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We study positive solutions to the two-point boundary value problem: $$\begin{aligned} \begin{matrix} -u''=\lambda h(t) f(u)~;~(0,1) \\ u(0)=0\\ u'(1)+c(u(1))u(1)=0,\end{matrix} \end{aligned}$$ where $$\lambda >0$$ is a parameter, $$h \in C^1((0,1],(0,\infty ))$$ is a decreasing function, $$f \in C^1((0,\infty ),\mathbb {R}) $$ is an increasing concave function such that $$\lim \limits _{s \rightarrow \infty }f(s)=\infty $$ , $$\lim \limits _{s \rightarrow \infty }\frac{f(s)}{s}=0$$ , $$\lim \limits _{s \rightarrow 0^+}f(s)=-\infty $$ (infinite semipositone) and $$c \in C([0,\infty ),(0,\infty ))$$ is an increasing function. For classes of such h and f, we establish the uniqueness of positive solutions for $$\lambda \gg 1$$ .

We are concerned with extinction properties for a p-Laplacian equation with gradient source and absorption terms under nonlinear boundary condition. Based on the integral norm estimate method and t...

A system governed by a one-dimensional hyperbolic equation with a mixing transport term and both ends being general nonlinear boundary conditions is considered in this paper. By using the snap-back...

We improve one of the methods for obtaining Liouville theorems for superlinear parabolic problems. In particular, if we consider a positively homogeneous gradient nonlinearity $$F:{{\mathbb {R}}}^m\rightarrow {{\mathbb {R}}}^m$$ of degree $$p>1$$, where $$n>2$$, $$p

This paper is devoted to the study of a system of nonlinear viscoelastic wave equations with nonlinear boundary conditions. Based on the Faedo-Galerkin method and standard arguments of density corresponding to the regularity of initial conditions, we first establish two local existence theorems of weak solutions. By the construction of a suitable Lyapunov functional, we next prove a blow up result and a decay result of global solutions.

We prove the existence of positive radial solutions to the problem \begin{document}$ \begin{cases} -\Delta _{p}u = \lambda \ K(|x|)f(u)\ \text{in } |x|>r_{0}, \\ \dfrac{\partial u}{\partial n}+\tilde{c}(u)u = 0\ \text{on }|x| = r_{0},\ \ u(x)\rightarrow 0\text{ as }|x|\rightarrow \infty ,\end{cases} $\end{document} where \begin{document}$ \ \Delta _{p}u = div(|\nabla u|^{p-2}\nabla u),\ N>p>1, \Omega = \{x\in \mathbb{R}^{N}:|x|>r_{0}>0\}, $\end{document} \begin{document}$ f:(0,\infty )\rightarrow \mathbb{R} $\end{document} is \begin{document}$ p $\end{document} -superlinear at \begin{document}$ \infty $\end{document} with possible singularity at \begin{document}$ 0, $\end{document} and \begin{document}$ \lambda $\end{document} is a small positive parameter. A nonexistence result is also established when \begin{document}$ f $\end{document} has semipositone structure at \begin{document}$ 0. $\end{document}

The main contribution of this paper is to prove the existence of extremal solutions for a novel class of ψ-Caputo fractional differential equation with nonlinear boundary conditions. For this purpose, we utilize the well-known monotone iterative technique together with the method of upper and lower solutions. Finally, we provide an example along with graphical representations to confirm the validity of our main results.

In this paper we study a free boundary problem modeling the growth of nonnecrotic tumors with angiogenesis. This model differs from the other tumor models studied in existing literatures at the point that it has a nonlinear boundary value condition for the nutrient concentration. We first study spherically symmetric version of this model. We prove that there exists a unique spherically symmetric stationary solution which is asymptotically stable under spherically symmetric perturbation. Next we make rigorous analysis to the spherically asymmetric version of this model. By using some abstract theory of parabolic differential equations in Banach manifold, we prove that this free boundary problem is locally well-posed in little Holder spaces and the radial stationary solution is asymptotically stable in case the surface tension coefficient γ is larger than a threshold value, whereas unstable in case γ is less than this threshold value.

