%%%%%%%%%%% %definiciones de TeX %%%%%%%%%%%%%%%%%%%%%%%% \define \ulu {\underline u} \define \olu {\overline u} \redefine \neq {\not\equiv} \define \qd {\qquad\qquad} \define \qdd {\qd\qd} \define \fp {\flushpar} \define \prf {\demo{Proof}} \define \mpb {\medpagebreak} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \topmatter \hsize = 6 truein \vsize = 8 truein \rightheadtext{Critical problems of fourth order} \title EXISTENCE AND MULTIPLICITY OF NONTRIVIAL SOLUTIONS \\ IN SEMILINEAR CRITICAL PROBLEMS OF FOURTH ORDER \endtitle \author F. BERNIS, J. GARCIA-AZORERO, I. PERAL \endauthor \affil Departamento de Matem\'aticas,\\ Universidad Aut\'onoma de Madrid, 28049 Madrid, SPAIN \endaffil \thanks{Work partially supported by D.G.I.C.Y.T. (M.E.C., Spain) Project PB90-0218} \endthanks \subjclass 35J35, 35J40, 35J70, 46E35, 58E05, 58E30 \endsubjclass \abstract In this paper we consider the equation $\bil u = \l |u|^{q-2} u + |u|^{\pss-2} u\equiv f(u)$ in a smooth bounded domain $\O\subset\ren$ with boundary conditions either $u|_{\p \O} =\frac{\p u}{\p n}|_{\p \O}=0$ or $u|_{\p \O}=\D u|_{\p \O}=0$, where $N>4$, $ 10$ and $\pss= 2N/(N-4)$. We prove the existence of $\l_0$ such that for $0<\l<\l_0$ the above problems have infinitely many solutions. For the problem with the second boundary conditions, we prove the existence of a positive solution also in the supercritical case,i.e. when we have an exponent larger than $ \pss $. Moreover, in the critical case, we show the existence of at least two positive solutions. \endabstract \endtopmatter %%%%%%%%%%%%%%%%%%%%%%%%%%% \document \heading {\bf 1.- INTRODUCTION } \endheading In this paper we study the following fourth order problems $$ \left\{\aligned \bil u &= \l |u|^{q-2} u + |u|^{\pss-2} u \quad \text{ in } \, \O , \\ \left.u\right|_{\p \O} &=0,\\ \left.\frac{\p u}{\p n}\right|_{\p \O} &=0, \endaligned \right.\tag {P1} $$ and $$ \left\{\aligned \bil u &= \l |u|^{q-2} u + |u|^{\pss-2} u\quad \text{ in } \, \O , \\ \left.u\right|_{\p \O} &=0,\\ \left.\D u \right|_{\p \O} &=0, \endaligned \right.\tag {P2} $$ where $\O\subset \ren$ is a smooth bounded domain, $N>4$, $ 10$ and $\pss=2N/(N-4)$, the critical Sobolev exponent for fourth order problems. Hereafter we denote $f(u)=\l |u|^{q-2} u + |u|^{\pss-2} u.$ The critical growth in semilinear and quasilinear problems of second order has been extensively studied in the last years, starting with the seminal paper [6]. See [10] and [11] for an extensive list of references. For fourth order equations there are some results for the case $q=2$ and $\pss>q>2$. See the references [9] and [16] for existence results and [18] for nonexistence theorems. Other results about existence and nonexistence, also for related systems, can be seen in [17] and in [12]. In this work one of the main points is to prove the existence of infinitely many solutions for problems $(P1)$ and $(P2)$, independently of the dimension. The proof of the existence of at least two positive solutions for problem $(P2)$ is another main point of our work. \ We study the existence of solutions understood as critical points of the energy functional, $$J(u)=\frac 12\io |\D u|^2 dx-\frac{\l}{q}\io |u|^q dx- \frac{1}{\pss}\io |u|^{\pss}dx.\tag 1.1$$ For the first problem, $(P1)$, $J$ is defined in $\esd$; for the problem $(P2)$, $J$ is defined in $\esc$. Then a critical point must be understood in the following way: \roster \item $u\in \esd$ is a critical point associated to problem $(P1)$ if $$0=\io \D u \D \phi dx-\io f(u)\phi dx\quad\text{for all}\quad \phi\in\esd,$$ \item $u\in \esc$ is a critical point associated to problem $(P2)$ if $$0=\io \D u \D \phi dx-\io f(u)\phi dx\quad\text{for all}\quad \phi\in\esc.$$ \endroster In (1) integration by parts shows that the critical points of $J$ are weak solutions of problem (P1). In (2) it is not directly clear why the second boundary condition must be satisfied by a critical point. We need some information about the regularity of such critical points. Then the organization of the paper is the following. In section 2 we prove the regularity results that we need. Section 3 will be devoted to the proof of a {\it Local Palais-Smale condition}. The main tool here is the P.L. Lions concentration-compactness result. See [13], [14]. We will use at this point the result about the best constant of the Sobolev inclusion in [22]. The application of Ljusternik-Schnirelmann methods, allows us to establish the existence of infinitely many solutions for $\l$ small enough. This is the contents of section 4. In section 5 we obtain a positive solution for (P2), also for {\it supercritical} problems, by classical methods and for $0<\l<\Lambda$. The section 6 contains the extension of a well known result by Brezis and Nirenberg [7] that we apply in section 7 to show the existence of a second positive solution for problem (P2). In the last section we obtain further results, for instance, some extensions to the quasilinear contexts. \heading{\bf 2.