Ambrosetti Prodi Type Research Articles

An Ambrosetti–Prodi type result for fractional spectral problems

AbstractWe consider the following class of fractional parametric problems where is a smooth bounded domain, , , is the fractional Dirichlet Laplacian, is a locally Lipschitz nonlinearity having linear or superlinear growth and satisfying Ambrosetti–Prodi type assumptions, , φ1 is the first eigenfunction of the Laplacian with homogenous boundary conditions, and is a bounded function. Using variational methods, we prove that there exists a such that the above problem admits at least two distinct solutions for any . We also discuss the existence of solutions for a fractional periodic Ambrosetti–Prodi type problem.

Periodic solutions to parameter-dependent equations with a $$\phi $$-Laplacian type operator

We study the periodic boundary value problem associated with the $$\phi $$ -Laplacian equation of the form $$(\phi (u'))'+f(u)u'+g(t,u)=s$$ , where s is a real parameter, f and g are continuous functions, and g is T-periodic in the variable t. The interest is in Ambrosetti–Prodi type alternatives which provide the existence of zero, one or two solutions depending on the choice of the parameter s. We investigate this problem for a broad family of nonlinearities, under non-uniform type conditions on g(t, u) as $$u\rightarrow \pm \infty $$ . We generalize, in a unified framework, various classical and recent results on parameter-dependent nonlinear equations.

A quasi-linear nonlocal Venttsel' problem of Ambrosetti–Prodi type on fractal domains

We investigate the solvability of the Ambrosetti-Prodi problem for the p-Laplace operator ∆p with Venttsel' boundary conditions on a twodimensional open bounded set with Koch-type boundary, and on an open bounded three-dimensional cylinder with Koch-type fractal boundary. Using a priori estimates, regularity theory and a sub-supersolution method, we obtain a necessary condition for the non-existence of solutions (in the weak sense), and the existence of at least one globally bounded weak solution. Moreover, under additional conditions, we apply the Leray-Schauder degree theory to obtain results about multiplicity of weak solutions.

A multiplicity result for periodic solutions of Liénard equations with an attractive singularity

A periodic problem of Ambrosetti–Prodi type is studied in this paper for the Liénard equation with a singularity of attractive typex″+f(x)x′+φ(t)xm+r(t)xμ=s,where f:(0,+∞)→R is continuous, r:R→(0,+∞) and φ: R → R are continuous with T−periodicity in the t variable, 0 < m ≤ 1, μ > 0, s ∈ R are constants. By using the method of upper and lower functions as well as some properties of topological degree, we obtain a new multiplicity result on the existence of periodic solutions for the equation.

Asymmetric critical p-Laplacian problems

We obtain nontrivial solutions for two types of asymmetric critical p-Laplacian problems with Ambrosetti–Prodi type nonlinearities in a smooth bounded domain in $${\mathbb {R}}^N,\, N \ge 2$$ . For $$1< p < N$$ , we consider an asymmetric problem involving the critical Sobolev exponent $$p^*= Np/(N - p)$$ . In the borderline case $$p = N$$ , we consider an asymmetric critical exponential nonlinearity of the Trudinger–Moser type. In the absence of a suitable direct sum decomposition, we use a linking theorem based on the $${\mathbb {Z}}_2$$ -cohomological index to prove existence of solutions.

Results of Ambrosetti–Prodi type for non-selfadjoint elliptic operators

The well-known Ambrosetti–Prodi theorem considers perturbations of the Dirichlet Laplacian by a nonlinear function whose derivative jumps over the principal eigenvalue of the operator. Various extensions of this landmark result were obtained for self-adjoint operators, in particular by Berger and Podolak, who gave a geometrical description of the solution set. In this text we show that similar theorems are valid for non-self-adjoint operators. In particular, we prove that the semilinear operator is a global fold. As a consequence, we obtain what appears to be the first exact multiplicity result for elliptic equations in non-divergence form. We employ techniques based on the maximum principle.

Ambrosetti-Prodi type result to a Neumann problem via a topological approach

We prove an Ambrosetti-Prodi type result for a Neumann problem associated to the equation \begin{document}$u''+f(x, u(x))=μ$\end{document} when the nonlinearity has the following form: \begin{document}$f(x, u):=a(x)g(u)-p(x)$\end{document} . The assumptions considered generalize the classical one, \begin{document}$f(x, u)\to+∞$\end{document} as \begin{document}$|u|\to+∞$\end{document} , without requiring any uniformity condition in \begin{document}$x$\end{document} . The multiplicity result which characterizes these kind of problems will be proved by means of the shooting method.