In this work, we derive a reliable and efficient residual-typed error estimator for the finite element approximation of a 2D cathodic protection problem governed by a steady-state diffusion equation with a nonlinear boundary condition. We propose a standard adaptive finite element method involving the Dörfler marking and a minimal refinement without the interior node property. Furthermore, we establish the contraction property of this adaptive algorithm in terms of the sum of the energy error and the scaled estimator. This essentially allows for a quasi-optimal convergence rate in terms of the number of elements over the underlying triangulation. Numerical experiments are provided to confirm this quasi-optimality.

Results appearing in this paper can be used to establish the solvability of nonlinear discrete time systems subject to generalized nonlinear boundary conditions. Two separate sets of result...

In this paper, we study the homogeneous nonlinear boundary value problem for the p-Laplacian equation u t − △ p u + a ( x , t ) | u | σ − 2 u − b ( x , t ) | u | ν − 2 u = h ( x , t ) . We prove the existence of weak solutions which is global or local in time in dependence on the relation between the exponent of nonlinear part in boundary value and p. Boundedness of weak solution is proved. We established conditions of uniqueness. We prove also the properties of extinction in a finite time, finite speed propagation and waiting time. Lastly, by using the energy method, we obtain sufficient conditions that the solutions of this problem with non-positive initial energy blow up in finite time.

The finite time blow-up of solutions for 1-D NLS with oscillating nonlinearities is shown in two domains: (1) the whole real line where the nonlinear source is acting in the interior of the domain and (2) the right half-line where the nonlinear source is placed at the boundary point. The distinctive feature of this work is that the initial energy is allowed to be non-negative and the momentum is allowed to be infinite in contrast to the previous literature on the blow-up of solutions with time dependent nonlinearities. The common finite momentum assumption is removed by using a compactly supported or rapidly decaying weight function in virial identities - an idea borrowed from [ 18 ]. At the end of the paper, a numerical example satisfying the theory is provided.

We study positive solutions to (singular) boundary value problems of the form: \begin{document}$\left\{ \begin{align} & -\left( {{\varphi }_{p}}(u') \right)'=\lambda h(t)\frac{f(u)}{{{u}^{\alpha }}},~\ \ t\in (0,1),~~ \\ & u'(1)+c(u(1))u(1)=0,~ \\ & u(0)=0, \\ \end{align} \right.$ \end{document} where \begin{document}$\varphi_p(u): = |u|^{p-2}u$\end{document} with \begin{document}$p>1$\end{document} is the \begin{document}$p$\end{document} -Laplacian operator of \begin{document}$u$\end{document} , \begin{document}$λ>0$\end{document} , \begin{document}$0≤α , \begin{document}$c:[0,∞)\rightarrow (0,∞)$\end{document} is continuous and \begin{document}$h:(0,1)\rightarrow (0,∞)$\end{document} is continuous and integrable. We assume that \begin{document}$f∈ C[0,∞)$\end{document} is such that \begin{document}$f(0) , \begin{document}$\lim_{s\rightarrow ∞}f(s) = ∞$\end{document} and \begin{document}$\frac{f(s)}{s^{α}}$\end{document} has a \begin{document}$p$\end{document} -sublinear growth at infinity, namely, \begin{document}$\lim_{s \rightarrow ∞}\frac{f(s)}{s^{p-1+α}} = 0$\end{document} . We will discuss nonexistence results for \begin{document}$λ≈ 0$\end{document} , and existence and uniqueness results for \begin{document}$λ \gg 1$\end{document} . We establish the existence result by a method of sub-supersolutions and the uniqueness result by establishing growth estimates for solutions.