- ABOUT REGULARITY }\endheading The regularity for problem $(P1)$ can be seen in the paper by S. Luckhaus [15]. The problem $(P2)$ must be considered in a different way because the second boundary condition is not included in the natural space $\esc$. Consider the linear problem $$ \left\{\aligned \bil u &= g(x) \text{ in } \, \O , \\ \left.u\right|_{\p \O} &=0,\\ \left.\D u \right|_{\p \O} &=0, \endaligned \right.\tag {PL2} $$ where we assume $g\in\ci (\overline{\O})$. It is well known by the standard $L^2$ theory that (PL2) has a unique $\ci$ solution. Then we can use, for instance, the classical Agmon-Douglis-Nirenberg {\it a priori} estimates, $$||u||_{W^{4,p}}\le c ||g||_p, \quad 10$ there exist $q_\e \in L^{N/4}(\O)$ and $g_\e \in L^{\infty}(\O)$ such that \item{i)} $||q_\e||_{N/4}\le \e$ \item{ii)} $q_\e (x)u(x)+g_\e(x)=a(x)u(x)+g(x)$, $x\in\O$}. Let $q_\e$ and $g_\e$ be as in the claim, so that $$\bil u=q_\e u+g_\e.$$ Consider the operator $(\bil)^{-1}$ defined in the space of functions verifying our boundary conditions. Put $A_\e u=(\bil)^{-1} (q_\e u)$ and $h_\e=(\bil)^{-1}g_\e$; then $$u-A_\e u=h_\e,\quad \text{and}\quad u\in L^{\pss}. $$ or in a equivalent way $$u=(I-A_\e)^{-1}h_\e.$$ Now, for $\e$ small enough, $$A_\e:L^p\apl L^p\quad \text{for all}\quad p\ge \pss,$$ with $$||A_\e||_{p,p}<\frac 12.$$ Indeed, by Hardy-Littlewood-Sobolev inequality we get $$||A_\e v||_p\le c(p)||q_\e v||_r,\quad\text{if}\quad \frac 1r=\frac 1p+\frac 4N,$$ and H\"older inequality provides the final estimate $$||A_\e v||_p\le c(p)||q_\e||_{N/4}||v||_p\le \frac 12||v||_p,$$ if $c(p)\e<1/2$. In conclusion, for such an $\e$, we obtain $$||(I-A_\e)^{-1}||_{p,p}\le 2,$$ and in turn, $$||u||_p\le ||(I-A_\e)^{-1}h_\e||_p\le 2||h_\e||_p\le 2C(p)||g_\e||_\infty,$$ in others words, $u\in L^p$ for all $p\in [1,\infty)$. This is sufficient to deduce the desired regularity, by the Sobolev inclusion and the Agmon-Douglis-Nirenberg estimates. \enddemo \heading{\bf 3.- THE LOCAL PALAIS-SMALE CONDITION BY THE LIONS CONCENTRATION-COMPACTNESS RESULTS}\endheading We denote $\bold E$ either $\esd$ or $\esc$ and define $||u||_{\bold E}=||\D u||_2$. A sequence $ \{u_j\}\subset \bold E $ is said to be a {\it Palais-Smale sequence} for $J$, defined by (1.1), if $$ \left\{ \aligned &J(u_j) \rightarrow c\\ &J'(u_j) \rightarrow 0 \quad \text{in} \quad \bold {E'}. \endaligned \right. \tag 3.1 $$ If (3.1) implies the existence of a subsequence $ \{u_{j_k}\} \subset\{u_j\} $ which converges strongly in $\bold E$, we say that $J$ {\it verifies the Palais-Smale condition.} If this strongly convergent subsequence exists only for some values of $c$, we say that $J$ verifies a {\it local Palais-Smale condition}. In our case, the main difficulty is the lack of compactness in the inclusion of $\bold E$ in $L^{\pss}.$ Here, we shall prove a local Palais-Smale condition, which is sufficient for the problem. The technical results used are based on a measure representation lemma, given by P.L. Lions in the proof of the concentration-compactness principle (see [13] and [14]). Let $\{u_j\}$ be a bounded sequence in $\bold E$. Then, there is a subsequence, such that $$u_j \db u \quad \text{weakly in}\quad \bold E,$$ and $$ \left. \aligned | \D u_j|^2 &\db d\mu \\ | u_j |^{\pss} & \db d \nu \endaligned \right\} \text{ weakly-* in the sense of measures}. $$ If we take $\phi \in \ci$, the Sobolev inequality applied to $ u_j \phi $ gives: $$ (\io |\phi |^{\pss} d\nu)^{\frac1{\pss}} S^{\frac 12} \le (\io |\phi|^2 d \mu)^{\frac 12} + 2(\io \langle \n \phi,\n u\rangle^2dx)^{\frac 12}+(\io |\D\phi|^2 |u|^2 dx)^\frac 12, \tag 3.2 $$ where $S= \inf \{ || u ||_{\bold E}^2 : u \in \bold E ,|| u ||_{\pss} =1 \}$ is the best constant in the Sobolev inclusion. The main idea is that if $u\equiv 0$ in (3.2), then we have a reverse H\"older inequality for two different measures, for which we have the