A quasi-linear Neumann problem of Ambrosetti–Prodi type on extension domains

Let Ω⊆RN be a bounded domain with the extension property, whose boundary Γ≔∂Ω is an upper d-set (with respect to a measure μ), for N≥2 and d∈(N−p,N). We investigate the solvability of the Ambrosetti–Prodi problem for the p-Laplace operator Δp, with Neumann boundary conditions. Using a priori estimates, regularity theory, a sub-supersolution method, and the Leray–Schauder degree theory, we obtain a necessary condition for the non-existence of solutions (in the weak sense), the existence of at least one minimal solution, and the existence of at least two distinct solutions. Moreover, we establish global Hölder continuity for weak solutions of the Neumann problem of Ambrosetti–Prodi type on a large class of “bad” domain, extending the solvability of this type of elliptic problem for the first time, to a wide class of non-smooth domains.

Concentration on submanifolds for an Ambrosetti–Prodi type problem

Given a smooth bounded domain \(\Omega \) of \(\mathbb {R}^n\) and consider the problem $$\begin{aligned} \left\{ \begin{array} {ll} - \Delta u = |u|^p - t\,\psi &{}\quad \hbox { in } \ \Omega \\ u = 0 &{}\quad \hbox { on }\ \partial \Omega \end{array}\right. \end{aligned}$$ (0.1) where t is a large positive parameter, \(p>1\) and \(\psi \) is an eigenfunction of \(- \Delta \) with Dirichlet boundary condition corresponding to the first eigenvalue \(\lambda _1\). Assuming that \(\Omega \) contains a k-dimensional compact submanifold K which is stationary and non-degenerate for the weighted functional $$\begin{aligned} \int _K \psi ^{(1 - \frac{1}{p})(\frac{p+1}{p-1}-\frac{n-k}{2})}dvol \end{aligned}$$ such that \(\mathrm{dist}(K,\partial \Omega )>\delta _0>0\), then for \(1< p < \frac{n+2-k}{n-2-k}\) we prove the existence of a sequence \(t=t_j\rightarrow \infty \) and solutions \(u_t\) that concentrate along K. This result proves in particular the validity of a conjecture by Hollman–Mckenna in full generality, see Hollman and McKenna (Commun Pure Appl Anal 10(2):785–802, 2011), extending the result by Manna and Santra (Discrete Contin Dyn Syst 36(10):5595–5626, 2016) where the case \(n=2\) and \(k=1\) has been considered.

On a singular periodic Ambrosetti–Prodi problem

We investigate the possibility of extending a classical multiplicity result by Fabry, Mawhin and Nkashama to a periodic problem of Ambrosetti–Prodi type having a nonlinearity with possibly one or two singularities. In the second part of the paper we study the existence of periodic rotating solutions for radially symmetric systems with nonlinearities of the same type.

MULTIPLICITY OF SOLUTIONS FOR SEMILINEAR ELLIPTIC PROBLEMS WITH COMBINED NONLINEARITIES

In this paper, we are concerned with the semilinear elliptic problem [Formula: see text] where Ω is a bounded smooth domain in Rm(m ≥ 2), h ∈ L∞(Ω), h(x) ≢ 0, 1 &lt; q &lt; 2, [Formula: see text]. When the nonlinearity f satisfies the Ambrosetti–Prodi type condition at infinity, two multiplicity results are established by using a combination of variational methods, upper and lower solutions, Leray–Schauder degree theory and suitable truncation techniques.

On a stronger Lazer-McKenna conjecture for Ambrosetti-Prodi type problems

We consider an elliptic problem of Ambrosetti-Prodi type involving critical Sobolev exponent on a bounded smooth domain. We show that if the domain has some symmetry, the problem has infinitely many (distinct) solutions whose energy approach to infinity even for a fixed parameter, thereby obtaining a stronger result than the Lazer-McKenna conjecture. Mathematics Subject Classification (2010): 35J65 (primary); 35B38, 47H15 (secondary).

Multiplicity of solutions for a superlinear [formula omitted]-Laplacian equation

We consider quasi-linear elliptic equations involving the p -Laplacian with nonlinearities which interfere asymptotically with the spectrum of the differential operator. We show that such equations have for certain forcing terms at least two solutions. Such equations are of the so-called Ambrosetti–Prodi type. In particular, our theorem is a partial generalization of corresponding results for the semi-linear case by Ruf and Srikanth (1986) [2] and de Figueiredo (1988) [9].

Exact multiplicity of solutions to a diffusive logistic equation with harvesting

An Ambrosetti–Prodi type exact multiplicity result is proved for a diffusive logistic equation with harvesting. We show that a modified diffusive logistic mapping has exactly either zero, or one, or two pre-images depending on the harvesting rate. It implies that the original diffusive logistic equation with harvesting has at most two positive steady state solutions.

