\input amstex \documentstyle{amsppt} \loadmsbm \magnification=\magstephalf \hcorrection{1cm} \vcorrection{6mm} \nologo \TagsOnRight \NoBlackBoxes \headline={\ifnum\pageno=1 \hfill\else% {\tenrm\ifodd\pageno\rightheadline \else \leftheadline\fi}\fi} \def\rightheadline{EJDE--1994/04\hfil Non-autonomous EllipticBoundary Value Problems\hfil\folio} \def\leftheadline{\folio\hfil V. Anuradha, S. Dickens, and R. Shivaji \hfil EJDE--1994/04} \def\pretitle{\vbox{\eightrm\noindent\baselineskip 9pt % Electronic Journal of Differential Equations\hfil\break Vol. {\eightbf 1994}(1994), No. 04, pp. 1-10. Published July 8, 1994.\hfil\break ISSN 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \hfil\break ftp (login: ftp) 147.26.103.110 or 129.120.3.113 \bigskip} } \topmatter \title Existence Results for Non-autonomous Elliptic Boundary Value Problems \endtitle \thanks \noindent Submitted: January 23, 1994.\hfil\break 1991 AMS {\eighti Subject Classification:} 35J65.\hfil\break Partially supported by NSF Grants DMS - 8905936 and DMS - 9215027. \hfil\break This work was completed while R. Shivaji was a Visiting Research Scientist at Georgia Institute of Technology in 1993.\hfil\break \copyright 1994 Southwest Texas State University and University of North Texas. \endthanks \author V. Anuradha, S. Dickens, and R. Shivaji \endauthor \address V. Anuradha, Department of Mathematics and Statistics, University of Arkansas at Little Rock, Little Rock, AR 72204-1099\endaddress \address S. Dickens, Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762 \endaddress \address R. Shivaji, Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762. E-mail rs1\@ra.msstate.edu \endaddress \abstract We study solutions to the boundary value problems $$-\Delta u(x) = \lambda f(x, u);\quad x \in \Omega$$ $$u(x) + \alpha(x) \frac{\partial u(x)}{\partial n} = 0;\quad x \in \partial \Omega$$ where $\lambda > 0$, $\Omega$ is a bounded region in $\Bbb{R}^{N}$; $N \geq 1$ with smooth boundary $\partial \Omega$, $\alpha(x) \geq 0$, $n$ is the outward unit normal, and $f$ is a smooth function such that it has either sublinear or restricted linear growth in $u$ at infinity, uniformly in $x$. We also consider $f$ such that $f(x, u) u \leq 0$ uniformly in $x$, when $|u|$ is large. Without requiring any sign condition on $f(x, 0)$, thus allowing for both positone as well as semipositone structure, we discuss the existence of at least three solutions for given $\lambda \in (\lambda_{n}, \lambda_{n + 1})$ where $\lambda_{k}$ is the $k$-th eigenvalue of $-\Delta$ subject to the above boundary conditions. In particular, one of the solutions we obtain has non-zero positive part, while another has non-zero negative part. We also discuss the existence of three solutions where one of them is positive, while another is negative, for $\lambda$ near $\lambda_{1}$, and for $\lambda$ large when $f$ is sublinear. We use the method of sub-super solutions to establish our existence results. We further discuss non-existence results for $\lambda$ small. \endabstract \endtopmatter \document \subhead{1.