following representation result (See P.L. Lions [13] and [14]): \proclaim{Lemma 3.1} Let $\mu, \nu $ be two non-negative and bounded measures on $\overline{\O}$, such that for $\mathbreak 1\le p0$ for which $$ (\io |\phi|^r d \nu)^\frac1r \le C (\io |\phi|^p d \mu )^\frac1p\quad \forall \phi \in \ci(\overline{\O}),\quad \text{with supp}(\phi)\quad \text{bounded}$$ Then, there exist $\{x_j\}_{j \in I} \subset \overline \O $ and $\{\nu_j\}_{j \in I } \subset (0,\infty) $, where $I$ is at most countable, such that: $$ \nu =\sum_{j \in I } \nu_j \d_{x_j} \quad , \quad \mu \ge C^{-p} \sum_{j \in I } \nu_j^{\frac pr} \d_{x_j}, $$ where $ \d_{x_j} $ is the Dirac mass supported at $ x_j $. \endproclaim \flushpar{\bf Remark} If $\O$ is bounded then $I$ is finite. By application of this result to $v_j=u_j-u$, P.L.Lions obtains (see [13] and [14]): \proclaim{Lemma 3.2} Let $\{u_j\}$ be a weakly convergent sequence in $\bold E$ with weak limit $u$. Assume also $$ \aligned i &) \quad | \D u_j |^2 \quad \text{converges in the weak-* sense of measures to a measure } \mu \\ ii &) \quad | u_j |^{\pss} \quad \text{converges in the weak-* sense of measures to a measure } \nu . \endaligned $$ Then, there exist $\{x_j\}\subset \overline{\O},\quad j=1,2,...,l$, such that: $$ \left\{ \aligned 1 &) \quad \nu = |u|^{\pss} + \sum_{j=1}^l \nu_j \d_{x_j}, \quad \nu_j >0, \\ 2 &) \quad \mu \ge | \D u |^2 + \sum_{j=1}^N \mu_j \d_{x_j}, \quad \mu_j>0,\\ 3 &) \quad \nu_j^{\frac 2{\pss}} \le \frac{\mu_j}S, \endaligned \right. \tag 3.3 $$ \endproclaim The Lemma 3.2 allows us to prove the following important result. \proclaim{Lemma 3.3} Let $\{v_j\}\subset\bold E$ be a Palais-Smale sequence for $J$, defined by (1.1), that is: $$J(v_j) \to c \tag 3.4 $$ $$J' (v_j) \to 0 \quad \text{in}\quad \bold{ E'}\tag 3.5$$ If $ c < \dfrac 2N S^{\frac N4} - K \l^{\b} $, where $ \b = \dfrac{\pss}{\pss -q} $ and $K$ depends on $ q,N $ and $\O$, then there exists a subsequence $ \{v_{j_k}\} \subset \{v_j\} $, which converges strongly in $ \bold E $. \endproclaim \demo{Proof} From (3.4), (3.5) we get that the sequence $\{v_j\}$ is bounded in $\bold E$. More precisely, by (3.5) we have $$0=\langle J'(v_j), v_j\rangle +{\pss}\langle \e_j ,v_j\rangle,\quad ||\e_j||_{{\bold E}'}\to 0\quad\text{as}\quad j\to \infty.$$ Therefore for $\d>0$ we have $$\aligned &c+\d\ge J(v_j)-\frac 1{\pss}\langle J'(v_j), v_j\rangle -\langle \e_j ,v_j\rangle\\ &=(\frac 12-\frac 1{\pss})\io |\D v_j|^2dx-\l(\frac 1q-\frac 1{\pss})\io |v_j|^qdx-\langle \e_j,v_j\rangle\endaligned$$ and the boundedness of $\{v_j\}$ in $\bold E$ follows easily from the Sobolev inequality. By Lemma 3.2 there exists a subsequence which we continue to denote by $\{v_j\}$ such that $$ \left \{ \aligned &v_j \db v \text{ weakly in } \bold E, \\ &v_j \to v \text{ in } L^r , 1 =0,$$ where $< , >$ is the duality pairing. Moreover $$\io \phi d\nu + \l \io |v|^q \phi dx = \lim_{j\to\infty} \io \D v_j \D (v_j \phi ) dx $$ By (3.6), (3.7), the weak convergence and the H\"older inequality, we can estimate: $$\lim_{j\to\infty} |\io \D v_j \D (v_j\phi) dx |= \io \phi d\mu + \lim_{j\to\infty} |\io \D v_j [2 \langle \n v_j,\n \phi\rangle+v_j \D\phi]dx|.$$ Now $$\aligned &0\le \lim_{j\to\infty} |\io \D v_j \langle \n v_j,\n \phi\rangle dx|\\ &\le\lim_{j\to\infty} (\io |\D v_j|^2 dx)^{1/2}(\io |\n \phi|^2 |\n v_j|^2 dx)^{1/2}\\ &\le C(\int_{B(x_k,\e)\cap \O} |\n \phi|^2 |\n v|^2 dx)^{1/2}\\ &\le C(\int_{B(x_k,\e)\cap \O}|\n\phi|^Ndx)^{1/N} (\int_{B(x_k,\e)\cap \O} |\n v|^{2N/(N-2)}dx)^{(N-2)/2N}\\ &\le C(\int_{B(x_k,\e)\cap \O} |\n v|^{2N/(N-2)}dx)^{(N-2)/2N}\to 0\quad\text {as}\quad \e\to 0, \endaligned$$ and $$\aligned &0\le\lim_{j\to\infty} |\io \D v_j v_j \D\phi dx|\\ &\le \lim_{j\to\infty} (\io |\D v_j|^2dx)^{1/2}(\io |\D \phi|^2 |v_j|^2 dx)^{1/2}\\ &\le C(\int_{B(x_k,\e)\cap \O} |\D \phi|^2 |v|^2 dx)^{1/2} \\ &\le C(\int_{B(x_k,\e)\cap \O}|\D \phi|^{N/2}dx)^{2/N} (\int_{B(x_k,\e)\cap \O} |v|^{\pss}dx)^{1/\pss}\\ &\le C(\int_{B(x_k,\e)\cap \O} |v|^{\pss}dx)^{1/\pss}\to 0\quad\text {as}\quad \e\to 0. \endaligned$$ Then, $$0=\lim_{\e \to 0}\{\io \phi d\nu+ \l \io |v|^q \phi dx - \io \phi d \mu \} = \nu_k - \mu_k.$$ By Lemma 3.2. $ \mu_k \ge S \nu_k^{\frac 2{\pss}}, \text{ i.e.