In this article we extend the known theory of solution regions to encompass nonlinear boundary conditions. We both provide results for new boundary conditions and recover some known results for the linear case This work was partially supported by NSERC Canada. Marcos Tella and F. Adrián F. Tojo were partially supported by Ministerio de Economía y Competitividad, Spain, and FEDER, project MTM2013-43014-P, and by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016-75140-P, co-financed by the European Community fund FEDER SI

This work presents a procedure for constructing the solution of a second order nonlinear partial differential equation subjected to nonlinear boundary conditions in a very simple and systematic way. The class of equations to be considered here includes, but is not limited to, the partial differential equations which govern the steady-state heat transfer process in bodies with temperature-dependent thermal conductivity and temperature-dependent internal heat source. In addition, it will be considered a large class of nonlinear boundary conditions, especially those arising from the description of the heat exchange process from/to bodies at high temperature levels. Proofs of the solution’s existence and uniqueness are presented.

We prove some uniqueness results for positive harmonic functions on the unit ball satisfying a nonlinear boundary condition.

In this paper we study quasilinear elliptic equations driven by the so-called double phase operator and with a nonlinear boundary condition. Due to the lack of regularity, we prove the existence of multiple solutions by applying the Nehari manifold method along with truncation and comparison techniques and critical point theory. In addition, we can also determine the sign of the solutions (one positive, one negative, one nodal). Moreover, as a result of independent interest, we prove for a general class of such problems the boundedness of weak solutions.

We investigate the following nonlinear parabolic equations with nonlocal source and nonlinear boundary conditions:$$ \textstyle\begin{cases} (g(u) )_{t} =\sum_{i,j=1}^{N} (a^{ij}(x)u_{x_{i}} ) _{x_{j}}+\gamma _{1}u^{m} (\int _{D} u^{l}{\,\mathrm{d}}x ) ^{p}-\gamma _{2}u^{r}& \mbox{in } D\times (0,t^{*}), \\ \sum_{i,j=1}^{N}a^{ij}(x)u_{x_{i}}\nu _{j}=h(u) & \mbox{on } \partial D\times (0,t^{*}), \\ u(x,0)=u_{0}(x)\geq 0 &\mbox{in } \overline{D}, \end{cases} $${(g(u))t=∑i,j=1N(aij(x)uxi)xj+γ1um(∫Duldx)p−γ2urin D×(0,t∗),∑i,j=1Naij(x)uxiνj=h(u)on ∂D×(0,t∗),u(x,0)=u0(x)≥0in D‾,wherepand$\gamma _{1}$γ1are some nonnegative constants,m,l,$\gamma _{2}$γ2, andrare some positive constants,$D\subset \mathbb{R}^{N}$D⊂RN($N\geq 2$N≥2) is a bounded convex region with smooth boundary∂D. By making use of differential inequality technique and the embedding theorems in Sobolev spaces and constructing some auxiliary functions, we obtain a criterion to guarantee the global existence of the solution and a criterion to ensure that the solution blows up in finite time. Furthermore, an upper bound and a lower bound for the blow-up time are obtained. Finally, some examples are given to illustrate the results in this paper.

We consider equilibrium problems for a cracked composite plate with a thin cylindrical rigid inclusion. Deformation of an elastic matrix is described by the Timoshenko model. The plate is assumed to have a through crack that does not touch the rigid inclusion. In order to describe mutual nonpenetration of the crack faces we impose a boundary condition in the form of inequality on the crack curve. For a family of appropriate variational problems, we analyze the dependence of their solutions on the location of the rigid inclusion. We formulate an optimal control problem with a cost functional defined by an arbitrary continuous functional on the solution space, while the location parameter of inclusion is chosen as the control parameter. The existence of a solution to the optimal control problem and a continuous dependence of the solutions in a suitable Sobolev space with respect to the location parameter are proved.

The $L^2$--gradient flow of the elastic energy of networks leads to a Willmore type evolution law with nontrivial nonlinear boundary conditions. We show local in time existence and uniqueness for this elastic flow of networks in a Sobolev space setting under natural boundary conditions. In addition we show a regularisation property and geometric existence and uniqueness. The main result is a long time existence result using energy methods.

In this paper we study quasilinear elliptic systems with nonlinear boundary condition with fully coupled perturbations even on the boundary. Under very general assumptions our main result says that each weak solution of such systems belongs to L ∞ ( Ω ‾ ) × L ∞ ( Ω ‾ ) . The proof is based on Moser's iteration scheme. The results presented here can also be applied to elliptic systems with homogeneous Dirichlet boundary condition.

