Nonlinear Robin Problems Research Articles

Global Multiplicity of Positive Solutions for Nonlinear Robin Problems with an Indefinite Potential Term

We consider a Robin problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction is parametric and exhibits the competing effects of a concave (sublinear) and of a convex (superlinear) terms (“concave-convex” problem). The parameter multiplies the convex term. We prove an existence and multiplicity theorem which is global in parameter.

Nonlinear Robin problems with double phase variable exponent operator

Nonlinear Robin problems with double phase variable exponent operator

Positive solutions for a class of nonlinear parametric Robin problems

We consider a nonlinear Robin problem driven by the p-Laplacian and a parametric concave-convex reaction with the parameter multiplying the convex (superlinear) term. We prove a multiplicity result for positive solutions which is global in the parameter λ>0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda >0$$\\end{document} (bifurcation-type theorem). We also show the existence of a minimal positive solution uλ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$u_{\\lambda }^*$$\\end{document} and determine the monotonicity and continuity properties of the map λ⟼uλ∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda \\longmapsto u_{\\lambda }^*$$\\end{document}

Multiplicity Results for Nonlinear Nonhomogeneous Robin Problems with Indefinite Potential Term

Multiplicity Results for Nonlinear Nonhomogeneous Robin Problems with Indefinite Potential Term

Nonexistence of positive solutions for nonlinear parabolic Robin problems and Hardy–Leray inequalities

Nonexistence of positive solutions for nonlinear parabolic Robin problems and Hardy–Leray inequalities

Positive solutions for nonlinear Robin problems with convection

We consider a Robin problem driven by the p-Laplacian and with a reaction that is gradient dependent (convection). Using truncations and perturbations, we show that the problem has at least one positive smooth solution.

Nonlinear nonhomogeneous Robin problems with gradient dependent reaction

We consider a Robin problem driven by a nonlinear, nonhomogeneous differential operator and a reaction term which depends on the gradient. Using a topological approach based on the Leray–Schauder Alternative Principle, we show the existence of a weak solution.

Concave-Convex Problems for the Robin p-Laplacian Plus an Indefinite Potential

We consider nonlinear Robin problems driven by the p-Laplacian plus an indefinite potential. In the reaction, we have the competing effects of a parametric concave (that is, ( p − 1 ) -sublinear) term and of a convex (that is, ( p − 1 ) -superlinear) term which need not satisfy the Ambrosetti–Rabinowitz condition. We prove a "bifurcation-type" theorem describing in a precise way the dependence the dependence of the set of positive solutions on the parameter λ > 0 . In addition, we show the existence of a smallest positive solution u λ * and determine the monotonicity and continuity properties of the map λ ↦ u λ * .

Existence and multiplicity of positive solutions for parametric nonlinear nonhomogeneous singular Robin problems

We consider nonlinear Robin problems driven by a nonhomogeneous differential operator and with a reaction that has a singular term and a parametric $$(p-1)$$-superlinear perturbation which need not satisfy the Ambrosetti–Rabinowitz condition. We are looking for positive solutions. Using variational arguments and a suitable truncation and comparison techniques, we prove a bifurcation-type theorem which describes the set of positive solutions as the parameter $$\lambda > 0$$ varies. Also we show the for every admissible value of the parameter $$\lambda >0$$, the problem has a smallest solution $${\bar{u}}_{\lambda }$$ and we determine the monotonicity and continuity properties of the map $$\lambda \rightarrow {\bar{u}}_{\lambda }$$.

Nonlinear Robin problems with indefinite potential

We consider a nonlinear Robin problem driven by the p-Laplacian plus an indefinite potential. The conditions on the source term are minimal. We prove two multiplicity theorems with sign information for all the solutions. In the semilinear case (p=2), we show that we can have multiple nodal solutions. We apply our results to a special class of logistic equations with equidiffusive reaction.

Nonlinear Nonhomogeneous Robin Problems with Almost Critical and Partially Concave Reaction

We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator, with reaction which exhibits the competition of two Carath\'eodory terms. One is parametric, $(p-1)$-sublinear with a partially concave nonlinearity near zero. The other is $(p-1)$-superlinear and has almost critical growth. Exploiting the special geometry of the problem, we prove a bifurcation-type result, describing the changes in the set of positive solutions as the parameter $\lambda>0$ varies.

Nonlinear nonhomogeneous Robin problems with convection

Nonlinear nonhomogeneous Robin problems with convection

Existence of positive solutions for nonlinear Robin problems with gradient dependence

Existence of positive solutions for nonlinear Robin problems with gradient dependence

Positive solutions for nonlinear Robin problems with indefinite potential and competing nonlinearities

We consider a nonlinear Robin problem associated to the p-Laplacian plus an indefinite potential. In the reaction we have the competing effects of two nonlinear terms. One is parametric and strictly $$(p-1)$$-sublinear. The other is $$(p-1)$$-linear. We prove a bifurcation-type theorem describing the dependence of the set of positive solutions on the parameter $$\lambda >0$$. We also show that for every admissible parameter the problem has a smallest positive solution $${\bar{u}}_{\lambda }$$ and we study monotonicity and continuity properties of the map $$\lambda \rightarrow {\bar{u}}_{\lambda }$$.

Nonlinear Robin problems with general potential and crossing reaction

We consider a nonlinear Robin problem driven by the p -Laplacian plus and indefinite potential. The reaction is parametric and exhibits an asymmetric behavior as x\to \pm\infty . Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we show that for all small values of the parameter \lambda > 0 , the problem admits at least four nontrivial smooth solutions, three with fixed sign and the fourth nodal. By introducing a local symmetry condition on the reaction, we show that we can have a whole sequence of distinct nodal solutions converging to zero in C^1(\overline{\Omega}) .

Solutions with sign information for nonlinear Robin problems with no growth restriction on reaction

We consider a parametric nonlinear Robin problem driven by a nonhomogeneous differential operator. The reaction is a Carathéodory function which is only locally defined (that is, the hypotheses concern only its behaviour near zero). The conditions on the reaction are minimal. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we show that for all small values of the parameter , the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal.

Positive solutions for nonlinear Robin problems with convection

We consider a nonlinear Robin problem driven by the p‐Laplacian and with a convection term f(z,x,y). Without imposing any global growth condition on f(z,·,·) and using topological methods (the Leray‐Schauder alternative principle), we show the existence of a positive smooth solution.

Nodal solutions for nonlinear nonhomogeneous Robin problems

We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator plus an indefinite potential. The reaction term is a Carathéodory function satisfying certain conditions only near zero. Using suitable truncation, comparison and cut-off techniques, we show that the problem has a sequence of nodal solutions converging to zero in the C^1(\overline{\Omega}) -norm.

Nodal Solutions for Nonlinear Non-Homogeneous Robin Problems with an Indefinite Potential

AbstractWe consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

Resonant Robin problems driven by the p-Laplacian plus an indefinite potential

We consider a nonlinear Robin problems driven by the $p$-Laplacian plus an indefinite potential. The reaction is resonant with respect to a variational eigenvalue. For the principal eigenvalue we assume strong resonance. Using variational tools and critical groups we prove existence and multiplicity theorems.