}\, \nu_k \ge S \nu_k^{\frac 2{\pss}}$. Hence, either $$\nu_k =0, $$ or $$ \nu_k \ge S^{\frac N4} \tag 3.8 $$ We shall prove that (3.8) does not occur. Assume for contradiction that there exists a $k_0$ with $\nu_{k_0} \ne 0,\quad i.e. \quad \nu_{k_0} \ge S^{\frac N4} $. From the hypotheses (3.4) and (3.5), $$ \align c = \lim_{j\to\infty} J(v_j ) &= \lim_{j\to\infty} \{J(v_j) - \frac 12 < J'(v_j), v_j> \} \\ &\ge \frac 2N \io |v|^{\pss} + \frac 2N S^{\frac N4} + \l (\frac 12 - \frac 1q ) \io |v|^q . \tag 3.9 \endalign $$ Because $ 10$) at the point $ x_0 = \left( \dfrac {\l c_2 q}{\pss c_1} \right) ^{\frac 1{\pss - q}}.$ Hence, $$f(x) \ge f(x_0) =-K \l^{\frac{\pss}{\pss - q}}, $$ which contradicts the hypothesis that $ c < \dfrac 2N S^{\frac 4N} - K \l^{\b} $. Hence, $\nu_k =0 \quad \forall k $, and the proof is completed. \enddemo \ \heading{\bf 4.- EXISTENCE OF INFINITELY MANY SOLUTIONS}\endheading Let $\bold E$ be the Hilbert space defined in section 3. Let $\sum $ be the class of subsets of $ \bold E - \{0\} $ which are closed and symmetric with respect to the origin. For $ A \in \sum $, we define the genus $ \g (A) $ by $$ \g (A) = \min \{k \in \ene : \exists \phi \in \bold C (A;\re^k -\{0\}) , \phi(x)=-\phi(-x)\}. $$ If such a minimum is not defined then we consider $\g (A)=+\infty $. The main properties of the genus are the following (see \cite{20} for the details): \proclaim{Proposition 4.1} Let $A,B \in \sum $. Then: \roster \item"1)" If there exists an odd function $ f \in \bold C (A,B) $, then $\g (A) \le\g (B) $. \item"2)" If $ A \subset B $ , then $ \g (A) \le \g (B) $. \item"3)" If there exists an odd homeomorphism between $A$ and $B$, then $\g (A) = \g (B) $. \item"4)" If $ S^{N-1} $ is the sphere in $ \re^N $, then $ \g (S^{N-1}) = N $. \item"5)" $ \g (A \cup B) \le \g (A) + \g (B) $. \item"6)" If $ \g (B) < + \infty , \quad \text{then} \quad \g (\overline{A-B}) \ge \g (A) - \g (B) $. \item"7)" If $A$ is compact, then $\g (A) < + \infty $, and there exists $\d >0 $ such that $ \g (A) =\g ( N_\d (A))$ where $ N_\d (A)= \{ x \in \bold E : d(x,A) \le \d \} $. \item"8)" If $ X_0 $ is a subspace of $\bold E$ with codimension $K$ , and $\g (A) > K $, then $ A \cap X_0 \ne \emptyset $. \endroster \endproclaim Assume that $10 $ such that, if $0< \l < \l_o $, $h$ attains a local minimum and a local maximum. Let $R_0$, $R_1$ be such that $r< R_0h(r)$. We make the following truncation of the functional $J$. Take $ \t : \re^{+} \to [0,1]$, nonincreasing and $ {\bold C }^{\infty} $, such that $$ \left\{ \aligned &\t (x)=1 \quad \text{if}\quad x \le R_0 \\ &\t (x)=0 \quad \text{if}\quad x \ge R_1. \endaligned \right. $$ Let $ \var (u) = \t ( \| \n u \|_p) $. We consider the truncated functional $$ \tilde {J}(u) = \frac 12 \io | \D u|^2dx - \frac 1{\pss} \io |u|^{\pss} \var (u)dx - \frac{\l}q \io |u|^q dx. \tag 4.2 $$ As in (4.1), $\tilde{J}(u) \ge \overline{h} (\| \D u \|_2 ) $, with $$ \overline{h} (x)= \dfrac 12 x^2 - \dfrac 1{ \pss S^{\frac {\pss}2}} x^{\pss}\t (x)- \dfrac {\l}q C_qx^q.\tag 4.3 $$ Observe that $ \overline h = h $, for $ x \le R_0 $, and $ \overline h (x) =\frac 12 x^2 - \frac {\l}q C_q x^q$ for $x \ge R_1$. The main properties of $\tilde{J}$, defined by (4.2), are the following: \proclaim{Lemma 4.2} \roster \item"1)" $\tilde{J} \in {\bold C}^1 ( \bold E , \re ). $ \item"2)" If $\tilde{J}(u) \le 0 $, then $ \| \D u \|_2 < R_0 $ , and $J(v) = \tilde{J}(v) $ for all $v$ in a small enough neighborhood of $u$. \item"3)" There exists $ A > 0 $, such that, if $ 0< \l < A $ , then $\tilde{J}$ verifies a local Palais-Smale condition for $ c \le 0 $. \endroster \endproclaim \demo{Proof} 1) and 2) are immediate. To prove 3), observe that all the Palais-Smale sequences for $\tilde{J}$ with $ c \le 0 $ must be bounded; then, by Lemma 3.1, if $ \l $ verifies $ \dfrac 1N S^{\frac Np} - K {\l}^\b \ge 0 $ there exists a convergent subsequence. \enddemo Note that, if we find some negative critical value for $\tilde{J}$, then by 2) we have a negative critical value of $J$. Now, we will construct an appropriate mini-max sequence of negative critical values for the functional $\tilde{J}$. \flushpar The next lemma uses the same idea as in [11], and we include here the proof for the sake of completeness. \proclaim{Lemma 4.3} Given $n \in \ene $, there is $ \e = \e (n) > 0 $ , such that $$ \g( \{ u \in \bold E : \tilde{J}(u) \le - \e \}) \ge n. $$ \endproclaim \demo{Proof} Fix $n$ and let $ E_n $ be a $n$-dimensional subspace of $ \bold E $. Take $ u_n \in E_n $, with $ || \D u_n||_2 = 1 $. For $ 0<\rho0, \\ \b _n & = \inf \{\io |u|^q : u \in E_n , \quad || \D u_n||_2 =1 \} >0. \endalign $$ Hence $ \tilde{J}(\rho u_n) \le \dfrac 12 \rho^2 -\dfrac {\a_n}{\pss} \rho^{\pss} - \dfrac {\l \b_n}q \rho^q $ , and we can choose $\e$ (which depends on $n$), and $\eta < R_0$, such that $ \tilde{J}(\eta u) \le -\e $ if $ || \D u ||_2 = 1 $. Let $S_{\eta} = \{ u \in \bold E : || \D u ||_2 = \eta \} $ so that $ S_{\eta} \cap E_n \subset \{ u \in \bold E : \tilde{J}(u) \le - \e \} $ ; therefore, by Proposition 4.1, $$ \g (\{ u \in \bold E : \tilde{J}(u) \le - \e \}) \ge \g ( S_{\eta} \cap E_n) = n. $$ \enddemo This lemma allows us to prove the existence of critical points. \proclaim{Lemma 4.4} Let $\Sig_k =\{C\subset \bold E -\{0\} : C \text{ is closed}, C=-C ,\,\g (C) \ge k \} $. Let $ c_k = \inf\limits_{ C \in \Sig_k} \sup\limits_{ u \in C } \tilde{J}(u) $, $ K_c = \{ u \in \bold E : \tilde{J}'(u) =0 , \tilde{J}(u)=c \} $ , and suppose $0< \l < A $, where $ A $ is the constant in Lemma 4.2. If $ c = c_k = c_{k+1} = ...= c_{k+r} $, then $ \g (K_c) \ge r+1$. (In particular, the $c_k$'s are critical values of $J$). \endproclaim \demo{Proof} In the proof, we will use Lemma 4.3, and a classical deformation lemma (see [20]). For simplicity, put $ \tilde{J}^{-\e} = \{ u \in \bold E : \tilde{J}(u) \le - \e \} $ . By Lemma 4.3, $ \forall k \in \ene , \quad \exists \e(k) >0 $ such that $ \g( \tilde{J}^{- \e } ) \ge k $. Because $\tilde{J}$ is continuous and even, $ \tilde{J}^{- \e} \in \Sig_k $ ; then, $ c_k \le - \e (k) < 0 , \forall k $. But $\tilde{J}$ is bounded from below; hence, $ c_k > - \infty $ $ \forall k $. Let us assume that $ c = c_k = ... = c_{k+r} $, and observe that $ c<0$; therefore, $\tilde{J}$ verifies the Palais-Smale condition in $ K_c $. It is easy to see that $ K_c $ is compact. Assume for contradiction that $ \g (K_c) \le r $. Thus there exists a closed and symmetric set $U$, with $ K_c \subset U $, such that $ \g (U) \le r$. ( We can choose $ U \subset \tilde{J}^0 $, because $ c<0 $). By the deformation lemma, we have an odd homeomorphism $ \eta : \bold E \to \bold E $, such that $ \eta ( \tilde{J}^{c+\d } -U ) \subset \tilde{J}^{c - \d } $ , for some $ \d >0 $. (Again, we must choose $ 0< \d < -c $, because $\tilde{J}$ verifies the Palais-Smale condition on $ \tilde{J}^0$ , and we need $ \tilde{J}^{c+ \d } \subset \tilde{J}^0 $ ). By definition, $$ c= c_{k+r} = \inf_{ C \in \Sig_{k+r}} \sup_{u \in C} \tilde{J}(u). $$ Then, there exists $ A \in \Sig_{k+r} $ , such that $ \sup\limits_{u \in A} \tilde{J}(u) < c+ \d $ ; i.e., $ A \subset \tilde{J}^{c+\d} $, and $$ \eta (A-U) \subset \eta (\tilde{J}^{c+ \d } - U) \subset \tilde{J}^{c- \d }. \tag 4.4 $$ But $ \g ( \overline{A-U} ) \ge \g (A) - \g (U) \ge k $ , and $ \g ( \eta ( \overline{A-U} ) \ge \g ( \overline{A-U} ) \ge k $. Consequently, $ \eta ( \overline{A-U} ) \in \Sig_k $. This contradicts (4.4) since $$ \eta ( \overline{A-U} ) \in \Sig_k \quad \text{implies} \quad \sup\limits_{ u \in \eta ( \overline{A-U} )} \tilde{J}(u) \ge c_k = c. $$ \enddemo The lemma above proves the following result: \proclaim{Theorem 4.5} Assume $10 $, such that for $ 0< \l < A $, problems (P1) and (P2) admit infinitely many solutions. \endproclaim \demo{Proof} Integration by parts shows that any critical point is a solution of (P1). In fact, the boundary conditions are included in the choice $\bold E=W_0^{2,2}(\O)$. For problem (P2), the result is a consequence of the regularity result given in section 2. \enddemo \heading{\bf 5.- EXISTENCE OF A POSITIVE SOLUTION }\endheading Consider the problem $$ \left\{\aligned \bil u &= \l |u|^{q-2} u + |u|^{r-2} u\equiv f(u) \quad \text{ in } \, \O , \\ \left.u\right|_{\p \O} &=0,\\ \left.\D u \right|_{\p \O} &=0, \endaligned \right.\tag {P3} $$ where $\O\subset \ren$ is a smooth bounded domain, $N>4$, $ 10$, $r>2$. This means that we consider also {\it supercritical } problems. (Obviously, problem (P2) is a particular case of (P3)). The Laplacian case of (P3) has been already treated in [5]. Notice that for $\bil$ with these boundary conditions, the maximum principle holds as a consequence of the maximum principle for the Laplacian. In this section, we will show the following result \proclaim{\bf Theorem 5.1} There is a constant $\l_0>0$ such that for $0< \l \le \l_0$, problem (P3) has a positive solution. \endproclaim The proof of the theorem is organized in several Lemmas. \proclaim{\bf Lemma 5.2} Let $v$ be the solution of the Dirichlet Problem $$ \left\{\aligned \bil v &= \l +1 \\ \left.v\right|_{\p \O} &=0\\ \left.