We show the existence of positive solutions for a singular superlinear fourth-order equation with nonlinear boundary conditions. u⁗x=λhxfux, x∈0,1,u0=u′0=0,u″1=0, u⁗1+cu1u1=0, where λ > 0 is a small positive parameter, f:0,∞⟶ℝ is continuous, superlinear at ∞, and is allowed to be singular at 0, and h: [0, 1] ⟶ [0, ∞) is continuous. Our approach is based on the fixed-point result of Krasnoselskii type in a Banach space.

In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian.

Bifurcation theory is used to prove the existence of connected components of positive solutions for some classes of singular superlinear semi-positone problems with nonlinear boundary conditions.

In this paper, for an one-dimensional semilinear wave equation we study a mixed problem with a nonlinear boundary condition. The questions of uniqueness and existence of global and blow-up solutions of this problem are investigated, depending on the nonlinearity nature appearing both in the equation and in the boundary condition.

The paper considers the blow-up solution for the following nonlinear reaction-diffusion equations under nonlinear boundary conditions: (h(u))t=∇⋅(ρ(|∇u|p)|∇u|p−2∇u)+b(x)k(t)f(u),(x,t)∈Ω×(0,...

We present a geometric formula of Poincaré type, which is inspired by a classical work of Sternberg and Zumbrun, and we provide a classification result of stable solutions of linear elliptic problems with nonlinear Robin conditions on Riemannian manifolds with nonnegative Ricci curvature. The result obtained here is a refinement of a result recently established by Bandle, Mastrolia, Monticelli and Punzo.

For a system of stationary solutions to a reaction–diffusion equations with small diffusion coefficient and nonlinear flux boundary condition we prove that the bulk balance law, not only on the domain but on its boundary as well, is a necessary condition for formation of internal and boundary layers.

Using Leray–Schauder alternative and convex analysis, we obtain geometric conditions for the existence of solutions of nonlocal nonlinear boundary value problems for N-dimensional n-order differenc...

By topological arguments, we prove new results on the existence, non-existence, localization and multiplicity of nontrivial solutions of a class of perturbed nonlinear integral equations. These type of integral equations arise, for example, when dealing with boundary value problems where nonlocal terms occur in the differential equation and/or in the boundary conditions. Some examples are given to illustrate the theoretical results., 29 pages

We consider the Cauchy problem for nonlinear abstract wave equations in a Hilbert space. Our main goal is to show that this problem has solutions with arbitrary positive initial energy that blow up in a finite time. The main theorem is proved by employing a result on growth of solutions of abstract nonlinear wave equation and the concavity method. A number of examples of nonlinear wave equations are given. A result on blow up of solutions with arbitrary positive initial energy to the initial boundary value problem for the wave equation under nonlinear boundary conditions is also obtained.

In this paper, we consider the existence and exactness of multiple positive solutions for the nonlinear boundary value problem \begin{document}$\left\{ \begin{array}{l} - u''(x) = \lambda f(u),\;\;\;\;0 where \begin{document}$λ>0$\end{document} is a bifurcation parameter, \begin{document}$f(u)>0$\end{document} for \begin{document}$u>0$\end{document} . We give complete descriptions of the structure of bifurcation curves and determine the existence and multiplicity of positive solutions of the above problem for \begin{document}$f(u) = e^{u},\ f(u) = a^{u}(a>0),\ f(u) = u^{p}(p>0),\ f(u) = e^{u}-1,\ f(u) = a^{u}-1(a>1)$\end{document} and \begin{document}$f(u) = (1+u)^{p}(p>0)$\end{document} . Our methods are based on a detailed analysis of time maps.