A one side superlinear Ambrosetti–Prodi problem for the Dirichlet p-laplacian

We study the solvability of the quasilinear elliptic problem of parameter s − Δ p u = g ( x , u ) + s φ ( x ) in Ω , u = 0 on ∂ Ω where Ω is a smooth bounded domain in R N , φ ⩾ 0 , g ( ⋅ , u ) / | u | p − 2 u lies for u < 0 below the first eigenvalue of the p-laplacian and g growths for u > 0 less than the lower Sobolev critical exponent p ∗ . We combine topological methods via upper and lower solutions and blow-up techniques to get a priori bounds to prove the following result of Ambrosetti–Prodi type: there exists s ∗ ⩽ s ∗ such that the problem possesses no solutions if s > s ∗ , it has at least one solution if s < s ∗ , and at least two solutions if s < s ∗ . We prove also that s ∗ = s ∗ in some cases.

Existence and multiplicity of solutions for a weakly coupled radial system in a ball

We prove a conjecture by Dalbono-McKenna on the number of solutions for a weakly coupled elliptic system. The system is of the Ambrosetti-Prodi type with a asymmetric nonlinearity. We consider the radial case in a ball. By applying a degree theoretic argument, we simplify the proof of the paper [3] and obtain the existence and multiplicity of solutions for this weakly coupled system.

Multiple solutions for a class of Ambrosetti-Prodi type problems for systems involving critical Sobolev exponents

In this work we study the existence of multiple solutions for the non-homogeneous system <p align="center"> $ - \Delta U = AU + (u^p_+, v^p_+)+ F$ in $\Omega$ <p align="left" class="times"> <p align="center"> $ U = 0 $ on $ \partial\Omega,$ <p align="left" class="times"> where $\Omega\subset \mathbb R^{N}$ is a bounded smooth domain; $U=(u,v), p=2^\star -1$, with $2^\star=\frac{2N}{N-2}, N \geq 3$; ${w_+}=$ max{ $w,0$} and $F \in L^s(\Omega)\times L^s(\Omega)$ for some $s>N$.<br> Using variational methods, we prove the existence of at least two solutions. The first is obtained explicitly by a direct calculation and the second via the Mountain Pass Theorem for the case $0< \mu_1 \leq \mu_2< \lambda_1$ or Linking Theorem if $\lambda_k < \mu_1 \leq \mu_2 < \lambda_{k+1}$, where $\mu_1, \mu_2$ are eigenvalues of symmetric matrix $A$ and $\lambda_j$ are eigenvalues of $(-\Delta, H_0^1(\Omega))$.

On some third order nonlinear boundary value problems: Existence, location and multiplicity results

We prove an Ambrosetti–Prodi type result for the third order fully nonlinear equation u ‴ ( t ) + f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) = s p ( t ) with f : [ 0 , 1 ] × R 3 → R and p : [ 0 , 1 ] → R + continuous functions, s ∈ R , under several two-point separated boundary conditions. From a Nagumo-type growth condition, an a priori estimate on u ″ is obtained. An existence and location result will be proved, by degree theory, for s ∈ R such that there exist lower and upper solutions. The location part can be used to prove the existence of positive solutions if a non-negative lower solution is considered. The existence, nonexistence and multiplicity of solutions will be discussed as s varies.

Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian

Using Leray–Schauder degree theory we obtain various existence and multiplicity results for nonlinear boundary value problems ( ϕ ( u ′ ) ) ′ = f ( t , u , u ′ ) , l ( u , u ′ ) = 0 where l ( u , u ′ ) = 0 denotes the Dirichlet, periodic or Neumann boundary conditions on [ 0 , T ] , ϕ : ] − a , a [ → R is an increasing homeomorphism, ϕ ( 0 ) = 0 . The Dirichlet problem is always solvable. For Neumann or periodic boundary conditions, we obtain in particular existence conditions for nonlinearities which satisfy some sign conditions, upper and lower solutions theorems, Ambrosetti–Prodi type results. We prove Lazer–Solimini type results for singular nonlinearities and periodic boundary conditions.

Critical Ambrosetti–Prodi type problems for systems of elliptic equations

In this paper we study multiple solutions for the non-homogeneous system (1) { − Δ u = a u + b v + 2 α α + β u + α − 1 v + β + f , Ω − Δ v = b u + d v + 2 β α + β u + α v + β − 1 + g , Ω u = v = 0 , ∂ Ω , where Ω ⊂ R N is a bounded smooth domain; α , β > 1 are real constants, α + β = 2 ∗ , where 2 ∗ = 2 N N − 2 , N ≥ 3 ; s + = max { s , 0 } and f , g ∈ L s ( Ω ) for some s > N .