\quad Introduction}\endsubhead We first consider the boundary value problem $$-\Delta u(x) = \lambda f(x, u);\quad x \in \Omega \tag{1.1}$$ $$u(x) + \alpha(x) \frac{\partial u(x)}{\partial n} = 0;\quad x \in \partial \Omega \tag{1.2}$$ where $\Delta$ is the Laplacian operator, $\Omega$ is a bounded region in $\Bbb{R}^{N}$; $N \geq 1$ with smooth boundary $\partial \Omega$, $\alpha(x) \geq 0$, $n$ is the unit outward normal, and $f$ is a smooth function such that $$\lim_{u \to \infty} \frac{f(x, u)}{u} = \sigma \text{ uniformly in }\, x, \tag{1.3}$$ $$\lim_{u \to - \infty} \frac{f(x, u)}{u} = \beta \text{ uniformly in }\, x, \tag{1.4}$$ and $$\frac{\partial f}{\partial u} (x, u) \geq 0. \tag{1.5}$$ Let $\lambda_{k}$, $\phi_{k}$ be the eigenvalues and corresponding eigenfunctions of the boundary value problem $$-\Delta \phi_{k} = \lambda_{k} \phi_{k};\quad x \in \Omega \tag{1.6}$$ $$\phi_{k}(x) + \alpha(x) \frac{\partial \phi_{k}(x)}{\partial n} = 0;\quad x \in \partial \Omega. \tag{1.7}$$ Let $I = [a, b] \subset (\lambda_{n}, \lambda_{n + 1})$ and for $\lambda \in I$ consider the unique solution $Z_{\lambda}$ of the boundary value problem $$-\Delta Z_{\lambda} - \lambda Z_{\lambda} = -1;\quad x \in \Omega \tag{1.8}$$ $$Z_{\lambda} (x) + \alpha (x) \frac{\partial Z_{\lambda} (x)}{\partial n} = 0.;\quad x \in \partial \Omega \tag{1.9}$$ Let $\mu_{1} = \inf_{x \in \bar{\Omega},\, \lambda \in I} Z_{\lambda}(x)$, $\mu_{2} = \sup_{x \in \bar{\Omega},\, \lambda \in I} Z_{\lambda}(x)$ and $\mu = \max \{|\mu_{1}|, \mu_{2}\}$. Note that $Z_{\lambda}^{+} \not\equiv 0$ (see Appendix 1). Also, $\exists$ $\delta (\Omega) > 0$ such that if $I \subset (\lambda_{1}, \lambda_1 +\delta)$ then $Z_{\lambda} > 0$; $x \in \Omega$ (anti-maximum principle: see Cl\'em\'ent and Peletier in [1]). Now assume that $$\exists\,\, m > 0 \text{ such that } y + m > f(x, y) > y - m \,\,\, \forall\,\, y \in [-bm\mu, b m\mu] \tag{1.10}$$ and $$\beta, \sigma < \frac{1}{b||w_{\alpha}||_{\infty}} \tag{1.11}$$ where $w_{\alpha}$ is the unique positive solution to $$-\Delta w_{\alpha} = 1; \quad x \in \Omega \tag{1.12}$$ $$w_{\alpha}(x) + \alpha(x) \frac{\partial w_{\alpha} (x)}{\partial n} = 0;\quad x \in \partial \Omega. \tag{1.13}$$ Then we prove: \proclaim{Theorem 1.1} Let $I = [a, b] \subset (\lambda_{n}, \lambda_{n + 1})$ and (1.3)--(1.5), (1.10)--(1.11) hold. Then these exists at least three solutions to (1.1)--(1.2) for $\lambda \in I$. One of these solutions is a non-negative or sign changing solution, while another is a non-positive or sign changing solution. \endproclaim \remark{Remark 1.1} Note that unlike in the literature of ``jumping nonlinearities" (see [2]--[3]), our results do not require $(\beta, \sigma)$ to include a part of the spectrum of $-\Delta$, nor require, a part of the spectrum of $-\Delta$ to lie between $f'(0)$ and $f'(\pm\infty)$ for autonomous nonlinearities as in [4] and the reference within. \endremark \proclaim{Theorem 1.2} Let $I = [a, b] \subset (\lambda_1, \lambda_{1} + \delta)$ and (1.3)--(1.5), (1.10)--(1.11) hold. Then there exists at least three solutions to (1.1)--(1.2) for $\lambda \in I$, where one is a positive solution while another is a negative solution. \endproclaim Next we consider the particular case when $\sigma = \beta = 0$ (sublinear case). We further assume that there exists $f_{1}(u) < f(x, u); \forall\, x \in \bar{\Omega}, u \geq 0$ such that $$f_{1}(r_{1})=0,\; f_{1}'(r_{1}) < 0,\; \int_{0}^{r_{1}} f(s)\, ds >0 \tag{1.14}$$ for some $r_{1} > 0$, and that there exists $f_{2}(u) > f(x, u); \forall\; x \in \bar{\Omega}, u \leq 0$ such that $$f_{2}(r_{2})=0,\; f_{2}'(r_{2}) < 0,\; \int_{r_{2}}^{0} f(s)\, ds <0 \tag{1.15}$$ for some $r_{2} < 0$. Then we prove: \proclaim{Theorem 1.3} Assume (1.14)--(1.15) and let (1.3)--(1.5) hold with $\sigma = \beta = 0$. Then there exists at least three solutions to (1.1)--(1.2) for $\lambda$ large, where one is a positive solution while another is a negative solution. \endproclaim \remark{Remark 1.2} We refer to [4] where a positive