This paper is devoted to the study of the blow-up solutions of the following weakly coupled degenerate parabolic systems with nonlinear boundary conditions: u t = div | ∇ u | p ∇ u + f ( x , u , v , t ) , v t = div | ∇ v | q ∇ v + g ( x , u , v , t ) , ( x , t ) ∈ D × ( 0 , T ∗ ) , ∂ u ∂ n = b ( u ) , ∂ v ∂ n = d ( v ) , ( x , t ) ∈ ∂ D × ( 0 , T ∗ ) , u ( x , 0 ) = u 0 ( x ) , v ( x , 0 ) = v 0 ( x ) , x ∈ D ¯ . Here p > 0 , q > 0 , D is a bounded spatial region in R n ( n ≥ 2 ) , and the boundary ∂ D is smooth. We mainly combine the maximum principles of the weakly coupled parabolic systems with the first-order differential inequality technique to discuss the blow-up phenomenon of the above problem. Sufficient conditions for the blow-up of the nonnegative solution ( u , v ) of this problem are given. In addition, for the nonnegative blow-up solution ( u , v ) , we also obtain an upper bound on the blow-up time and an upper estimate of the blow-up rate.

In this paper, we consider the following double phase problem with a nonlinear boundary condition − div | ∇ u | p − 2 ∇ u + μ ( x ) | ∇ u | q − 2 ∇ u = f ( x , u ) − | u | p − 2 u − μ ( x ) | u | q − 2 u in Ω , | ∇ u | p − 2 ∇ u + μ ( x ) | ∇ u | q − 2 ∇ u ⋅ ν = g ( x , u ) on ∂ Ω . First of all, we prove the existence of a solution and infinitely many solutions for this problem with superlinear nonlinearity (without A–R condition). In addition, under the sublinear assumptions on f and g , we establish the existence of infinitely many solutions via Clark’s theorem for the aforementioned problem.

We discuss existence and nonexistence results of positive radial solutions to the problem { − Δ u = λ K ( | x | ) f ( u ) in | x | > r 0 , ∂ u ∂ n + c ˜ ( u ) u = 0 on | x | = r 0 , u ( x ) → 0 as | x | → ∞ , where Ω = { x ∈ R N : | x | > r 0 > 0 } , N > 2 , f : ( 0 , ∞ ) → R is continuous, superlinear at ∞, and is allowed to be singular at 0 with no sign conditions near 0.

Let \begin{document}$ D\subset R^{d}$\end{document} be a bounded domain in the \begin{document}$d- $\end{document} dimensional Euclidian space \begin{document}$R^{d} $\end{document} with smooth boundary $Γ=\partial D.$ In this paper we consider exponential boundary stabilization for weak solutions to the wave equation with nonlinear boundary condition: \begin{document}$\left\{ \begin{gathered}u_{tt}(t)-ρ(t)Δ u(t)+b(x)u_{t}(t)=f(u(t)), \\ u(t)=0\ \ \text{on }Γ_{0}×(0,T), \\ \dfrac{\partial u(t)}{\partialν}+γ(u_{t}(t))=0\ \ \text{on }Γ _{1}×(0,T), \\ u(0)=u_{0},u_{t}(0)=u_{1},\end{gathered} \right.$ \end{document} where \begin{document}$\left\| {{u_0}} \right\| \begin{document}$ E(0) where \begin{document}$λ_{β}, $\end{document} \begin{document}$d_{β} $\end{document} are defined in (21), (22) and \begin{document}$Γ=Γ_{0}\cupΓ_{1} $\end{document} and \begin{document}$\bar{Γ}_{0}\cap\bar{Γ}_{1}=φ. $\end{document}

This paper is concerned with positive blowup solutions for the heat equation with a nonlinear boundary condition. The result in this paper provides a precise description of the blowup profile. A key part of the proof is a pointwise estimate of solutions near the blowup points. This estimate is obtained by the study of a stationary problem of some rescaled equation. Furthermore we provide uniform upper bounds of positive blowup solutions by the Liouville type theorem for a rescaled equation in similarity variables.