\D v\right|_{\p \O} &=0. \endaligned \right.\tag S $$ Then there exists a constant $\l_0$, such that for $\l<\l_0$ there is $T=T(\l)>0$ for which the function $\olu=T v$ is a supersolution of (P3). \endproclaim \demo{Proof} By simplicity we write $$|u|^{r -2} u +\l |u|^{q-2}u = F(u). $$ Fixed $\l$ consider the solution $v$ of the Dirichlet problem (S). Then $00$. Then $$\lim_{q\to 2}||u_q||_{\infty}=0.$$ It suffices to observe that the constant $c_5$ in the proof of Lemma 5.2 goes to zero as $q\to 2$. \proclaim{\bf Proposition 5.4} There is $ \b > 0 $ such that for all $ \l > \b $ problem (P3) has no positive solution. \endproclaim \demo{Proof} Let $\phi_1>0$ be as in Lemma 5.3. Integration by parts shows $$\io(\l |u|^{q-1} +u^{r-1})\phi_1 dx=\io \bil u \phi_1 dx=\l_1\io u\phi_1 dx .$$ But, for some $\a>0$, $c\l^\a u\le u^{q-1}+u^{r-1}\quad \forall u>0$, and then if $u$ is a positive solution of (P3) necessarily $c\l^\a<\l_1$. \enddemo \heading{\bf 6.- ON A RESULT BY BREZIS-NIRENBERG } \endheading In the paper [7], Brezis and Nirenberg obtain a remarkable result showing that for very general functionals related with semilinear problems involving the Laplacian, the local minima in $\cu$ are also local minima in $W_0^{1,2}$. We need an extension of this result of Brezis and Nirenberg to our problems. The next theorem is stated solely for problem (P2) because later applications will be done only for (P2). The proof follows that of Brezis and Nirenberg for second order problems, but is included for the sake of completeness. We remark that this first result is also true for problem (P1). Consider the functional $$J(u)=\frac 12\io |\D u|^2 dx-\frac{\l}{q}\io |u|^q dx- \frac{1}{\pss}\io |u|^{\pss}dx\tag 6.1$$ and put $f(u)=\l|u|^{q-2}u+|u|^{\pss-2}u$, $F(u)=\int_0^u f(s)ds$. We recall that $u\in \esc$ is a critical point of $J$ associated to problem $(P2)$ if $$0=\io \D u \D \phi dx-\io f(u)\phi dx,\quad\text{for all}\quad \phi\in\esc.$$ Define the class of functions $$\Cal E_0=\{v\in \cdos(\overline{\O})\quad |\, v(x)=0,\, x\in \p\O \}.$$ Then $u_0$ is a local minimizer of $J$ in $\Cal E_0$, (respectively $\esc$) if there is $\d>0$ such that $J(u_0)\le J(u_0+v)$, for all $v\in \Cal E_0$ such that $||v||_{\cdos}<\d$ (respectively for all $v\in \esc $ such that $||v||_{W^{2,2}}<\d.$) We denote by $B_\e$ the ball of radius $\e$ in $\esc$, centered at the origin. \proclaim{ Theorem 6.1} Let $u_0\in\esc$ be a local minimizer of $J$ in the $\cdos$ topology, then $u_0$ is a local minimizer in the $\esc$ topology. \endproclaim \demo{Proof} The regularity Theorem 2.1 shows that the minimizer $u_0\in \Cal C^3(\overline{\O})$. By linearity on the differential operator, without loss of generality, we can assume that $u_0=0$. Then if the conclusion does not hold, $$\text{for all}\quad \e>0, \quad \text{there exists}\quad v_\e \in B_\e,\quad \text{such that}\quad J(v_\e) < J(0). \tag 6.2$$ We shall proof that $0$ is not a minimum in the $\cdos$ topology. Consider the truncation $$ T_k(s)=\cases -k \quad &\text{if}\quad s\le -k\\ s \quad &\text{if}\quad -k0$ there is a constant $k=k(\e)$, such that $$J_{k(\e)}(v_\e)0$, $$u_1(x)+\e d(x,\p\O)\le u(x)\le u_2(x)-\e d(x,\p\O).\tag 6.12$$ Thus $J(u)=\overline J(u)$. Moreover, if we choose a ball in $\Cal E_0$ of radius less than $\e$, i.e., $v\in \cdos(\overline{\O})\cap \Cal C_0^1(\O)$ such that $||v-u||_{\cdos}<\e$ we have that $$J(v)=\overline J(v)\ge \overline J(u)=J(u),$$ that is, $u$ is a local minimum of $ J $ in $\cdos$. But then, by Theorem 6.1, $u$ is a local minimum of $ J $ in $\esc$. \enddemo We will use Theorems 6.1 and 6.2 in the next section. \flushpar{\bf Remark.-} Similar methods have been used by De Figueiredo, [8]. \heading{\bf 7.- EXISTENCE OF AT LEAST TWO POSITIVE SOLUTIONS} \endheading The existence of a second positive solution for (P2) depends on when we can apply some version of the {\it Mountain Pass Lemma}. In fact for $\l>0$ small enough, we can proceed as in [11] and obtain in this case a second positive solution. This result takes as starting point the minimum of the truncated functional discussed in section 4. The Palais-Smale condition is obtained depending on this minimum value. We concentrate the attention in a global result, in the spirit of [1]. More precisely, if we define $\Lambda=\sup\{\l>0 \, | (P2) \, \text{ has a positive solution}\}$, we obtain the following result. \proclaim{Theorem 7.1} If $\l\in I=(0,\Lambda)$, then problem (P2) has at least two positive solutions. \endproclaim \demo{Proof} The proof of the theorem will be done in several steps. \flushpar{\smc - Step 1.