solution for $\lambda \in I = [a, b] \subset (\lambda_{1}, \lambda_{1} + \delta)$ was discussed in the case of \underbar{autonomous}, \underbar{sublinear} $(\sigma = 0)$, and \underbar{semipositone} $(f(0) < 0)$ problems with \underbar{Dirichlet} boundary conditions. In [5] it was assumed that there exists $m > 0$ such that $f(y) \geq y - m$ $\forall y \in [0, bm\mu]$ to obtain a positive solution for $\lambda \in I$. Further by constructing a function $f_{1}(u)$ satisfying (1.14) a positive solution for $\lambda$ large was discussed. See also [6] where the authors study the existence of a positive solution via \underbar{degree theory} arguments. Hence they allow non-autonomous problems with Robin boundary condition, but consider only the \underbar{sublinear} case, and study the existence of positive solutions for \underbar{$\lambda$ large} with the assumption $f(x, 0) \leq 0$ $\forall x \in \bar{\Omega}$. \endremark We also consider $f$ such that $$\exists\,\, r > 0 \text{ for which } f(x, r) \leq 0 \text{ while } f(x, -r) \geq 0 \text{ for every } x \in \bar{\Omega}. \tag{1.16}$$ Then we prove: \proclaim{Theorem 1.4} Let $I = [a, b] \subset (\lambda_{n}, \lambda_{n + 1})$ and (1.10), (1.16) hold. Assume $r \geq bm\mu$. Then there exists at least three solutions to (1.1)--(1.2) for $\lambda \in I$. One of these solutions is a non-negative or sign changing solution, while another is a non-positive or sign changing solution. \endproclaim \proclaim{Theorem 1.5} Let $I = [a, b] \subset (\lambda_{1}, \lambda_{1} + \delta)$ and (1.10), (1.16) hold. Assume $r \geq bm\mu$. Then there exists at least three solutions to (1.1)--(1.2) for $\lambda \in I$. One of these is a positive solution, while another is a negative solution. \endproclaim \proclaim{Theorem 1.6} Let (1.16) hold, and assume that (1.14) holds for all $0 \leq u \leq r$ with $r_{1} \leq r$ while (1.15) holds for all $-r \leq u \leq 0$ with $r_{2} \geq -r$. Then there exists at least three solutions to (1.1)--(1.2) for $\lambda$ large, where one is a positive solution while another is a negative solution. \endproclaim \remark{Remark 1.3} Consider the boundary value problem $$-\Delta u(x) = \lambda f(||x||, u);\quad x \in B_{N}$$ $$u(x) + \alpha \frac{\partial u(x)}{\partial n} = 0;\quad x \in \partial B_{N}$$ where $\alpha \geq 0$ is a constant and $B_{N}$ is the unit ball in $\Bbb{R}^{N}$. Then under the corresponding hypothesis on $f(||x||, u)$ instead of $f(x, u)$, one can generate all the solutions obtained in Theorems 1.1--1.6 to be \underbar{radial}. This follows from the fact that all the sub and super solutions we will use in the proofs of these theorems will turn out to be radial. \endremark We next discuss non-existence results under the assumption that there exists $\gamma > 0$ such that $$\frac{\partial f}{\partial u} \leq \gamma. \tag{1.17}$$ We recall that $\lambda_{1}$ is the principal eigenvalue and $\phi_1 > 0$ is a corresponding eigenfunction of $-\Delta$ subject to boundary Robin conditions (1.2). Then we prove: \proclaim{Theorem 1.7} Assume $\int_{\Omega} f(x, 0) \phi_1 (x) \leq 0$ and $\lambda < \lambda_{1}/\gamma$. If $u(\not\equiv 0)$ is any solution of (1.1)--(1.2) then $u^{-} \not\equiv 0$ (i.e. there does not exist any solution $u(\not\equiv 0)$ which is non-negative). \endproclaim \proclaim{Theorem 1.8} Assume $\int_{\Omega} f(x, 0) \phi_1(x) \geq 0$ and $\lambda < \lambda_{1}/\gamma$. If $u(\not\equiv 0)$ is any solution of (1.1)--(1.2) then $u^{+} \not\equiv 0$ (i.e. there does not exist any solution $u(\not\equiv 0)$ which is