On elliptic problems in $\mathbf R^N$ with supercritical exponent of nonlinearity

S. I. Pokhozhaev

Sbornik. Mathematics, Tome 72 (1992) no. 2, pp. 447-466

Résumé

Elliptic problems of the form $$ \begin{cases} \Delta u+f(x,u)=h(x),\quad x\in\mathbf R^N\ \ (N\geqslant 3), \\ \displaystyle\lim_{|x|\to\infty}u(x)=0 \end{cases} $$ are considered under appropriate conditions. This class of problems includes the inhomogeneous Emden–Fowler problem $$ \begin{cases} \Delta u+|u|^{p-2}u=h(x),\quad x\in\mathbf R^N\ \ (N\geqslant 3), \\ \displaystyle\lim_{|x|\to\infty}u(x)=0 \end{cases} $$ with $p>p_c=\dfrac{2N}{N-2}$. The first part of this article is concerned with radial solutions, where $$ f(x,u)=f(|x|,u) \quad\text{and}\quad h(x)=h(|x|). $$ The second part considers solvability in classes of functions with prescribed bound on decay at infinity, but without assumptions on radial symmetry.

Bibliographie

[1] Ako K., Kusano T., “On bounded solutions of second order elliptic differential equations”, J. Foc. Sci. Univ. Tokyo, XI:1 (1964), 29–37 | MR

[2] Gidas B., Ni W.-M., Nirenberg L., “Symmetry and related properties via the maximum principle”, Comm. Math. Ph., 68:3 (1979), 209–243 | DOI | MR | Zbl

[3] Berestycki H., Lions P. L., Peletier L. A., “On $ODE$-approach to the existence of positive solutions for semilinear problems in $\mathbf{R}^n$”, Indiana Univ. Math. J., 30:1 (1981), 141–157 | DOI | MR | Zbl

[4] Gidas B., Spruck J., “Global and local behavior of positive solutions of nonlinear elliptic equations”, Comm. Pure Appl. Math., 34 (1981), 525–598 | DOI | MR | Zbl

[5] Ni W.-M., “On the elliptic equation $\Delta u+K(x)u^{\frac{n+2}{n-2}}=0$, its generalizations, and applications in geometry”, Indiana Univ. Math. J., 31:4 (1982), 493–529 | DOI | MR | Zbl

[6] Berestycki H., Lions P. L., “Nonlinear scalar field equations. I: Existence of a ground state”, Arch. Rat. Mech. Anal., 82:4 (1983), 313–345 | MR | Zbl

[7] Berestycki H., Lions P. L., “Nonlinear scalar field equations. II: Existence of infinitely many solutions”, Arch. Rat. Mech. Anal., 82:4 (1983), 347–375 | MR | Zbl

[8] Kawano N., “On bounded entire solutions of semilinear elliptic equations”, Hiroshima Math. J., 14:1 (1984), 125–158 | MR | Zbl

[9] Kusano T., Ohary Sh., “Bounded entire solutions of second order semilinear elliptic equations with application to a parabolic initial value problem”, Indiana Univ. Math. J., 34:1 (1985), 85–95 | DOI | MR | Zbl

[10] Weiyne D., “On a conformally invariant elliptic equation in $\mathbf{R}^n$”, Comm. Math. Ph., 107:2 (1986), 331–335 | DOI

[11] Furusho Y., Kusano I., “On the existence of positive decaying entire solutions for a class of sublinear elliptic equations”, Canadian J. Math., XL:5 (1988), 1156–1173 | MR

[12] Allegretto W., Huang Y. X., “Existence of positive global solutions of mixed sublinear-superlinear problems”, Canadian J. Math., XL:5 (1988), 1222–1242 | MR

[13] Furusho Y., “On decaying entire positive solutions of semilinear elliptic equations”, Japan J. Math., 14:1 (1988), 97–118 | MR | Zbl

[14] Budd C. J., Qi Y.-W., “The existence of bounded solutions of a semilinear elliptic equation”, J. Diff. Equations, 82:2 (1989), 207–218 | DOI | MR | Zbl

[15] Edelson A. L., “Entire solutions of singular elliptic equations”, J. Math. Anal. Appl., 139:2 (1989), 523–532 | DOI | MR | Zbl

[16] Noussair E. S., Swanson Ch. A., “An $L^q(\mathbf{R}^N)$-theory of subcritical semilinear elliptic problems”, J. Dif. Equations, 84:1 (1990), 52–61 | DOI | MR | Zbl

[17] Benci V., Cerami G., “Existence of positive solutions of the equation $-\Delta u+a(x)u=u^{(N+2)/(N-2)}$ in $\mathbf{R}^N$”, J. Funct. Anal., 88:1 (1990), 90–117 | DOI | MR | Zbl

[18] Pokhozhaev S. I., “O sobstvennykh funktsiyakh uravneniya $\Delta u+\lambda f(u)=0$”, DAN SSSR, 165:1 (1965), 36–39 | MR | Zbl