-\rm} Fix $\l\in I$ and consider the solution of (P2), $u_0$, obtained in Theorem 6.2, that is, $u_0$ is a local minimum of the functional $$J(u)=\frac 12\io |\D u|^2dx-\io F_\l (u)dx.\tag 7.1$$ in $ \esc $. Define $$ g_{\l } (x,s)=\cases f_\lambda(u_0(x)+s)-f_\lambda(u_0(x)),\quad&\text{if}\quad s>0\\ 0,\quad&\text{if}\quad s\le 0, \endcases $$ consider the truncation of $g_\l(x,s)$, $$ \overline g_{\l } (x,s)=\cases f_\lambda(u_2(x))-f_\lambda(u_0(x)),\quad&\text{if}\quad s\ge u_2(x)-u_0(x)\\ g_\lambda(x,s),\quad&\text{if}\quad u_2(x)-u_0(x) >s>0\\ 0,\quad&\text{if}\quad s\le 0, \endcases $$ and $G_\l(x,u)=\int_0^u g_\l(x,s)ds$, $\overline G_\l(x,u)=\int_0^u \overline g_\l(x,s)ds$, the respective primitives. Here $u_2\equiv u_{\l_2}$ as in Theorem 6.2. It is easy to check that the functional $$\Phi (v)=\frac 12\io |\D v|^2dx-\io \overline G_\l (x,v)dx\tag 7.2$$ attains its absolute minimum in $\esc$ in some point $v_0\in \esc$. By the maximum principle we have that $$u_2-u_0\ge v_0\ge 0. \tag 7.3$$ If $v_0\ne 0$ in $\O$ we are done since $$\bil v_0 =\overline g_\l (x,v_0)=g_\l (x,v_0),$$ and $u_1=u_0+v_0$ is the second positive solution of (P2), being $$\bil u_1=\bil v_0+\bil u_0=f_\l(u_0+v_0)-f_\l(u_0)+f_\l(u_0)=f_\l(u_1).$$ Our problem is now reduced to the case when $v_0\equiv 0$. \flushpar{\smc - Step 2.-\rm} Assume that the minimum of $\Phi$ is attained only in $v_0\equiv 0$. First, we want to show that $v_0=0$ is a local minimum in $\esc$ of the functional $$\Psi (v)=\frac 12\io |\D v|^2dx-\io G_\l (x,v)dx.\tag 7.4$$ Now, we have $\bil (u_2-u_0)\ge f_\l(u_2)-f_\l(u_0)\ge 0$ and $u_2-u_0=0$, $\D (u_2-u_0)=0$ on $\p \O$. Thus, by the Hopf Lemma for the Laplacian we can conclude that for $\e>0$ small enough, if $h\in \cdos \cap \Cal C_0^1$ and $||h||_{\cdos}\le \e$, then $$\Psi(h)=\Phi(h)\ge \Phi(v_0)\equiv\Phi(0)=\Psi(0).$$ This means that $v_0=0$ is a local minimum of $\Psi$ in $ \cdos \cap \Cal C_0^1$ and by Theorem 6.1 it is a local minimum in $\esc$. In the Step 2 above, we reduced the problem to the case where the minimum of $\Psi$ is zero. All the critical points of $\Psi$ are nonnegative. To finish the proof, assume that $ v_0 = 0 $ is the unique critical point of $ \Psi $. We shall prove that in this case the Mountain-Pass Theorem applies, giving a nontrivial solution, and hence a contradiction. This argument is developed in several steps. \flushpar{\smc - Step 3.-\rm} We will prove the following lemma. \proclaim{Lemma 7.2} If $v_0=0$ is the only critical point of $\Psi$, then $\Psi$ satisfies the Palais-Smale condition under the level $c_0=\frac{2}{N}S^{N/4}$. \endproclaim \demo{Proof} Assume that $\{ v_j\}_{j\in\ene}\subset \esc$ is a Palais-Smale sequence under the level $c_0$ for the functional $\Psi$ defined by (7.4), i.e., \item{i)} $\lim\limits_{j\to\infty}\Psi(v_j)=cc = \lim_{j\to\infty} \Psi (v_j ) = \lim_{j\to\infty} \{\Psi (v_j)- \frac12 < \Psi'(v_j), v_j> \} \ge \frac 2N S^{\frac N4}, $$ and this is a contradiction. Thus the subsequence converges strongly in $L^{\pss}(\O)$, and in turn in $\esc$. \enddemo \ The last result that we have to show is that there exists a Palais-Smale sequence below the critical level $ \frac{2}{N}S^{N/4} $. More precisely \proclaim{Lemma 7.3} If $v_0=0$, local minimum of $\Psi$, is its unique critical point, then there exists a Palais-Smale sequence such that $$\lim_{j\to\infty}\Psi(v_j)=c0, \quad K_1=[N(N-4)(N^2-4)]^{(N-4)/8}, $$ and $V_\e$ verifies the problem $\bil u=u^{\frac{N+4}{N-4}}$ in $\ren$ with $N>4$ and $$S^\frac N4=\irn |\D V_1|^2dx=\irn |V_1|^{\pss}dx\tag 7.5$$ The best constant is the same for equivalent norms. (See [13]). The idea is to perform a truncation with a cutoff function $\rho(x)\ge 0$, smooth, such that, $\rho(x)=1$ if $|x|2R$; where we take $R>0$ in such way that all $x$ verifying $|x|\le 2R$ belong to $\O$. More precisely, define $$v_\e (x)=\rho(x)V_\e (x).