non-positive). \endproclaim \proclaim{Theorem 1.9} Assume $f(x, 0) \leq 0$ and $\lambda < \lambda_{1}/\gamma$. If $u$ is any solution of (1.1)--(1.2) then $u \leq 0$. \endproclaim \proclaim{Theorem 1.10} Assume $f(x, 0) \geq 0$ and $\lambda < \lambda_{1}/\gamma$. If $u$ is any solution of (1.1)--(1.2) then $u \geq 0$. \endproclaim We give detailed proofs of our results in section 2. For literature on autonomous semipositone problems with Dirichlet boundary conditions see [7], while for the positone case see [8] and the references cited in [8]. Our existence results are based on sub-super solutions. Namely, a \underbar{super solution} is defined as a smooth function $\phi$ such that $$-\Delta \phi \geq \lambda f(x, \phi);\quad x \in \Omega \tag{1.18}$$ $$\phi(x) + \alpha(x) \frac{\partial \phi(x)}{\partial n} \geq 0;\quad x \in \partial \Omega \tag{1.19}$$ and a \underbar{subsolution} is a smooth function $\psi$ that satisfies (1.18)--(1.19) with the inequalities reversed. If $\psi \leq \phi$, then it follows that (1.1)--(1.2) has a solution $u$ such that $\psi \leq u \leq \phi$ (see [9]--[10]). Further if $\psi_{1}$ is a subsolution, $\psi_{2}$ is a strict subsolution, $\phi_{1}$ is a strict supersolution and $\phi_{2}$ is a supersolution such that $\psi_{1} \leq \psi_{2} \leq \phi_{2}$, $\psi_{1} \leq \phi_{1} \leq \phi_{2}$, $\psi_{2}(x_{0}) > \phi_{1}(x_{0})$ for some $x_{0} \in \bar{\Omega}$, then (1.1)--(1.2) has at least \underbar{three} distinct solutions $u_{1}$, $u_{2}$, $u_{3}$ such that $u_{1} \leq u_{2} \leq u_{3}$. See [11] for this multiplicity result which was proved for Dirichlet boundary conditions. However, it is easy to see that this result holds for the Robin boundary conditions (1.2) as well. \subhead{2.\quad Proofs of Theorems 1.1--1.10}\endsubhead \demo{\bf Proof of Theorem 1.1} Let $v_{1}(x) := bmZ_{\lambda}(x)$ and $u_{2}(x) = -v_{1}(x)$. Then $-\Delta v_{1} = bm(-\Delta Z_{\lambda}) = bm (\lambda Z_{\lambda} - 1)$ $\leq bm(\lambda Z_{\lambda} - \lambda/b) (\text{ since } \lambda \leq b) = \lambda [bm Z_{\lambda} - m]$ $= \lambda [v_{1} - m] < \lambda f(x, v_{1})$ by (1.10), and $-\Delta u_{2} = \Delta v_{1} \geq \lambda [-v_{1} + m] = \lambda[u_{2} + m]$ $> \lambda f(x, u_{2})$ again by (1.10). Thus $v_{1}$ is a strict subsolution while $u_{2}$ is a strict supersolution. Note that $v_{1}^{+} \not \equiv 0$ and $u_{2}^{-} \not \equiv 0$. Now let $u_{1} (x) := J w_{\alpha}(x)$ where $J > 0$ is large enough so that $$\frac{1}{b||w_{\alpha}||_{\infty}} \geq \frac{f(x, J||w_{\alpha}||_{\infty})}{J||w_{\alpha}||_{\infty}} \tag{2.1}$$ and $$u_{1} \geq v_{1},\quad u_{1} \geq u_{2}. \tag{2.2}$$ Here (2.1) is possible since $\sigma < \frac{1}{||w_{\alpha}||_{\infty}b}$, and (2.2) is possible by the Hopf's maximum principle. Then $-\Delta u_{1} = J \geq \lambda f(x, J||w_{\alpha}||_{\infty}); \,\, x \in \Omega$ (by (2.1) since $\lambda \leq b)$ $\geq \lambda f(x, J w_{\alpha})$ (since $\frac{\partial f}{\partial u} \geq 0) = \lambda f(x, u_{1})$. Next let $v_{2}(x) = - \tilde{J} w_{\alpha} (x)$ where $\tilde{J} > 0$ is large enough so that $$\frac{1}{b||w_{\alpha}||_{\infty}} \geq \frac{f(x, - \tilde{J} ||w_{\alpha}||_{\infty})}{- \tilde{J} ||w_{\alpha}||_{\infty}} \tag{2.3}$$ and $$v_{2} \leq v_{1},\quad v_{2} \leq u_{2}. \tag{2.4}$$ Here again (2.3) is possible since $\beta < \frac{1}{||w_{\alpha}||_{\infty}b}$ and (2.4) is possible by the Hopf's maximum principle. Then $-\Delta v_{2} = -\tilde{J} \leq \lambda f(x, -\tilde{J}||w_{\alpha}||_{\infty});\quad x \in \Omega$ (by (2.3) since $\lambda \leq b) \leq \lambda f(x, - \tilde{J} w_{\alpha})$ (since $\frac{\partial f}{\partial u} \geq 0) = \lambda f(x, v_{2})$. Thus $u_{1}$ is a supersolution while $v_{2}$ is a subsolution such that $v_{2} \leq v_{1} \leq u_{1}$ and $v_{2} \leq u_{2} \leq u_{1}$, where $v_{1}$ is a strict subsolution, $u_{2}$ is a strict supersolution with $v_{2} \leq 0$, $v_{1}^{+} \not \equiv 0$, $u_{2}^{-} \not \equiv 0$ and $u_{1} \geq 0$. Hence the result. \qed\enddemo \demo{\bf Proof of Theorem 1.2} If $\lambda \in (\lambda_{1}, \lambda_{1} + \delta)$ then $Z_{\lambda} > 0$; $x \in \Omega$. Then $v_{1}(x) > 0$; $x \in \Omega$ while $u_{2} < 0$; $x \in \Omega$ and hence the result follows by the proof in Theorem~1.1. \qed\enddemo \demo{\bf Proof of Theorem 1.3} Consider the autonomous Dirichlet problem $$-\Delta v = \lambda f_{1}(v);\quad x \in \Omega \tag{2.5}$$ $$v = 0;\quad x \in \partial \Omega. \tag{2.6}$$ Then by (1.14) it follows that there exists $\bar{\lambda}_{1} > 0$ such that for $\lambda \geq \bar{\lambda}_{1}$ (2.5)-(2.6) has a positive solution $v_{\lambda}$ (see [12]). Clearly since $\frac{\partial v_{\lambda}}{\partial n} \leq 0$; $x \in \partial \Omega$ and $f_{1}(S) < f(x, S)$; $\forall x \in \bar{\Omega}$, $S \geq 0$ $v_{\lambda}$ is a strict subsolution to (1.1)--(1.2) for $\lambda \geq \bar{\lambda}_{1}$. Next consider $$-\Delta u = \lambda f_{2} (u);\quad x \in \Omega \tag{2.7}$$ $$u = 0;\quad x \in \partial \Omega. \tag{2.8}$$ Then setting $w = -u$ we see that $w$ satisfies $$-\Delta w = \lambda [-f_{2}(-w)];\quad x \in \Omega \tag{2.9}$$ $$w = 0;\quad x \in \partial \Omega. \tag{2.10}$$ Let $g(w) = -f_{2}(-w)$. Then $g(-r_{2}) = 0$, $g'(-r_{2}) = f_{2}'(r_{2}) < 0$ and $\int_{0}^{-r_{2}} g(s) ds$ $= \int_{0}^{-r_{2}} - f_{2}(-s) ds = -\int_{\gamma_{2}}^{0} f_{2}(s) ds > 0$. Hence again by [12], there exists $\bar{\lambda}_{2} > 0$ such that for $\lambda \geq \bar{\lambda_{2}}$, (2.9)--(2.10) has a positive solution $w_{\lambda}$. Equivalently, (2.7)--(2.8) has a negative solution $u_{\lambda} = -w_{\lambda}$ for $\lambda \geq \bar{\lambda}_{2}$. Also since $\frac{\partial u_{\lambda}}{\partial n} \geq 0$; $x \in \partial \Omega$ and $f_{2}(S) > f(x, S)$; $\forall x \in \bar{\Omega}$, $S \geq 0$, $u_{\lambda}$ is a strict super solution to (1.1)--(1.2) for $\lambda \geq \bar{\lambda_{2}}$. Let $\bar{\lambda} = \max \{\bar{\lambda_{1}}, \bar{\lambda}_{2}\}$ and $\lambda > \bar{\lambda}$ be fixed. Consider $u_{1}(x) := J w_{\alpha} (x)$ where $J > 0$ is large enough so that $$\frac{1}{\lambda ||w_{\alpha}||_{\infty}} \geq \frac{f(x, J||w_{\alpha}||_{\infty})}{J||w_{\alpha}||_{\infty}} \tag{2.11}$$ and $$u_{1} \geq v_{\lambda}. \tag{2.12}$$ Here (2.11) is possible since $\sigma = 0$ and (2.12) is possible by the Hopf's maximum principle. Then $-\Delta u_{1} = J \geq \lambda f(x, J||w_{\alpha}||_{\infty}) \geq \lambda f(x, Jw_{\alpha})$ (since $\frac{\partial f}{\partial u} \geq 0) = \lambda f(x, u_{1})$. Next consider $v_{2}(x) := - \tilde{J} w_{\alpha} (x)$ when $\tilde{J} > 0$ is large enough so that $$\frac{1}{\lambda ||w_{\alpha}||_{\infty}} \geq \frac{f(x, - \tilde{J} ||w_{\alpha}||_{\infty})}{-\tilde{J} ||w_{\alpha}||_{\infty}} \tag{2.13}$$ and $$v_{2} \leq u_{\lambda}. \tag{2.14}$$ Here again (2.13) is possible since $\beta = 0$ and (2.14) is possible by the Hopf's maximum principle. Then $-\Delta v_{2} = - \tilde{J} \leq \lambda f(x, -\tilde{J}||w_{\alpha}||_{\infty})$ $\leq \lambda f(x_{1}, - \tilde{J} w_{\alpha})$ (since $\frac{\partial f}{\partial u} \geq 0) = \lambda f(x, v_{2})$. Thus $u_{1}$ is a supersolution while $v_{2}$ is a subsolution such that $v_{2} \leq u_{\lambda} \leq 0 \leq v_{\lambda} \leq u_{1}$, where $u_{\lambda}$ is a strict supersolution and $v_{\lambda}$ is a strict subsolution. Hence the result. \qed\enddemo \demo{\bf Proof of Theorem 1.4} Let $v_{1}(x)$ and $u_{2}(x)$ be the strict sub and strict super solutions to (1.1)--(1.2) as in the proof of Theorem 1.1. Then since $r \geq bm\mu$, both $v_{1}(x)$ and $u_{2}(x)$ satisfy $-r \leq v_{1}(x) \leq r$, $-r \leq u_{2}(x) \leq r$. But by (1.16) $u_1(x) = r$ and $v_{2}(x) \equiv -r$ are super and sub solutions respectively to (1.1)--(1.2). Hence the result. \qed\enddemo \demo{\bf Proof of Theorem 1.5} Let $v_{1}(x)$ and $u_{2}(x)$ be as in the proof of Theorem 1.4. But $\lambda \in (\lambda_{1}, \lambda_{1} + \delta)$. Hence $v_{1}(x) \geq 0$ while $u_{2}(x) \leq 0$. The rest of the proof is identical to the proof of Theorem 1.4. \qed\enddemo \demo{\bf Proof of Theorem 1.6} Let $u_{\lambda}$ and $v_{\lambda}$ be respectively the strict super and strict subsolutions to (1.1)--(1.2) as in the proof of Theorem 1.3. But $-r_{2} \leq u_{\lambda} \leq 0 \leq v_{\lambda} \leq r_{1}$ (see [12]) while $u_{1}(x) \equiv r$ are $v_{2}(x) = -r$ are super and subsolutions respectively to (1.1)--(1.2). Hence the result. \qed\enddemo \demo{\bf Proof of Theorem 1.7} Assume $u \geq 0$, $u \not \equiv 0$. Then $$\align -\Delta u &= \lambda f(x, u)\\ &= \lambda [f(x, u) - f(x, 0)] + \lambda f(x, 0)\\ &= \lambda \frac{\partial f}{\partial u} (x, \eta) u + \lambda f(x, 0) (\text{ where } 0 \leq \eta \leq u)\\ &\leq \lambda \gamma u + \lambda f(x, 0) (\text{ by } (1.17)).\endalign$$ Thus $u$ satisfies $$-\Delta u - \lambda \gamma u \leq \lambda f(x, 0);\quad x \in \Omega \tag{2.15}$$ $$u(x) + \alpha(x) \frac{\partial u(x)}{\partial n} = 0;\quad x \in \partial \Omega. \tag{2.16}$$ Multiplying (2.15) by $\phi_1$ and integrating we obtain $$\int_{\Omega} - \Delta u \phi_1\, dx - \int_{\Omega} \lambda \gamma u \phi_1\, dx \leq \int_{\Omega} \lambda f(x, 0) \phi_1\, dx \leq 0.$$ Applying Green's second identity we obtain $$\int_{\partial \Omega} \{-\phi_1 \frac{\partial u}{\partial n} + u \frac{\partial \phi_1}{\partial n} \}\, ds + \int_{\Omega} u \lambda_{1} \phi_1\, dx - \int_{\Omega} \lambda \gamma u \phi_1\, dx \leq 0.$$ But applying the boundary conditions we see that $$\int_{\partial \Omega} \{-\phi_1 \frac{\partial u}{\partial n} + u \frac{\partial \phi_1}{\partial n}\}\, ds = 0.$$ Thus we obtain $$\int_{\Omega} u \phi_1 (\lambda_{1} - \lambda \gamma)\, dx \leq 0.$$ But $u \geq 0$, $\phi_1 > 0$ in $\Omega$ and this is a contradiction if $\lambda < \lambda_{1}/\gamma$. Hence the result. \qed\enddemo \demo{\bf Proof of Theorem 1.8} Assume $u \leq 0$, $u \not \equiv 0$. Proceeding as in the proof of Theorem 1.7 we obtain $$\align -\Delta u &= \lambda \frac{\partial f}{\partial u} (x, z) u + \lambda f(x, 0)\\ & \geq \lambda \gamma u + \lambda f(x, 0) (\text{ by } (1.17)). \endalign$$ Thus $u$ satisfies $$-\Delta u - \lambda \gamma u \geq \lambda f(x, 0);\quad x \in \Omega \tag{2.17}$$ $$u(x) + \alpha(x) \frac{\partial u(x)}{\partial n} = 0;\quad x \in \partial \Omega. \tag{2.18}$$ Multiplying (2.17) by $\phi_1$, using $\int_{\Omega} f(x, 0) \phi_1 dx \geq 0$, and proceeding as in the proof of Theorem 1.7 we obtain $$\int_{\Omega} u\phi_1 (\lambda_{1} - \lambda \gamma)\, dx \geq 0.