This paper studies the following two non-Newtonian equations with nonlinear boundary conditions. Firstly, we show that finite time blow up occurs on the boundary and we get a blow up rate and an estimate for the blow up time of the equation $k_{t}=(\left \vert k_{x}\right \vert ^{r-2}k_{x})_{x}$, $(x,t)\in (0,L)\times (0,T)\ $with $k_{x}(0,t)=k^{\alpha }(0,t)$, $k_{x}(L,t)=k^{\beta }(L,t)$,$\ t\in (0,T)\ $and initial function $k\left(x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $r\geq 2$, $\alpha ,\beta \ $and $L\ $are positive constants. Secondly, we show that finite time blow up occurs on the boundary, and we get blow up rates and estimates for the blow up time of the equation $k_{t}=(\left \vert k_{x}\right \vert ^{r-2}k_{x})_{x}+k^{\alpha }$, $(x,t)\in (0,L)\times (0,T)\ $with $k_{x}(0,t)=0$, $k_{x}(L,t)=k^{\beta }(L,t)$,$\ t\in (0,T)\ $ and initial function $k\left( x,0\right) =k_{0}\left( x\right) $,$\ x\in \lbrack 0,L]\ $where $r\geq 2$, $\alpha ,\beta$ and $L$ are positive constants.

This paper is concerned with the convergence of solutions for a class of functional integro-differential equations with nonlinear boundary conditions. New comparison principles are obtained. By using the comparison principles and quasilinearization method, we present two monotone iterative sequences uniformly and monotonically converging to the unique solution with rate of order 2. Meanwhile, an example is given to demonstrate applications of the result reported.

In this paper we analyze the porous medium equation \begin{document}$ \begin{equation} u_t = \Delta u^m + a\int_\Omega u^p-b u^q -c\lvert\nabla\sqrt{u}\rvert^2 \quad {\rm{in}}\quad \Omega \times I,\;\;\;\;\;\;(◇) \end{equation} $\end{document} where \begin{document}$ \Omega $\end{document} is a bounded and smooth domain of \begin{document}$ \mathbb R^N $\end{document} , with \begin{document}$ N\geq 1 $\end{document} , and \begin{document}$ I = [0,t^*) $\end{document} is the maximal interval of existence for \begin{document}$ u $\end{document} . The constants \begin{document}$ a,b,c $\end{document} are positive, \begin{document}$ m,p,q $\end{document} proper real numbers larger than 1 and the equation is complemented with nonlinear boundary conditions involving the outward normal derivative of \begin{document}$ u $\end{document} . Under some hypotheses on the data, including intrinsic relations between \begin{document}$ m,p $\end{document} and \begin{document}$ q $\end{document} , and assuming that for some positive and sufficiently regular function \begin{document}$ u_0({\bf x}) $\end{document} the Initial Boundary Value Problem (IBVP) associated to (◇) possesses a positive classical solution \begin{document}$ u = u({\bf x},t) $\end{document} on \begin{document}$ \Omega \times I $\end{document} : \begin{document}$ \triangleright $\end{document} when \begin{document}$ p>q $\end{document} and in 2- and 3-dimensional domains, we determine a lower bound of \begin{document}$ t^* $\end{document} for those \begin{document}$ u $\end{document} becoming unbounded in \begin{document}$ L^{m(p-1)}(\Omega) $\end{document} at such \begin{document}$ t^* $\end{document} ; \begin{document}$ \triangleright $\end{document} when \begin{document}$ p and in \begin{document}$ N $\end{document} -dimensional settings, we establish a global existence criterion for \begin{document}$ u $\end{document} .

The first part of this paper deals with existence of solutions to the quasilinear elliptic problem (P) − div a ( x , ∇ u ) = f ( x , u , ∇ u ) in Ω , a ( x , ∇ u ) ⋅ ν = g ( x , u ) − ζ | u | p − 2 u on ∂ Ω , involving a general nonhomogeneous differential operator, namely div a , and Caratheodory functions f : Ω × R × R N → R and g : ∂ Ω × R → R . Under appropriate conditions on the perturbations, we show that (P) possesses a bounded solution. In the second part, we consider the special case when div a is the ( p , q ) -Laplacian with a parameter μ > 0 , and study the asymptotic behavior of solutions as μ goes to zero or to infinity. A uniqueness result is also provided.

