scholarly journals Nontrivial solutions for partial discrete Dirichlet problems via a local minimum theorem for functionals

2019 ◽  
Vol 2019 (1) ◽  
Keyword(s):  
Local Minimum ◽  
Nontrivial Solutions ◽  
Dirichlet Problems ◽  
Minimum Theorem

scholarly journals NONTRIVIAL SOLUTIONS FOR STURM–LIOUVILLE SYSTEMS VIA A LOCAL MINIMUM THEOREM FOR FUNCTIONALS

2013 ◽  
Vol 89 (1) ◽  
pp. 8-18 ◽  
Author(s):  
GABRIELE BONANNO ◽  
SHAPOUR HEIDARKHANI ◽  
DONAL O’REGAN
Keyword(s):  
Nontrivial Solution ◽  
Local Minimum ◽  
Second Order ◽  
Nontrivial Solutions ◽  
Sturm Liouville ◽  
Minimum Theorem

AbstractIn this paper, employing a very recent local minimum theorem for differentiable functionals, the existence of at least one nontrivial solution for a class of systems of $n$ second-order Sturm–Liouville equations is established.

Fractional problems with critical nonlinearities by a sublinear perturbation

2020 ◽  
Vol 23 (2) ◽  
pp. 484-503 ◽  
Author(s):  
Lin Li ◽  
Stepan Tersian
Keyword(s):  
Critical Exponent ◽  
Variational Approach ◽  
Local Minimum ◽  
Numerical Estimate ◽  
Nontrivial Solutions ◽  
The Real ◽  
Critical Nonlinearities ◽  
Minimum Theorem ◽  
Sublinear Perturbation

AbstractIn this paper, the existence of two nontrivial solutions for a fractional problem with critical exponent, depending on real parameters, is established. The variational approach is used based on a local minimum theorem due to G. Bonanno. In addition, a numerical estimate on the real parameters is provided, for which the two solutions are obtained.

scholarly journals Multiplicity results for Kirchhoff type elliptic problems with Hardy potential

2019 ◽  
Vol 38 (4) ◽  
pp. 31-50
Author(s):  
M. Bagheri ◽  
Ghasem A. Afrouzi
Keyword(s):  
Local Minimum ◽  
Nonlinear Term ◽  
Existence Of Solutions ◽  
Mountain Pass Theorem ◽  
Elliptic Problems ◽  
Mountain Pass ◽  
Hardy Potential ◽  
Multiplicity Results ◽  
Kirchhoff Type ◽  
Minimum Theorem

In this paper, we are concerned with the existence of solutions for fourth-order Kirchhoff type elliptic problems with Hardy potential. In fact, employing a consequence of the local minimum theorem due to Bonanno and mountain pass theorem we look into the existence results for the problem under algebraic conditions with the classical Ambrosetti-Rabinowitz (AR) condition on the nonlinear term. Furthermore, by combining two algebraic conditions on the nonlinear term using two consequences of the local minimum theorem due to Bonanno we ensure the existence of two solutions, applying the mountain pass theorem given by Pucci and Serrin we establish the existence of third solution for our problem.

scholarly journals Fourth-order elliptic problems with critical nonlinearities by a sublinear perturbation

2021 ◽  
Vol 26 (2) ◽  
pp. 227-240
Author(s):  
Lin Li ◽  
Donal O’Regan
Keyword(s):  
Boundary Condition ◽  
Positive Solutions ◽  
Fourth Order ◽  
Local Minimum ◽  
Elliptic Problems ◽  
Order Problem ◽  
Critical Nonlinearities ◽  
Fourth Order Problem ◽  
Minimum Theorem ◽  
Sublinear Perturbation

In this paper, we get the existence of two positive solutions for a fourth-order problem with Navier boundary condition. Our nonlinearity has a critical growth, and the method is a local minimum theorem obtained by Bonanno.

Multiple solutions for parametric double phase Dirichlet problems

2020 ◽  
pp. 2050006 ◽  
Author(s):  
N. S. Papageorgiou ◽  
C. Vetro ◽  
F. Vetro
Keyword(s):  
Dirichlet Problem ◽  
Multiple Solutions ◽  
Critical Parameter ◽  
Constant Sign ◽  
Nontrivial Solutions ◽  
Dirichlet Problems ◽  
Double Phase ◽  
Variational Tools

We consider a parametric double phase Dirichlet problem. Using variational tools together with suitable truncation and comparison techniques, we show that for all parametric values [Formula: see text] the problem has at least three nontrivial solutions, two of which have constant sign. Also, we identify the critical parameter [Formula: see text] precisely in terms of the spectrum of the [Formula: see text]-Laplacian.

scholarly journals Nonexistence of Solutions for Dirichlet Problems with Supercritical Growth in Tubular Domains