$$ But $u \leq 0$ while $\phi_1 > 0$ for $x \in \Omega$, thus this is a contradiction if $\lambda < \lambda_{1}/\gamma$. Hence the result. \qed\enddemo \demo{\bf Proof of Theorem 1.9} Suppose $u > 0$ somewhere in $\bar{\Omega}$. Then $\exists$ some $\Omega_{1} \subseteq \Omega$ such that $u > 0$ in $\Omega$, and either $$u(x) = 0 \text{ or } u(x) + \alpha(x) \frac{\partial u(x)}{\partial n} = 0;\quad x \in \partial \Omega. \tag{2.19}$$ Let $\tilde{\lambda}_{1}$ be the principal eigenvalue and $\tilde{\phi}_1 > 0$ be a corresponding eigenfunction of $-\Delta$ on the region $\Omega_{1}$ subject to Robin conditions (1.2) on $\partial \Omega_1$. Note that $\tilde{\lambda}_{1} \geq \lambda_{1}$. Now since $f(x, 0) \leq 0$ we have $\int_{\Omega} f(x, 0) \tilde{\phi}_1\, dx \leq 0$. Thus following the steps in the proof of Theorem 1.7 we obtain $$\int_{\Omega_{1}} - \Delta u \tilde{\phi}_1\, dx - \int_{\Omega_{1}} \lambda \gamma u \tilde{\phi}_1\, dx \leq 0$$ and then $$\int_{\partial \Omega_{1}} [-\tilde{\phi}_1 \frac{\partial u}{\partial n} + u \frac{\partial \tilde{\phi}_1}{\partial n}]\, ds + \int_{\Omega_{1}} [\tilde{\lambda}_{1} - \lambda \gamma] u \tilde{\phi}_1 \, dx \leq 0.$$ But if $u = 0$ on $\partial \Omega_{1}$, then $\frac{\partial u}{\partial n} \leq 0$ on $\partial \Omega_{1}$, while if $u + \alpha \frac{\partial u}{\partial n} = 0$ on $\partial \Omega_{1}$, since $\tilde{\phi}_1 + \alpha \frac{\partial \tilde{\phi}_1}{\partial n} = 0$, we have $-\tilde{\phi}_1 \frac{\partial u}{\partial n} + u \frac{\partial \tilde{\phi}_1}{\partial n} = 0$ on $\partial \Omega_{1}$. Thus in any case (see (2.19)) $-\tilde{\phi}_1 \frac{\partial u}{\partial n} + u \frac{\partial \tilde{\phi}_1}{\partial n} \geq 0$. Hence $$\int_{\Omega_{1}} [\tilde{\lambda}_{1} - \lambda \gamma] u \tilde{\phi}_1\, dx \leq 0$$ which is a contradiction since $u > 0$, $\tilde{\phi}_1 > 0$ for $x \in \Omega_{1}$ while $\lambda < \lambda_{1}/\gamma \leq \tilde{\lambda_{1}}/\gamma$. Hence the result. \qed\enddemo \demo{\bf Proof of Theorem 1.10} Suppose $u < 0$ somewhere in $\bar{\Omega}$. Then $\exists$ some $\Omega_{1} \subseteq \Omega$ such that $u < 0$ in $\Omega_{1}$ and either $$u(x) = 0 \text{ or } u(x) + \alpha(x) \frac{\partial u(x)}{\partial n} = 0;\quad x \in \partial \Omega_{1} \tag{2.20}$$ Let $\tilde{\lambda}_{1}$ and $\tilde{\phi}_1 > 0$ be as in the proof of Theorem 1.9. Now $f(x, 0) \geq 0$ and hence $\int_{\Omega_{1}} f(x, 0) \tilde{\phi}_1 dx \geq 0$. Thus following the steps in the proof of Theorem 1.8 we obtain $$\int_{\Omega_{1}} - \Delta u \tilde{\phi}_1\, dx - \int_{\Omega_{1}} \lambda \gamma u \tilde{\phi}_1\, dx \geq 0,$$ and then $$\int_{\partial \Omega} [-\tilde{\phi}_1 \frac{\partial u}{\partial n} + u \frac{\partial \tilde{\phi}_1}{\partial n}]\, ds + \int_{\Omega_1} [\tilde{\lambda}_{1} - \lambda \gamma] u \tilde{\phi}_1\, dx \geq 0.$$ But if $u = 0$ on $\partial \Omega_{1}$, then $\frac{\partial u}{\partial n} \geq 0$ on $\partial \Omega_{1}$, while if $u + \alpha \frac{\partial u}{\partial n} = 0$ on $\partial \Omega_{1}$, since $\tilde{\phi}_1 + \alpha \frac{\partial \tilde{\phi}_1}{\partial n} = 0$ on $\partial \Omega$, we have $-\tilde{\phi}_1 \frac{\partial u}{\partial n} + u \frac{\partial \tilde{\phi}_1}{\partial n} = 0$ on $\partial \Omega_{1}$. Thus in any case (see (2.20)) $-\tilde{\phi}_1 \frac{\partial u}{\partial n} + u \frac{\partial \tilde{\phi}_1}{\partial n} \leq 0$. Hence $$\int_{\Omega_{1}} [\tilde{\lambda}_{1} - \lambda \gamma] u \tilde{\phi}_1 \, dx \geq 0$$ which is a contradiction since $u < 0$, $\tilde{\phi}_1 > 0$ for $x \in \Omega$, while $\lambda < \lambda_1/\gamma \leq \tilde{\lambda}_{1}/\gamma$. Hence the result. \qed\enddemo \subhead{Appendix 1} \endsubhead Let $\lambda \in (\lambda_{n}, \lambda_{n + 1})$ and consider the unique solution to the boundary value problem $$-\Delta Z_{\lambda} - \lambda Z_{\lambda} = -1;\quad x \in \Omega$$ $$Z_{\lambda}(x) + \alpha(x) \frac{\partial Z_{\lambda}(x)}{\partial n} = 0;\quad x \in \partial \Omega.