\tag 7.6$$ For $\e$ small enough, the concentration produced will give us that $$\sup_{t\ge 0}\Psi(tv_\e)=c_\e<\frac{2}{N}S^{N/4},\tag 7.7$$ that is sufficient to have the result. We proceed to prove (7.7). We can get the following estimates, $$ \io |\D v_\e|^2 dx=\int_{\ren}|\D V_1|^2dx+O(\e^{N-4})\tag7.8 $$ $$ \io |v_\e|^{\pss} dx=\int_{\ren}|V_1|^{\pss}dx+O(\e^{N})\tag7.9 $$ and for some positive $k$, $$ \io |v_\e|^{r} dx=\left\{\aligned &k\e^\frac{(N-4)r}{2}+o(\e^\frac{(N-4)r}{2})\quad\text{if}\quad r<\frac{N}{N-4}\\ &k\e^{N-\frac{(N-4)r}{2}}|\log \e|+o(\e^{N-\frac{(N-4)r}{2}}|\log\e|)\quad\text{if}\quad r=\frac{N}{N-4}\\ &k\e^{N-\frac{(N-4)r}{2}}+o(\e^{N-\frac{(N-4)r}{2}})\quad\text{if}\quad r>\frac{N}{N-4} \endaligned\right.\tag 7.10 $$ The key for the estimate (7.7) is $$G_\l(x,s)\ge \dfrac 1{\pss}s^{\pss}+u_0(x)s^{\pss-1}+C u_0(x)^{\pss-\gamma}s^\gamma,\quad \gamma\in(\frac N{N-4},\frac{N+4}{N-4}),\tag 7.11 $$ which is a consequence of the following calculus inequality: {\it If $ r > 2 $ then given $ \g \in (1, r-1) $ there exists a constant $ C>-\infty $ such that $$ \inf_{t>0}\{\dfrac{(1+t)^r -( 1+ t^r + r t + r t^{r-1})}{ t^{\g}}\}\ge C. $$} Now, from (7.11) we have $$\Psi(tv_\e)\le \frac {t^2}{2} \io |\D v_\e|^2 dx - \frac {t^{\pss}}{\pss}\io |v_\e|^{\pss} dx-m_1t^{\pss-1}\io |v_\e|^{\pss-1}dx+|C|m_1^{\pss-\gamma}t^\gamma\io |v_\e(x)|^\gamma dx,$$ (here we use that $00}h_{\e} (t) < h_o(t_o) = \dfrac 2 N S^{N/4}. $$ To finish the proof, we need to analyze the influence of the error term. If we denote by $ t_{\e} $ the point where $ h_{\e} $ attaint its maximum, it is easily seen that \newline $ 0< t_{\e} < t_o $ and $ t_{\e} \to t_o $ as $ \e \to 0 $. Therefore, we can write $ t_{\e} = t_o x_{\e}$, where $ x_{\e} \to 1 $ as $ \e \to 0 $. Taking into account that $ h_{\e}'(t_{\e})=0 $, we get $$ t_o x_{\e}\int_{\ren} |\D V_1|^2 dx - t_o^{\pss-1} x_{\e}^{\pss-1} \int_{\ren} |V_1|^{\pss} dx = C (\pss - 1) t_o^{\pss-2} x_{\e}^{\pss-2} \e^{\frac{N-4}2}. $$ Using the precise value of $ t_o $, after some computations we arrive to $$ 1- x_{\e}^{\pss-2} = A x_{\e}^{\pss-3} \e^{\frac{N-4}4}, $$ where $$ A= C(\pss-1) \dfrac {( \int_{\ren} |\D V_1|^2 dx)^{\frac{-1}{\pss-2}}}{( \int_{\ren} |V_1|^{\pss} dx)^{1- \frac 1{\pss-2}}}. $$ By Taylor's expansion: $$ (1-x_{\e})(\pss-2) x_{\e}^{\pss-3}+ o(1-x_{\e}) = A x_{\e}^{\pss-3} \e^{\frac{N-4}2}. $$ Therefore, $ 1- x_{\e}= M \e^{\frac{N-4}2} + o(\e^{\frac{N-4}2}) $, for $ M= \dfrac A{\pss-2} $. Finally, this identity allows us to prove that $$ h_{\e}(t_{\e} )= \dfrac 2 N S^{N/4} - C t_o^{\pss -1 } \e^{\frac{N-4}2} + o( \e^{\frac{N-4}2}) $$ and the conclusion follows at once. \ \flushpar{\smc - Step 4.-\rm} Assume that $v_0$ is the unique critical point of $\Psi$. Consider the function $w_\e=r_\e v_\e$ , with $r_\e$ large enough, such that $\Psi(w_\e)<0$ and the mini-max value $$c_\e=\inf_{\gamma\in \Cal P}\max_{t\in [0,1]}\Psi(\gamma(t)),$$ where $$\Cal P=\{\gamma:[0,1]\apl \esc\,:\,\text{continuous,}\,\gamma(0)=0,\, \gamma(1)=w_\e\}.$$ Because $v_0=0$ is the local minimum, then $0\le c_\e<\frac 2N S^{\frac N4}$. If $c_\e>0$ the Mountain Pass Lemma by Ambrosetti and Rabinowitz, [4], give us a second positive critical point, in contradiction with the hypothesis. In the case $c_\e=0$, we get the same contradiction by using the result by Pucci-Serrin, [19]. This contradiction finishes the proof. \enddemo \enddemo \ \flushpar{\bf Remark} We can say that the solutions constructed in sections 4 and 5 correspond to the {\it sublinear} term, because, for instance, when $\l\to 0$ they converge to the trivial solution. The same behavior is obtained for $q\to 2$ in the case of the minimal positive solution obtained in section 5. The second positive solution obtained in Theorem 7.1, however, tends to a Dirac mass as $\l \to 0$. This behavior was obtained for solutions of the p-laplacian in [11]. \ \heading{\bf 8.- FURTHER RESULTS} \endheading The results given above can be generalized without difficulty to second members of the form $f(x,u)$, where $f$ is increasing in $u$, verifies some regularity, growth and oddness properties. We prefer to avoid more technicalities. We have, on the other hand, several remarks about possible applications of the methods above to similar cases. \ \subheading{A) Subcritical Problems} Obviously all the results obtained above also hold for the problems $$ \left\{\aligned \bil u &= \l |u|^{q-2} u + |u|^{r-2} u\equiv f(u) \quad \text{ in } \, \O , \\ \left.u\right|_{\p \O} &=0,\\ \left.\frac{\p u}{\p n}\right|_{\p \O} &=0, \endaligned \right.\tag {S1} $$ and $$ \left\{\aligned \bil u &= \l |u|^{q-2} u + |u|^{r-2} u\equiv f(u) \quad \text{ in } \, \O , \\ \left.u\right|_{\p \O} &=0,\\ \left.\D u \right|_{\p \O} &=0, \endaligned \right.\tag {S2} $$ where $12p$, $1