2021 ◽  
Vol 21 (1) ◽  
pp. 189-198
Author(s):  
Riccardo Molle ◽  
Donato Passaseo
Keyword(s):  
Bounded Domain ◽  
Sobolev Embedding ◽  
Nontrivial Solutions ◽  
Dirichlet Problems ◽  
Tubular Domain ◽  
Nonexistence Of Solutions ◽  
Compact Submanifold

Abstract We deal with Dirichlet problems of the form { Δ ⁢ u + f ⁢ ( u ) = 0 in  ⁢ Ω , u = 0 on  ⁢ ∂ ⁡ Ω , \left\{\begin{aligned} \displaystyle{}\Delta u+f(u)&\displaystyle=0&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right. where Ω is a bounded domain of ℝ n {\mathbb{R}^{n}} , n ≥ 3 {n\geq 3} , and f has supercritical growth from the viewpoint of Sobolev embedding. In particular, we consider the case where Ω is a tubular domain T ε ⁢ ( Γ k ) {T_{\varepsilon}(\Gamma_{k})} with thickness ε > 0 {{\varepsilon}>0} and center Γ k {\Gamma_{k}} , a k-dimensional, smooth, compact submanifold of ℝ n {\mathbb{R}^{n}} . Our main result concerns the case where k = 1 {k=1} and Γ k {\Gamma_{k}} is contractible in itself. In this case we prove that the problem does not have nontrivial solutions for ε > 0 {{\varepsilon}>0} small enough. When k ≥ 2 {k\geq 2} or Γ k {\Gamma_{k}} is noncontractible in itself we obtain weaker nonexistence results. Some examples show that all these results are sharp for what concerns the assumptions on k and f.

scholarly journals Triple solutions for a Dirichlet boundary value problem involving a perturbed discrete p(k)-Laplacian operator

2017 ◽  
Vol 15 (1) ◽  
pp. 1075-1089 ◽  
Author(s):  
Mohsen Khaleghi Moghadam ◽  
Johnny Henderson
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Local Minimum ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Laplacian Operator ◽  
Discrete Problem ◽  
Dirichlet Boundary Value Problem ◽  
One Dimensional ◽  
Minimum Theorem

Abstract Triple solutions are obtained for a discrete problem involving a nonlinearly perturbed one-dimensional p(k)-Laplacian operator and satisfying Dirichlet boundary conditions. The methods for existence rely on a Ricceri-local minimum theorem for differentiable functionals. Several examples are included to illustrate the main results.

Existence of non-trivial solutions for systems of n fourth order partial differential equations

2014 ◽  
Vol 64 (5) ◽  
Author(s):  
Shapour Heidarkhani
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Nontrivial Solution ◽  
Fourth Order ◽  
Local Minimum ◽  
Partial Differential ◽  
Navier Boundary Conditions ◽  
Minimum Theorem

AbstractIn this paper, employing a very recent local minimum theorem for differentiable functionals due to Bonanno, the existence of at least one nontrivial solution for a class of systems of n fourth order partial differential equations coupled with Navier boundary conditions is established.

scholarly journals Nonlinear Dirichlet problems with unilateral growth on the reaction

2019 ◽  
Vol 31 (2) ◽  
pp. 319-340
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Dirichlet Problem ◽  
Critical Groups ◽  
Constant Sign ◽  
Nodal Solution ◽  
Nontrivial Solutions ◽  
Dirichlet Problems ◽  
Multiplicity Results ◽  
Subcritical Growth ◽  
Nonlinear Dirichlet Problem ◽  
The Third

AbstractWe consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, {p=2}), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multiplicity results existing for a class of parametric, coercive Dirichlet problems.

scholarly journals Existence and uniqueness of positive solution for nonlinear difference equations involving p(k)-Laplacian operator

2019 ◽  
Vol 27 (1) ◽  
pp. 141-167
Author(s):  
Mohsen Khaleghi Moghadam ◽  
Renata Wieteska
Keyword(s):  
Positive Solution ◽  
Existence And Uniqueness ◽  
Local Minimum ◽  
Linear Problem ◽  
Dirichlet Boundary ◽  
Laplacian Operator ◽  
Nonlinear Difference Equations ◽  
Technical Approach ◽  
Lipschitzian Continuous ◽  
Minimum Theorem

Abstract In this paper, we deal with the existence of at least one and of at least two positive solutions as well the uniqueness of a positive solution for an anisotropic discrete non-linear problem involving p(k)-Laplacian with Dirichlet boundary value conditions. The technical approach for the existence part is based on a local minimum theorem and on a two critical points theorem for differentiable functionals, and for uniqueness part is based on a Lipschitzian continuous condition on the nonlinearity term.

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