$$ Let $\phi_{1} > 0$ satisfy (1.6)--(1.7) for $k = 1$. Then $$\int_{\Omega} - \Delta Z_{\lambda} \phi_{1}\, dx - \int_{\Omega} \lambda Z_{\lambda} \phi_{1}\, dx = \int_{\Omega} - \phi_{1}\, dx$$ which implies $$\int_{\partial \Omega} \{-\phi_{1} \frac{\partial Z_{\lambda}}{\partial n} + Z_{\lambda} \frac{\partial \phi_{1}}{\partial n}\}\, ds + \int_{\Omega} Z_{\lambda} \lambda_{1} \phi_1\, dx - \int_{\Omega} \lambda Z_{\lambda}\phi_{1} dx = \int_{\Omega} - \phi_{1}\, dx\,.$$ But $$\int_{\partial \Omega} \{-\phi_1 \frac{\partial Z_{\lambda}}{\partial n} + Z_{\lambda} \frac{\partial \phi_{1}}{\partial n}\}\, ds = \int_{\partial \Omega} \{\alpha \frac{\partial \phi_1}{\partial n} \frac{\partial Z_{\lambda}}{\partial n} - \alpha \frac{\partial Z_{\lambda}}{\partial n} \frac{\partial \phi_{1}}{\partial n}\} \, ds = 0\,.$$ Thus $$\int_{\Omega} (\lambda_{1} - \lambda) Z_{\lambda} \phi_{1}\, dx = \int_{\Omega} - \phi_{1}\, dx.$$ But $\lambda > \lambda_1$ and $\phi_1 > 0$ for $x \in \Omega$. Hence $Z_{\lambda}^{+} \not \equiv 0$. \Refs \ref \no 1 \by Ph. Cl\'ement and L. A. Peletier \paper An anti-maximum principle for second order elliptic operators \jour J. Diff. Eqns. \vol 34 \yr 1979 \pages 218--229 \endref \ref \no 2 \by A. C. Lazer and P. J. McKenna \paper Large-Amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis \jour SIAM Rev. \vol 32(4) \yr 1990 \pages 537--578 \endref \ref \no 3 \by A. Castro and S. Gadam \paper The Lazer-McKenna conjecture for radial solutions in the $R^{N}$ ball \jour Electronic J. Diff. Eqns. \vol 7 \yr 1993 \pages 1--6 \endref \ref \no 4 \by A. Castro and J. Cossio \paper Multiple solutions for a nonlinear Dirichlet problem \jour SIAM Journal of Math. Anal. \toappear \endref \ref \no 5 \by A. Castro, J. B. Garner and R. Shivaji \paper Existence results for classes of sublinear semipositone problems \jour Results in Mathematics \vol 23 \yr 1993 \pages 214--220 \endref \ref \no 6 \by W. Allegretto and P. Nistri \paper Existence and stability for non-positone elliptic problems \jour J. Nonlinear Analysis TMA \toappear \endref \ref \no 7 \by A. Castro and R. Shivaji \paper Semipositone problems \jour Semigroups of Linear and Nonlinear Operations and applications \eds G. R. Goldstein and J. A. Goldstein \yr 1993 \pages 109--119 \endref \ref \no 8 \by K. J. Brown, M. M. A. Ibrahim and R. Shivaji \paper S-shaped bifurcation curves \jour J. Nonlinear Analysis, TMA \vol 5(5) \yr 1981 \pages 475--486 \endref \ref \no 9 \by H. Amann \paper Fixed point equations and nonlinear eigenvalue problems in ordered Bannach spaces \jour SIAM Rev. \vol 18 \yr 1976 \pages 620--709 \endref \ref \no 10 \by D. H. Sattinger \paper Monotone methods in nonlinear elliptic and parabolic boundary value problems \jour Indiana University Mathematics Journal \vol 21(11) \yr 1972 \pages 979--1000 \endref \ref \no 11 \by R. Shivaji \paper A remark on the existence of three solutions via sub-super solutions \jour Nonlinear Analysis and Applications \eds V. Lakshmikantham \yr 1987 \pages 561--566 \endref \ref \no 12 \by P. Clement and G. Sweers \paper Existence and multiplicity results for a semilinear elliptic eigenvalue problem \jour Ann. Scuola Norm. Sup. Pisa. Cl. Sci. \vol (4)14 \yr 1987 \pages 97--121 \endref \endRefs \enddocument \bye