We study positive solutions to singular boundary value problems of the form: \begin{equation*} \begin{cases} -u'' = h(t) \dfrac{f(u)}{u^\alpha} &\text{for } t \in (0,1), \\ u(0) = 0, \\ u'(1) + c(u(1)) u(1) = 0, \end{cases} \end{equation*} where $0< \alpha< 1$, $h\colon(0,1]\rightarrow(0,\infty)$ is continuous such that $h(t)\leq {d}/{t^\beta}$ for some $d> 0$ and $\beta\in[0,1-\alpha)$ and $c\colon [0,\infty)\rightarrow [0,\infty)$ is continuous such that $c(s)s$ is nondecreasing. We assume that $f\colon[0,\infty)\rightarrow(0,\infty)$ is continuously differentiable such that $[(f(s)-f(0))/s^\alpha]+\tau s$ is strictly increasing for some $\tau\geq 0$ for $s\in(0,\infty)$. When there exists a pair of sub-supersolutions $(\psi,\phi)$ such that $0\leq \psi\leq\phi$, we first establish a minimal solution $\underline u$ and a maximal solution $\overline u$ in $[\psi,\phi]$. When there exist two pairs of sub-supersolutions $(\psi_1,\phi_1)$ and $(\psi_2,\phi_2)$ where $0\leq \psi_1 \leq \psi_2 \leq \phi_1$, $\psi_1 \leq \phi_2 \leq \phi_1$ with $\psi_2\not \leq \phi_2$, and $\psi_2$, $\phi_2$ are not solutions, we next establish the existence of at least three solutions $u_1$, $u_2$ and $u_3$ satisfying $u_1\in [\psi_1,\phi_2], u_2\in [\psi_2,\phi_1]$ and $u_3\in [\psi_1,\phi_1]\setminus ([\psi_1,\phi_2]\cup [\psi_2,\phi_1])$.

In this paper we study a free boundary problem modeling the growth of a tumor under the action of an inhibitor. Unlike other tumor models in the existing literature, the model under this study contains nonlinear boundary conditions which induces many new difficulties in rigorous mathematical analysis. Under the assumption that the tumor is spherically symmetric, we first establish global well-posedness of this model by proving that it admits a unique global classical solution. Next we study large-time behavior of the solution. We prove that the tumor cannot expand unboundedly, and give sufficient conditions to guarantee the tumor finally vanishing or persisting. We also give sufficient conditions to ensure that the model has at least one stationary solution ( σ s ( r ) , β s ( r ) , R s ) , and study asymptotic stability of the stationary solution.

We consider one-dimensional ( p , q ) -Laplace problems: { − ( φ ( u ′ ) ) ′ = λ h ( t ) f ( u ) , t ∈ ( 0 , 1 ) , u ( 0 ) = 0 = a u ′ ( 1 ) + g ( λ , u ( 1 ) ) u ( 1 ) , where λ > 0 , a ≥ 0 , φ ( s ) : = | s | p − 2 s + | s | q − 2 s , 1 p q ∞ , h ∈ C ( ( 0 , 1 ) , ( 0 , ∞ ) ) , f ∈ C ( ( 0 , ∞ ) , R ) with lim s → 0 + ⁡ f ( s ) ∈ ( − ∞ , 0 ) ∪ { − ∞ } , and g ∈ C ( ( 0 , ∞ ) × [ 0 , ∞ ) , ( 0 , ∞ ) ) such that g ( r , s ) s is nondecreasing with respect to s ∈ [ 0 , ∞ ) . Classifying the behaviors of f near infinity, we establish the existence, multiplicity and nonexistence of positive solutions. In particular, we provide a sufficient condition on f to obtain a multiplicity result for the case when lim s → ∞ ⁡ f ( s ) s r − 1 ∈ ( 0 , ∞ ) , 1 r q , which is new even in semilinear problems ( p = q = 2 ). The proofs are based on a Krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity.